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At which value of x is f continuous but not differentiable?
A. a
B. b
C. c
D. d
E. e | D | Math/1 | Math | null |
The table shows values of $f'$, the derivative of function $f$. Although $f'$ is continuous over all real numbers, only selected values of $f'$ are shown. If $f'$ has exactly two real zeros, then $f$ is increasing over which of the following intervals?
A. $ 1<x<5 $
B. $x<1$ or $x>5$
C. $x > 5$ only
D. $x > 1$
E. $x > 3$ | B | Math/2 | Math | null |
Shown is a graph of $f''$, the second derivative of function $f$. The curve is given by the equation $f'' = (x - a)^2 (x - d)$. The graph of $f$ has inflection points at which values of $x$?
A. b only
B. c only
C. a and d
D. a and c
E. d only | E | Math/3 | Math | null |
Shown in the diagram is a graph of $f'$, the derivative of function $f$. If $f(3) = 3$, then $f(1)=$?
A. \frac{10}{3}
B. \frac{13}{3}
C. -\frac{5}{3}
D. 4
E. 6 | B | Math/4 | Math | null |
Shown is a graph of $f'$, the derivative of function $f$. Which of the following statements is true?
A. $f$ is not differentiable at $x = -2$.
B. $f$ has a local minimum at $x = -1$.
C. $f$ is increasing from $x = 1$ to $x = 3$.
D. $f$ is increasing from $x = -3$ to $x = -1$.
E. $f$ is decreasing from $x = -2$ to $x = 1$. | D | Math/5 | Math | null |
Which of the graphs shown does the $\lim_{x \to 3} f(x)$ exist?
A. I only
B. II only
C. I and II only
D. III only
E. I and III only | E | Math/6 | Math | null |
A graph of function $f(x)$ on the interval $[-1, 4]$ is shown. Regions A, B, and C have areas of 1, 2, and 3 respectively. What is $\int_{-1}^{4} (f(x) + 2) dx$?
A. 2
B. 4
C. 10
D. 12
E. 16 | D | Math/7 | Math | null |
Shown above is a slope field for which of the following differential equations?
A. $\frac{dy}{dx} = \frac{x}{y}$
B. $\frac{dy}{dx} = xy$
C. $\frac{dy}{dx} = x + y$
D. $\frac{dy}{dx} = x - y$ | C | Math/8 | Math | null |
The graph of a piecewise linear function $f(x)$ is above. Evaluate $\int_{3}^{8} f'(x) , dx$.
A. 2
B. -2
C. 5
D. 0 | B | Math/9 | Math | null |
The table above gives selected values for the twice-differentiable function f. In which of the following intervals must there be a number c such that $f'(c) = -2$.
A. (0,2)
B. (2,4)
C. (4,6)
D. (6,8) | B | Math/10 | Math | null |
The graph of a function, $f$, is shown above. Let $h(x)$ be defined as $h(x) = (x + 1) \cdot f(x)$. Find $h'(4)$.
A. -6
B. -2
C. 4
D. 14 | A | Math/11 | Math | null |
Two differentiable functions, $f$ and $g$ have the property that $f(x) \geq g(x)$ for all real numbers and form a closed region $R$ that is bounded from $x = 1$ to $x = 7$. Selected values of $f$ and $g$ are in the table above. Estimate the area between the curves $f$ and $g$ between $x = 1$ and $x = 7$ using a Right Riemann sum with the three sub-intervals given in the table.
A. 13
B. 19
C. 21
D. 27 | B | Math/12 | Math | null |
The table above gives values of a differentiable function $f(x)$ at selected $x$ values. Based on the table, which of the following statements about $f(x)$ could be false?
A. There exists a value $c$, where $-5 < c < 3$ such that $f(c) = 1$
B. There exists a value $c$, where $-5 < c < 3$ such that $f'(c) = 1$
C. There exists a value $c$, where $-5 < c < 3$ such that $f(c) = -1$
D. There exists a value $c$, where $-5 < c < 3$ such that $f'(c) = -1$ | B | Math/13 | Math | null |
The function $f$ is continuous on the closed interval $[-2,2]$. The graph of $f'$, the derivative of $f$, is shown above. On which interval(s) is $f(x)$ increasing?
A. $[-1,1]$
B. $[-2, -1]$ and $[1, 2]$
C. $[0, 2]$
D. $[-2, 0]$ | C | Math/14 | Math | null |
The graph of $f$ is shown above. Which of the following statements is false?
A. $f(1) = \lim_{x \to 1} f(x)$
B. $f(3) = \lim_{x \to 3} f(x)$
C. $f(x)$ has a jump discontinuity at $x = 2$
D. $f(x)$ has a removable discontinuity at $x = 4$ | A | Math/15 | Math | null |
Consider the following back-to-back stemplot:
Which of the following are true statements?
I. The distributions have the same mean.
II. The distributions have the same range.
III. The distributions have the same standard deviation.
A. II only
B. I and II
C. I and III
D. II and III
E. I, II, and III | D | Math/16 | Math | null |
The graph below shows cumulative proportions plotted against land values (in dollars per acre) for farms on sale in a rural community. What is the median land value?
A. $2000
B. $2250
C. $2500
D. $2750
E. $3000 | A | Math/17 | Math | null |
Following is a histogram of ages of people applying for a particular high-school teaching position. Which of the following statements are true?
I. The median age is between 24 and 25.
II. The mean age is between 22 and 23.
III. The mean age is greater than the median age.
A. I only
B. II only
C. III only
D. All are true
E. None is true | C | Math/18 | Math | null |
A study was conducted to determine the effectiveness of varying amounts of vitamin C in reducing the number of common colds. A survey of 450 people provided the following information:
Is there evidence of a relationship between catching a cold and taking vitamin C?
A. The data prove that vitamin C reduces the number of common colds.
B. The data prove that vitamin C has no effect on the number of common
colds.
C. There is sufficient evidence at the 1% significance level of a relationship
between taking vitamin C and catching fewer colds.
D. There is sufficent evidence at the 10% significance level, but not at the 1%
significance level, of a relationship between taking vitamin C and catching
fewer colds.
E. There is not sufficient evidence at the 10% level of a relationship between
takng vitamin C and catching fewer colds. | E | Math/19 | Math | null |
The boxpolts below summarize the distribution of SAT verbal and math scores among students at an upstate New York high school. Which of the following statements are true?
I. The range of the math scores equals the range of the verbal scores.
II. The highest math score equals the median verbal score.
III. The verbal scores appear to be roughly symmetric, while the math scores appear to be skewed to the right.
(A) I only
(B) III only
(C) I and II
(D) II and III
(E) I, II, and III | C | Math/20 | Math | null |
The prices, in thousands of dollars, of 304 homes recently sold in a city are summarized in the histogram below. Based on the histogram, which of the following statements must be true?
(A) The minimum price is $250,000.
(B) The maximum price is $2,500,000.
(C) The median price is not greater than $750,000.
(D) The mean price is between $500,000 and $750,000.
(E) The upper quartile of the prices is greater than $1,500,000. | C | Math/21 | Math | null |
As part of a study on the relationship between the use of tanning booths and the occurrence of skin cancer, researchers reviewed the medical records of 1,436 people. The table below summarizes tanning booth use for people in the study who did and did not have skin cancer. Of the people in the study who had skin cancer, what fraction used a tanning booth?
(A) 190/265
(B) 190/896
(C) 190/1,436
(D) 265/1,436
(E) 896/1,436 | B | Math/22 | Math | null |
In northwest Pennsylvania, a zoologist recorded the ages, in months, of 55 bears and whether each bear was male or female. The data are shown in the back-to-back stemplot below. Based on the stemplot, which of the following statements is true?
(A) The median age and the range of ages are both greater for female bears than for male bears.
(B) The median age and the range of ages are both less for female bears than for male bears.
(C) The median age is the same for female bears and male bears, and the range of ages is the same for female bears and male bears.
(D) The median age is less for female bears than for male bears, and the range of ages is greater for female bears than for male bears.
(E) The median age is greater for female bears than for male bears, and the range of ages is less for female bears than for male bears. | A | Math/23 | Math | null |
Nutritionists examined the sodium content of different brands of potato chips. Each brand was classified as either healthy or regular based on how the chips were marketed to the public. The sodium contents, in milligrams (mg) per serving, of the chips are summarized in the boxplots below.
Based on the boxplots, which statement gives a correct comparison between the two classifications of the sodium content of the chips?
(A) The number of brands classified as healthy is greater than the number of brands classified as regular.
(B) The interquartile range (IQR) of the brands classified as healthy is greater than the IQR of the brands classified as regular.
(C) The range of the brands classified as healthy is less than the range of the brands classified as regular.
(D) The median of the brands classified as healthy is more than twice the median of the brands classified as regular.
(E) The brand with the least sodium content and the brand with the greatest sodium content are both classified as healthy. | B | Math/24 | Math | null |
A factory has two machines, A and B, making the same part for refrigerators. The number of defective parts produced by each machine during the first hour of operation was recorded on 19 randomly selected days. The scatterplot below shows the number of defective parts produced by each machine on the selected days. Which statement gives the best comparison between the number of defective parts produced by the machines during the first hour of operation on the 19 days?
(A) Machine A always produced the same number of defective parts as machine B.
(B) Machine A always produced fewer defective parts than machine B.
(C) Machine A always produced more defective parts than machine B.
(D) Machine A usually, but not always, produced fewer defective parts than machine B.
(E) Machine A usually, but not always, produced more defective parts than machine B. | D | Math/25 | Math | null |
A state educational agency was concerned that the salaries of public school teachers in one region of the state, region A, were higher than the salaries in another region of the state, region B. The agency took two independent random samples of salaries of public school teachers, one from region A and one from region B. The data are summarized in the table below. Assuming all conditions for inference are met, do the data provide convincing statistical evidence that the salaries of public school teachers in region A are, on average, greater than the salaries of public school teachers in region B?
(A) Yes, there is evidence at the significance level of a= 0.001.
(B) Yes, there is evidence at the significance level of a=0.01 but not at a=0.001.
(C) Yes, there is evidence at the significance level of a=0.05 but not at a=0.01.
(D) Yes, there is evidence at the significance level of a=0.10 but not at a=0.05.
(E) No, there is no evidence at the significance level of a=0.10. | C | Math/26 | Math | null |
As part of a community service program, students in three middle school grades (grade 6, grade 7, grade 8) each chose to participate in one of three school-sponsored volunteer activities. The graph below shows the distribution for each class for the three activities. Based on the graph, which statement must be true?
(A) Of all the students who chose activity B, the greatest number of students were in grade 6.
(B) Grade 7 and grade 8 had the same number of students who did not choose activity A.
(C) The grade with the greatest percentage of students who chose activity C was grade 8.
(D) For students in grade 7, the number who chose activity C was greater than the number who chose activity B.
(E) For students in grade 8, the number who chose activity A was greater than the number who chose activity B. | D | Math/27 | Math | null |
The number of siblings was recorded for each student of a group of 80 students. Some summary statistics and a histogram displaying the results are shown below. An outlier is often defined as a number that is more than 1.5 times the interquartile range below the first quartile or above the third quartile. Using the definition of an outlier and the given information, which of the following can be concluded?
(A) The median is greater than the mean, and the distribution has no outliers.
(B) The median is greater than the mean, and the distribution has only one outlier.
(C) The median is greater than the mean, and the distribution has two outliers.
(D) The median is less than the mean, and the distribution has only one outlier.
(E) The median is less than the mean, and the distribution has two outliers. | E | Math/28 | Math | null |
Extra study sessions were offered to students after the midterm to help improve their understanding of statistics. Student scores on the midterm and the final exam were recorded. The following scatterplot shows final test scores against the midterm test scores. Which of the following statements correctly interprets the scatterplot?
A. All students have shown significant improvement in the final exam scores as a result of the extra study sessions.
B. The extra study sessions were of no help. Each student’s final exam score was about the same as his or her score on the midterm.
C. The extra study sessions further confused students. All student scores decreased from midterm to final exam.
D. Students who scored below 55 on the midterm showed considerable improvement on the final exam; those who scored between 55 and 80 on the midterm showed minimal improvement on the final exam; and those who scored above 80 on the midterm showed almost no improvement on the final exam.
E. Students who scored below 55 on the midterm showed minimal improvement on the final exam; those who scored between 55 and 80 on the midterm showed moderate improvement on the final exam; and those who scored above 80 on the midterm showed considerable improvement on the final exam. | D: The scatterplot shows three different groups. The group of students on the far left of the scatterplot scored lowest on the midterm, in a range of 30-50, but then scored in a range of 65-85 on the final, showing considerable improvement. The middle group scored between 55-80 on the midterm and then between 60-85 on the final, showing very little improvement. The third group, the one on the far right of the scatterplot, scored above 85 on the midterm and then above 85 on the final, showing almost no improvement. To assist you in seeing this, you can add a straight line to the graph, going from a point on the x-axis at 50 (midterm = 50) and the bottom of the y-axis (final = 50) to a point at midterm = 100 and final = 100. Then, people on the upper left of that line did better on the final, whereas those on the lower right did better on the midterm. | Math/29 | Math | null |
A resident of Auto Town was interested in finding the cheapest gas prices at nearby gas stations. On randomly selected days over a period of one month, he recorded the gas prices (in dollars) at four gas stations near his house. The boxplots of gas prices are as follows: Based on this display, which of the stations had the closest mean and median during the month?
A. Station 1
B. Station 2
C. Station 3
D. Station 4
E. We cannot tell without knowing the standard deviation of the gas prices from each station. | B The mean is strongly affected by skew and outliers, while the median is not. Therefore the mean will be closest to the median when the box-and-whisker plot is most symmetric; the most symmetric of the box-and-whisker plots is from station 2. | Math/30 | Math | null |
During flu season, a city medical center needs to keep a large supply of flu shots. A nurse’s aid compiles data on the number of flu shots given per day in the past few years during flu season. A cumulative probability chart of the collected data is as follows: How many flu shots should the center store every day to meet the demand on 95 percent of the days?
A. At most 190
B. At most 140
C. Exactly 170
D. At least 150
E. At least 200 | E Look closely at the chart. For example, the point (160, 0.4) indicates that on 40 percent of the days, up to 160 shots were given. To find the number of shots needed to meet the demand on 95 percent of the days, we need to find the 95th percentile. Draw a horizontal line from the cumulative probability of 0.95 to the curve. From the point at which the line meets the curve, draw a vertical line down to the x-axis, and then read the number of flu shots given there. It should be approximately 200. | Math/31 | Math | null |
In the northern United States, schools are sometimes closed during winter due to severe snowstorms. At the end of the school year, schools have to make up for the days missed. The following graph shows the frequency distribution of the number of days missed due to snowstorms per year using data collected from the past 75 years. Which of the following should be used to describe the center of this distribution?
A. Mean, because it is an unbiased estimator
B. Median, because the distribution is skewed
C. IQR, because it excludes outliers and includes only the middle 50 percent of data
D. First quartile, because the distribution is left skewed
E. Standard deviation, because it is unaffected by the outliers | B IQR, first quartile, and standard deviation are not measures of central tendency. Median and mean are measures of central tendency. The mean is affected by extreme observations—large observations tend to make the mean higher. Because this distribution is skewed, the median should be used to describe the center of the distribution. Note that the median is not affected by the extreme measurements. | Math/32 | Math | null |
X and Y are independent random variables; the probability distributions for each are given in the table above. Let Z = X + Y. What is P(Z ≤ 2)?
A. 0.07
B. 0.15
C. 0.24
D. 0.40
E. 0.80 | B There are three mutually exclusive ways to get a sum less than or equal to two: x = 0 and y = 1, x = 0 and y = 2, x = 1 and y = 1. Since the random variables are independent, we can multiply each pair’s probabilities to get the probability of the pair. That is:
P(x = 0 and y = 1) = P(x = 0) × P(y = 1) = 0.4 × 0.1 = 0.04 | Math/33 | Math | null |
Two hundred students were classified by sex and hostility level (low, medium, high), as measured by an HLT-test. The results were the following. If the hostility level among students were independent of their sex, then how many female students would we expect to show the medium HLT score?
A. 25
B. 45
C. 54
D. 60
E. 75 | C The total number of females in the sample = 120.
The total number of students with medium HLT score = 90.
The total number of students in the sample = 200.
The expected number of females with medium = (120)*(90)/200 = 54 | Math/34 | Math | null |
The following boxplot summarizes the prices of books at a bookstore. Which of the following is true about the prices of books at this store?
A. This store carries more books priced above the mean than below the mean.
B. This store carries more books priced below the mean than above the mean.
C. This store carries about the same number of books at different prices in the entire price range.
D. The mean price of books at this store is the same as the median price of books.
E. The mean price of books at this store is lower than the median price of books. |
B This boxplot is highly
right-skewed. The book prices range from about $8 to $80, but the median
price is around $12. This means that 50 percent of the books are priced
around $12 or lower. Even the third quartile is around $25, which means
that about 75 percent of the books are priced at $25 or below. When a
distribution is right-skewed, the mean is above the median, and
therefore, more books are below the mean than above it. | Math/35 | Math | null |
The following graph summarizes data collected on annual rainfall in two cities for the past 150 years. Which of the following conclusions can be made from this graph?
A. The cities have different mean annual rainfalls, but the range of their annual rainfalls is approximately the same.
B. On average, City B gets more rain than City A, but it has a smaller range of annual rainfall.
C. On average, City B gets more rain than City A, and it has a larger range of annual rainfall.
D. On average, City A gets more rain than City B, but it has a smaller range of annual rainfall.
E. On average, City A gets more rain than City B, and it has a larger range of annual rainfall. | B The approximate range of annual rainfall for City A is five to 45 inches, with a mean of approximately 23 inches. The approximate range of annual rainfall for City B is 20 to 40 inches, with a mean of approximately 30 inches. This means that, on the average, City B gets more rain than City A, but the range for City A is larger than that for City B. | Math/36 | Math | null |
The following cumulative graph gives the electricity demand of a certain town.The power plant that provides electricity to this town is capable of generating 12,000 kW daily. On what percent of days will the power plant not be able to meet the demand for electricity?
A. About 3 percent
B. About 60 percent
C. About 80 percent
D. About 92 percent
E. About 97 percent | A In this cumulative graph, we need to find P(Demand > 12,000), because that is the point at which the power company will not be able to meet the demand for electricity. Draw a vertical line at 12,000 until it reaches the curve. From the point at which it crosses the curve, draw a horizontal line to the Y-axis. The line reaches the Y-axis at approximately 97 percent, which means that on 97 percent of days the demand for electricity will be, at most, 12,000. In other words, on about 3 percent of days, the power plant will have more demand for electricity than what they produce. Because none of the other choices are close to 3 percent, you do not need to be accurate with your drawing. | Math/37 | Math | null |
Sixty pairs of measurements were taken at random to estimate the relation between variables X and Y. A least squares regression line was fitted to the collected data. The resulting residual plot is as follows. Which of the following conclusions is appropriate?
A. A line is an appropriate model to describe the relation between X and Y.
B. A line is not an appropriate model to describe the relation between X and Y.
C. The assumption of normality of errors has been violated.
D. The assumption of constant sample standard deviations has been violated.
E. The variables X and Y are not related at all. | B The curvature in the residual plot indicates that a line is not an appropriate model to describe the relation between X and Y. If a line is an appropriate model, then the residual plot will show randomly scattered residuals without any pattern. | Math/38 | Math | null |
Use the following computer output for a least-squares regression for Questions below. What is the equation of the least-squares regression line?
A. ŷ= -0.6442x+ 22.94
B. ŷ= 22.94 + 0.5466x
C. ŷ= 22.94 + 2.866x
D. ŷ= 22.94 - 0.6442x
E. ŷ= -0.6442 + 0.5466x | D The correct answer is (d). The slope of the regression line. –0.6442, can be found under “Coef” to the right of “x.” The intercept of the regression line, 22.94, can be found under “Coef” to the right of “Constant.” | Math/39 | Math | null |
Given that the analysis is based on 10 datapoints, what is the P-value for the t-test of the hypothesis $H_0 : \beta = 0$ vesus $H_A : \beta ≠ 0$.
A. 0.02 < P < 0.03
B. 0.20 < P < 0.30
C. 0.01 < P < 0.05
D. 0.15 < P < 0.20
E. 0.10 < P < 0.15 | B The correct answer is (b). The t statistic for H0: β = 0 is given in the printout as –1.18. We are given that n = 10 ⇒ df = 10 – 2 = 8. From the df = 8 row of Table B (the t Distribution Critical Values table), we see, ignoring the negative sign since it’s a two-sided test,
1.108 < 1.18 < 1.397 ⇒ 2(0.10) < P < 2(0.15),
which is equivalent to 0.20 < P < 0.30. | Math/40 | Math | null |
If three fair coins are flipped, P(0 heads) = 0.125, P(exactly 1 head) = 0.375, P(exactly 2 heads) = 0.375, and P(exactly 3 heads) = 0.125. The following results were obtained when three coins were flipped 64 times. What is the value of the X2statistic used to test if the coins are behaving as expected, and how many degrees of freedom does the determination of the P-value depend on?
A. 3.33, 3
B. 3.33, 4
C. 11.09, 3
D. 3.33, 2
E. 11.09, 4 | A $\chi^2 = \frac{(10-8)^2}{8} + \frac{(28-24)^2}{24} + \frac{(22-24)^2}{24} + \frac{(4-8)^2}{8} = 3.33$In a chi-square goodness-of-fit test, the number of degrees of freedom equals one less than the number of possible outcomes. In this case, df = n–1 = 4 – 1 = 3. | Math/41 | Math | null |
For the histogram pictured above, what is the class interval (boundaries) for the class that contains the median of the data?
A. (5, 7)
B. (9, 11)
C. (11, 13)
D. (15, 17)
E. (7, 9) | E The correct answer is (e). There are 101 terms, so the median is located at the 51st position in an ordered list of terms. From the counts given, the median must be in the interval whose midpoint is 8. Because the intervals are each of width 2, the class interval for the interval whose midpoint is 8 must be (7, 9). | Math/42 | Math | null |
Thirteen large animals were measured to help determine the relationship between their length and their weight. The natural logarithm of the weight of each animal was taken and a least-squares regression equation for predicting weight from length was determined. The computer output from the analysis is given below. Give a 99% confidence interval for the slope of the regression line. Interpret this interval.
A. (0.032, 0.041); the probability is 0.99 that the true slope of the regression line is between 0.032 and 0.041.
B. (0.032, 0.041); 99% of the time, the true slope will be between 0.032 and 0.041.
C. (0.032, 0.041); we are 99% confident that the true slope of the regression line is between 0.032 and 0.041.
D. (0.81, 1.66); we are 99% confident that the true slope of the regression line is between 0.032 and 0.041.
E. (0.81, 1.66); the probability is 0.99 that the true slope of the regression line is between 0.81 and 1.66. | C The correct answer is (c). df = 13 – 2 = 11 ⇒ t* = 3.106 (from Table B; if you have a TI-84 with the invT function, t* = invT(0.995,11)). Thus, a 99% confidence interval for the slope is:
0.0365 ± 3.106(0.0015) = (0.032, 0.041).
We are 99% confident that the true slope of the regression line is between 0.032 units and 0.041 units. | Math/43 | Math | null |
For the following observations collected while doing a chi-square test for independence between the two variables Aand B, find the expected value of the cell marked with "X."
A. 4.173
B. 9.00
C. 11.56
D. 8.667
E. 9.33 | D The correct answer is (d). There are 81 observations total, 27 observations in the second column, 26 observations in the first row. The expected number in the first row and second column equals 27/81*26=8.667. | Math/44 | Math | null |
The following is a probability histogram for a discrete random variable X. What is $\mu_x$.
A. 3.5
B. 4.0
C. 3.7
D. 3.3
E. 3.0 | D The correct answer is (d). 2(0.3) + 3(0.2) + 4(0.4) + 5(0.1) = 3.3. | Math/45 | Math | null |
You are developing a new strain of strawberries (say, Type X) and are interested in its sweetness as compared to another strain (say, Type Y). You have four plots of land, call them A, B, C, and D, which are roughly four squares in one large plot for your study (see the figure below). A river runs alongside of plots C and D. Because you are worried that the river might influence the sweetness of the berries, you randomly plant type Xin either A or B (and Yin the other) and randomly plant type Xin either C or D (and Yin the other). Which of the following terms best describes this design?
A. A completely randomized design
B. A randomized study
C. A randomized observational study
D. A block design, controlling for the strain of strawberry
E. A block design, controlling for the effects of the river | E The correct answer is (e). The choice is made here to treat plots A and B as a block and plots C and D as a block. That way, we are controlling for the possible confounding effects of the river. Hence the answer is (c). If you answered (e), be careful of confusing the treatment variable with the blocking variable. | Math/46 | Math | null |
A set of test scores has the following 5-number summary. Which statement about outliers must be true?
A. There is exactly one outlier on the lower end.
B. There is at least one outlier on the lower end.
C. There is exactly one outlier on the higher end.
D. There is at least one outlier on the higher end.
E. There are no outliers. | B Using the outlier guideline, any value above Q3 + 1.5 ⋅ IQR = 28.5 + 1.5 ⋅ 10.5 = 44.25, or below Q1 − 1.5 ⋅ IQR = 18 − 1.5 ⋅ 10.5 = 2.25, is considered an outlier. Because the maximum is 33, there are no values above 44.25. The minimum of 2 is an outlier. There may be more than one value below 2.25, so all we can say is that there is at least one outlier on the lower end. | Math/47 | Math | null |
The table above shows the regression output for predicting the number of calories from the number of milligrams of sodium for items at a fast-food chain. What percent of the variation in calories is explained by the regression model using sodium as a predictor?
A. 7.36%
B. 54.2%
C. 53.8%
D. 73.4%
E. 73.6% | B You should recognize that this is the interpretation of $r_2$. According to the printout, that value is 0.542313, or about 54.2%. | Math/48 | Math | null |
Researchers in the Southwest are studying tortoises-a species of animal that is affected by habitat loss due to human development of the desert. A total of 78 tortoises are being studied by researchers at several sites. The data on gender and species of all these tortoises is organized in the table shown below. If a tortoise from this study is to be selected at random, let A = the tortoise is femaleand B = the tortoise is a Morafka. Which of the following appropriately interprets the value of P(B|A)?
A. 31.3% is the probability that a randomly selected Morafka tortoise is a female tortoise.
B. 31.3% is the probability that a randomly selected female tortoise is a Morafka tortoise.
C. 19.2% is the probability that a randomly selected Morafka tortoise is a female tortoise.
D. 19.2% is the probability that a randomly selected female tortoise is a Morafka tortoise.
E. 31.3% is the probability that a randomly selected tortoise is a Morafka tortoise. | B This is asking for a conditional probability. P(Morafka|Female). Out of the 48 female tortoises, 15 are Morafka. 15/48 = 0.3125, or about 31.3%. | Math/49 | Math | null |
Breakfast cereals have a wide range of sugar content. Some cereals contain High Fructose Corn Syrup (HFCS) as a source of sugar and some do not. The boxplots above show the total sugar content of different types of cereal for those containing HFCS and for those that do not. Which statement is true based on the boxplots?
A. The number of cereals with HFCS is about the same as the number of cereals without HFCS.
B. The cereals with HFCS have a greater interquartile range than the cereals without HFCS.
C. The cereals without HFCS have a greater range than the cereals with HFCS.
D. About half the cereals without HFCS have less sugar than about three-fourths of the cereals with HFCS.
E. About half the cereals with HFCS have more sugar than about three-fourths of the cereals without HFCS. | E The median of the HFCS group is the same as Q3 of the No HFCS group. | Math/50 | Math | null |
Researchers were gathering data on alligators in an attempt to estimate an alligator's weight from its length. They captured 29 alligators and measured their length and weight. They created three regression models. Each model, along with its residual plot, is shown below. yrepresents the weight in pounds and xrepresents the length in inches.Which statement is true?
A. There is a linear relationship between weight and length, and model I is most appropriate.
B. There is a linear relationship between weight and length, and model II is most appropriate.
C. There is a nonlinear relationship between weight and length, and model I is most appropriate.
D. There is a nonlinear relationship between weight and length, and model II is most appropriate.
E. There is a nonlinear relationship between weight and length, and model III is most appropriate. | D Because there is a curve in residual plot I, which is for weight vs. length, the relationship between those two variables is not linear. Because residual plot II shows a no curve and plot III does, plot II represents a better model. | Math/51 | Math | null |
Listed above are the summary statistics for the daily high temperatures (in degrees F) for the month of September in a Midwestern U.S. city. One particular day has a z-score of 0.70. What was the likely temperature that day?
A. 68
B. 70
C. 74
D. 76
E. 80 | E z = 0.70 = (x-74.9)/7.29, x = 80. | Math/52 | Math | null |
In American football, the running back carries the football forward in a quest to score points. Many people create fantasy football teams by selecting players as their "team." These fans analyze data to judge which players are outstanding. The following regression output analyzes the linear relationship between the number of times a running back is given the football (carries) and the total number of yards that player gains in a season for a random sample of 18 running backs. The appropriate calculation for a 99% confidence interval for the slope of the least squares regression line for the total yards gained versus number of carries in a season for all running backs is:
A. 4.3441 ± 2.576.(1.811)\sqrt{18}
B. 4.3441 ± 2.921.(1.811)
C. 4.3441 ± 2.921.(1.811)\sqrt{18}
D. 4.3441 ± 2.898.(1.811)
E. 4.3441 ± 2.898.(1.811)\sqrt{18} | B The formula for the confidence interval for the slope is $b_1 ± t* · s_{b1}$. From the printout b1 = 4.3441 and sb1 = 1.811. t* for 18 – 2 = 16 degrees of freedom and 99% confidence is 2.921. | Math/53 | Math | null |
Above is a cumulative relative frequency plot for the scores of a large university class on a 90-point statistics exam. Which of the following observations is correct?
A. The median score is at least 60 points.
B. The distribution of scores is skewed to the right.
C. The distribution of scores is skewed to the left.
D. The distribution is roughly symmetric.
E. If a passing score is 60, most students passed the test. | B The steeper part of the graph corresponds to the higher bars on a histogram. So the higher bars would be on the left with a tail stretching toward the right. | Math/54 | Math | null |
Some health professionals suspect that doctors are more likely to order cardiac tests for men than women, even when women describe exactly the same symptoms. Young doctors, training to work in the emergency room, were randomly assigned to two groups-Group A and Group B-and presented with the exact same description of common symptoms of heart disease. However, those in Group A were told that the patient was a 55-year-old female and those in Group B were told that the patient was a 55-year-old male. All other patient characteristics were exactly the same. Which of the following is the appropriate statistic to test whether doctors are less likely to order cardiac tests when the patient is female?
Here's the content with the mathematical expressions enclosed in dollar signs:
A.$ z = \frac{(0.70) - (0.85)}{\sqrt{(0.70)(0.85)\left(\frac{1}{60} + \frac{1}{40}\right)}} $
B.$ z = \frac{(0.70) - (0.85)}{\sqrt{(0.76)(0.24)\left(\frac{1}{60} + \frac{1}{40}\right)}} $
C.$ z = \frac{(0.70) - (0.85)}{\sqrt{(0.76)(0.24)\left(\frac{1}{42} + \frac{1}{34}\right)}} $ | B Under the null hypothesis, the proportions are the same, so the standard error formula is based on the pooled proportion (x1+x2)/(n1+n2) | Math/55 | Math | null |
The graph of the function \( f \) is shown below:
If $g(x) = \int_{0}^{x^2} f(t) \, dt$, what is $g'(2)$?
A. 32
B. 16
C. 8
D. 4
| C: The Second Fundamental Theorem of Calculus says that $\frac{d}{dx} \int_{c}^{x} f(t)dt = f'(x)$. Here, we get: $g'(x) = \frac{d}{dx} \int_{0}^{x^2} f(t)dt = f(x^2) \cdot 2x$ (the $2x$ term comes from applying the Chain Rule to the limit in the integral). Therefore, $g'(2) = f(4) \cdot 2 \cdot 2$. From the graph, we can see that $f(4) = 2$, so $g'(2) = 2 \cdot 2 \cdot 2 = 8$. | Math/56 | Math | null |
The graph of $f'$ is shown above. Which statements about $f$ must be true for $a < x < b$?
I. $f$ is increasing.
II. $f$ is continuous.
III. $f$ is differentiable.
A. all three
B. II only
C. I and II only
D. I and III only
| A: $f'(x) > 0$; the curve shows that $f'$ is defined for all $a < x < b$, so $f$ is differentiable and therefore continuous. | Math/57 | Math | null |
The table shows some of the values of differentiable functions $f$ and $g$ and their derivatives. If $h(x) = f(g(x))$, then $h'(2)$ equals
A. -2
B. -1
C. 0
D. 1 | B: Since $h(x) = f(g(x))$, $h'(x) = f'(g(x))g'(x)$ and $h'(2) = f'(g(2))g'(2) = f'(3) \cdot 1 = -1$. | Math/58 | Math | null |
The graph of $f$ is shown above, and $f$ is twice differentiable. Which of the following has the largest value?
I. $f(0)$
II. $f'(0)$
III. $f''(0)$
A. I
B. II
C. III
D. I and II | C: Since $f$ is decreasing, $f' < 0$ and since $f$ is concave up, $f'' > 0$. The graph also shows that $f(0) < 0$. Thus $f''(0)$ has the largest value.
| Math/59 | Math | null |
A slope field for a differential equation is shown above. Which of the following could be the differential equation?
A. $ \frac{dx}{dy} = 2x $
B. $ \frac{dx}{dy} = -2x $
C. $ \frac{dx}{dy} = y $
D. $ \frac{dx}{dy} = x - y $ | B: Note that for each column, all the tangents have the same slope. For example, when $ x = 0 $ all tangents are horizontal, which is to say their slopes are all zero. This implies that the slope of the tangents depends solely on the $ x $-coordinate of the point and is independent of the $ y $-coordinate. Also note that when $ x > 0 $ slopes are negative, and when $ x < 0 $ slopes are positive. Thus, the only differential equation that satisfies these conditions is $ \frac{dy}{dx} = -2x $.
| Math/60 | Math | null |
The graph of $f''$, the second derivative of $f$ is shown above. The graph of $f''$ has horizontal tangents at $x = -2$ and $x = 2$. For what values of $x$ does the graph of the function $f$ have a point of inflection?
A. $-4$, $0$ and $4$
B. $-2$, $0$ and $2$
C. $-4$ and $4$ only
D. $0$ only
| A: Note that $f'' > 0$ on the intervals $(-\infty, -4)$ and $(0, 4)$. Thus the graph of the function $f$ is concave up on these intervals. Similarly, $f'' < 0$ and concave down on the intervals $(-4, 0)$ and $(4, \infty)$. Therefore $f$ has a point of inflection at $x = -4$, $0$, and $4$. | Math/61 | Math | null |
The domain of the function $f$ is $0 \leq x \leq 9$ as shown above. If $g(x) = \int_{0}^{x} f(t) , dt$, at what value of $x$ is $g(x)$ the absolute maximum?
A. 0
B. 1
C. 2
D. 8
| C: First, the graph of $f(x)$ is above the x-axis on the interval $[0, 2]$ thus $f(x) \geq 0$, and $\int_{0}^{x} f(t) , dt > 0$ Secondly, $f(x) \leq 0$ on the interval $[2, 8]$ and $\int_{2}^{8} f(x) , dx < 0$, and thus $\int_{0}^{2} f(t) , dt$ maximum. Note that the area of the region bounded by $f(x)$ and the x-axis on $[2, 8]$ is greater than the sum of the areas of the two regions above the x-axis. Therefore,
$\int_{0}^{2} f(x) , dx + \int_{2}^{8} f(x) , dx + \int_{8}^{x} f(x) , dx < 0$ and $\int_{0}^{x} f(x) , dx < 0$ and
Consequently, $\int_{0}^{x} f(x) , dx$ is the absolute maximum value. | Math/62 | Math | null |
The graph of a function $f$ is shown above. Which of the following statements is/are true?
I. $\lim_{x \to 1} f(x)$ exists
II. $f(1)$ exists
III. $\lim_{x \to 1} f(x) = f'(1)$
A. I only
B. II only
C. I and II only
D. I, II, and III | C: Since $\lim_{x \to 1^+} f(x) = \lim_{x \to 1^-} f(x) = 4$, $\lim_{x \to 1} f(x)$ exists. The graph shows that at $x = 1$, $f(x) = 1$ and thus $f(1)$ exists. Lastly, $\lim_{x \to 1} f(x) \neq f'(1)$. | Math/63 | Math | null |
The base of a solid is a region bounded by the lines $y = x$, $y = -x$, and $x = 4$, as shown above. What is the volume of the solid if the cross sections perpendicular to the x-axis are equilateral triangles?
A. $\frac{16\sqrt{3}}{3}$
B. $\frac{32\sqrt{3}}{3}$
C. $\frac{64\sqrt{3}}{3}$
D. $\frac{256\pi}{3}$ | C: Area of a cross section $= \frac{\sqrt{3}}{4} (2x)^2 = \sqrt{3}x^2$.
Using your calculator, you have:
Volume of solid $= \int_{0}^{4} \sqrt{3}(x^2)dx = \frac{64\sqrt{3}}{3}$.
| Math/64 | Math | null |
Let $f$ be a continuous function on $[0, 6]$ and have selected values as shown below.
If you use the subintervals $[0, 2]$, $[2, 4]$, and $[4, 6]$, what is the trapezoidal approximation of $\int_{0}^{6} f(x) , dx$?
A. 9.5
B. 12.75
C. 19
D. 25.5 | B: $\int_{0}^{6} f(x)dx \approx \frac{6 - 0}{2} \left[ 0 + 2(1) + 2(2.25) + 6.25 \right] \approx 12.75$ | Math/65 | Math | null |
The graph of $f$ for $-1 \leq x \leq 3$ consists of two semicircles, as shown above. What is the value of $\int_{-1}^{3} f(x) , dx$?
A. 0
B. $\pi$
C. $2\pi$
D. $4\pi$
| A: $\int_{-1}^{3} f(x) , dx = \int_{-1}^{1} f(x) , dx + \int_{1}^{3} f(x) , dx = \frac{1}{2} \pi (1)^2 - \frac{1}{2} \pi (1)^2 = 0$ | Math/66 | Math | null |
The graph of the polar curve $r = 3 - \sin \theta$ is shown above. Which of the following expressions gives the area of the region enclosed by the curve?
A. $\frac{1}{2} \int_{0}^{2\pi} (3 - \sin \theta) d\theta$
B. $\frac{1}{2} \int_{0}^{2\pi} (3 - \sin \theta)^2 d\theta$
C. $\int_{0}^{2\pi} (3 - \sin \theta) d\theta$
D. $\int_{0}^{2\pi} (3 - \sin \theta)^2 d\theta$ | B: To enclose the area, $\theta$ must sweep through the interval from $0$ to $2\pi$. The area of the region enclosed by $r = 3 - \sin \theta$ is $A = \frac{1}{2} \int_{0}^{2\pi} (3 - \sin \theta)^2 d\theta$. | Math/67 | Math | null |
The graph of the velocity function of a moving particle is shown above. What is the total displacement of the particle during $0 \leq t \leq 20$?
A. $20 , \text{m}$
B. $50 , \text{m}$
C. $100 , \text{m}$
D. $500 , \text{m}$
| B: $\int_{0}^{20} v(t)dt = \frac{1}{2}(40)(5) + \frac{1}{2}(10)(-20) + \frac{1}{2}(5)(20) = 50$ | Math/68 | Math | null |
A particular solution of the differential equation whose slope field is shown abovecontains point $P$. This solution may also contain which other point?
A. E
B. B
C. C
D. D
| A: The slope field suggests the curve shown above as a particular solution. | Math/69 | Math | null |
Using the left rectangular method and four subintervals of equal width, estimate $\int_{0}^{8} |f(t)| , dt$, where $f$ is the function graphed below.
A. 16
B. 5
C. 8
D. 15 | A: $2(3) + 2(0) + 2(4) + 2(1)$.
| Math/70 | Math | null |
Use the graph of $f$ shown on $[0,7]$. Let $G(x) = \int_{2}^{3x-1} f(t) , dt$.
$G'(1)$ is
A. 1
B. 2
C. 3
D. 6
| D: $G'(x) = f(3x - 1) \cdot 3$. | Math/71 | Math | null |
Use the graph of $f$ shown on $[0,7]$. Let $G(x) = \int_{2}^{3x-1} f(t) , dt$.
$G$ has a local maximum at $x =$
A. 1
B. $\frac{4}{3}$
C. 2
D. 5 | B: Since $f$ changes from positive to negative at $t = 3$, $G'$ does also where $3x - 1 = 3$. | Math/72 | Math | null |
Use the graph below, consisting of two line segments and a quarter-circle. The graph shows the velocity of an object during a 6-second interval.
During what time interval (in sec) is the speed increasing?
A. $0 < t < 3$
B. $3 < t < 5$
C. $3 < t < 6$
D. $5 < t < 6$ | B: Speed is the magnitude of velocity; its graph is shown in the answer explanation for question 20. | Math/73 | Math | null |
The graph of $f$ is shown above. Let $G(x) = \int_{0}^{x} f(t)dt$ and $H(x) = \int_{2}^{x} f(t) dt$. Which of the following is true?
A. $G(x) = H(x) + 3$
B. $G'(x) = H'(x + 2)$
C. $G(x) = H(x + 2)$
D. $G(x) = H(x) - 2$
| A: $G(x) = H(x) + \int_{0}^{2} f(t) dt$, where $\int_{0}^{2} f(t) dt$ represents the area of a trapezoid.
| Math/74 | Math | null |
Which function could be a particular solution of the differential equation whose slope field is shown above?
A. $y = x^3$
B. $y = \frac{2x}{x^2 + 1}$
C. $y = \frac{x^2}{x^2 + 1}$
D. $y = \sin x$ | B: Solution curves appear to represent odd functions with a horizontal asymptote. In the figure above, the curve in (B) of the question has been superimposed on the slope field. | Math/75 | Math | null |
The work done in lifting an object is the product of the weight of the object and the distance it is moved. A cylindrical barrel 2 feet in diameter and 4 feet high is half-full of oil weighing 50 pounds per cubic foot. How much work is done, in foot-pounds, in pumping the oil to the top of the tank?
A. $100\pi$
B. $200\pi$
C. $300\pi$
D. $400\pi$ | D: The graph shown has the x-axis as a horizontal asymptote. | Math/76 | Math | null |
The graph below shows the velocity of an object moving along a line, for $0 \leq t \leq 9$.
The object is farthest from the starting point at $t =$
A. 2
B. 5
C. 6
D. 8
| C: Velocity $v$ is the derivative of position; because $v > 0$ until $t = 6$ and $v < 0$ thereafter, the position increases until $t = 6$ and then decreases; since the area bounded by the curve above the axis is larger than the area below the axis, the object is farthest from its starting point at $t = 6$. | Math/77 | Math | null |
The equation of the curve shown below is $y = \frac{4}{1+x^2}$. What does the area of the shaded region equal?
A. $4 - \frac{\pi}{4}$
B. $8 - 2\pi$
C. $8 - \pi$
D. $8 - \frac{\pi}{2}$ | B: The required area, $A$, is given by the integral：$2 \int_{0}^{1} \left( \frac{4}{1+x^2} \right) dx = 2[4 \arctan(x)]_{0}^{1} = 2 \left( 4 - \frac{\pi}{4} \right)$
| Math/78 | Math | null |
Use the following table, which shows the values of the differentiable functions $f$ and $g$.
The average rate of change of function $f$ on $[1,4]$ is
A. $\frac{7}{6}$
B. $\frac{4}{3}$
C. $\frac{15}{8}$
D. $\frac{9}{4}$ | B: $f(4) - f(1) = \frac{6 - 2}{4 - 1} = \frac{4}{3}$ | Math/79 | Math | null |
Use the following table, which shows the values of the differentiable functions $f$ and $g$f $h(x) = g(f(x))$ then $h'(3) =$
A. $\frac{1}{2}$
B. $1$
C. $4$
D. $6$ | B: $h'(3) = g'(f(3)) \cdot f'(3) = g'(\frac{1}{4}) \cdot f'(3) = \frac{1}{2} \cdot 2$. | Math/80 | Math | null |
Which equation has the slope field shown below?
A. $\frac{dy}{dx} = \frac{5}{y}$
B. $\frac{dy}{dx} = \frac{5}{x}$
C. $\frac{dy}{dx} = \frac{x}{y}$
D. $\frac{dy}{dx} = 5y$ | A: Note that (1) on a horizontal line the slope segments are all parallel, so the slopes there are all the same and $\frac{dy}{dx}$ must depend only on $y$; (2) along the x-axis (where $y = 0$) the slopes are infinite; and (3) as $y$ increases, the slope decreases. | Math/81 | Math | null |
The graph below shows the velocity of an object moving along a line, for $0 \leq t \leq 9$.
At what time does the object attain its maximum acceleration?
A. $8 < t < 9$
B. $5 < t < 8$
C. $t = 6$
D. $t = 8$ | A: Acceleration is the derivative (the slope) of velocity $v$; $v$ is largest on $8 < t < 9$. | Math/82 | Math | null |
Here is the content from the image with the mathematical parts denoted by $:
Water is poured at a constant rate into the conical reservoir shown above. If the depth of the water, $h$, is graphed as a function of time, the graph is
A. concave downward
B. constant
C. linear
D. concave upward | A; As the water gets deeper, the depth increases more slowly. Hence, the rate of change of depth decreases: $\frac{d^2 h}{dt^2} < 0$. | Math/83 | Math | null |
If $P(x) = g^2(x)$, then $P'(3)$ equals
A. 4
B. 6
C. 9
D. 12 | D: $P'(x) = 2g(x) \cdot g'(x)$ | Math/84 | Math | null |
In the table above, if $h(x) = g(f(x))$, then $h'(2) =$
A. 4
B. 2
C. -6
D. -18 | D: We can find $h'(x)$ using the Chain Rule. We get: $h'(x) = g'(f(x)) \cdot f'(x)$. Now, we just have to evaluate $h'(2)$. We get: $h'(2) = g'(f(2)) \cdot f'(2) = 6 \cdot -3 = -18$. | Math/85 | Math | null |
Using the subintervals $[0, 2]$, $[2, 5]$, $[5, 7]$, and $[7, 10]$, approximate $\int_{0}^{10} f(x)dx$ using a left Riemann sum.
A. 102
B. 204
C. 176
D. 242 | C: We need to add up the areas of the four rectangles whose widths are the distances between the x-coordinates and whose heights are determined by evaluating $f$ at the left coordinate of each interval. We get: $R = (2)f(0) + (3)f(2) + (2)f(5) + (3)f(7)$. Now we can use the table to find the appropriate values of $f$. We get: $R = (2)(8) + (3)(11) + (2)(20) + (3)(29) = 16 + 33 + 40 + 87 = 176$. | Math/86 | Math | null |
Which of the following is the differential equation of the slope field above?
A. $\frac{dy}{dx} = x^2 + y^2$
B. $\frac{dy}{dx} = x^2 - y^2$
C. $\frac{dy}{dx} = x - y$
D. $\frac{dy}{dx} = (x + y)^2$ | A: We can find the correct differential equation by testing some values of $x$ and $y$ and seeing what the slope is at those values. At the origin, the slope is $0$, which unfortunately, does not eliminate any of the choices. At $(1, 1)$, the slope is positive, which eliminates (B) and (D). At $(-1, -1)$, the slope is again positive, which eliminates (C). This leaves (A). | Math/87 | Math | null |
If $g(x) = \int_{0}^{x^2} f(t) , dt$, what is $g'(2)$?
A. 32
B. 16
C. 8
D. 4 | C: The Second Fundamental Theorem of Calculus says that $\frac{d}{dx} \int_{c}^{x^2} f(t),dt = f'(x)$. Here, we get: $g'(x) = \frac{d}{dx} \int_{0}^{x^2} f(t),dt = f'(x) \cdot x^2$. Therefore, $g'(2) = f'(4) \cdot 4$. From the graph, we can see that $f(4) = 2$, so $g'(2) = 2 \cdot 4 = 8$. | Math/88 | Math | null |
The graph of $f$ is shown in the figure above. If $g(x) = \int_{0}^{x} f(t) , dt$, for what positive value of $x$ does $g(x)$ have a minimum?
A. 1
B. 2
C. 3
D. 4 | D: If we want to find where $g(x)$ is a minimum, we can look at $g'(x)$. The Second Fundamental Theorem of Calculus tells us how to find the derivative of an integral: $\frac{d}{dx} \int_{c}^{x} f(t) , dt = f(x)$, where $c$ is a constant. Thus, $g'(x) = f(x)$. The graph of $f$ is zero at $x = 0$, $x = 2$, and $x = 4$. We can eliminate $x = 0$ because we are looking for a positive value of $x$. Next, notice that $f$ is negative to the left of $x = 4$ and positive to the right of $x = 4$. Thus, $g(x)$ has a minimum at $x = 4$.
We also could have found the answer geometrically. The function $g(x) = \int_{0}^{x} f(t) , dt$ is called an accumulation function and stands for the area between the curve and the x-axis to the point $x$. Thus, the value of $g$ grows from $x = 0$ to $x = 2$. Then, because we subtract the area under the x-axis from the area above it, the value of $g$ shrinks from $x = 2$ to $x = 4$. The value begins to grow again after $x = 4$. | Math/89 | Math | null |
In the northern United States, schools are sometimes closed during winter due to severe snowstorms. At the end of the school year, schools have to make up for the days missed. The following graph shows the frequency distribution of the number of days missed due to snowstorms per year using data collected from the past 75 years. Which of the following should be used to describe the center of this distribution?
A. Mean, because it is an unbiased estimator
B. Median, because the distribution is skewed
C. IQR, because it excludes outliers and includes only the middle 50 percent of data
D. First quartile, because the distribution is left skewed
E. Standard deviation, because it is unaffected by the outliers
| B:QR, first quartile, and standard deviation are not measures of central tendency. Median and mean are measures of central tendency. The mean is affected by extreme observations—large observations tend to make the mean higher. Because this distribution is skewed, the median should be used to describe the center of the distribution. Note that the median is not affected by the extreme measurements.
| Math/90 | Math | null |
2. The table shows some of the values of differentiable functions $f$ and $g$ and their derivatives. If $h(x) = f(g(x))$, then $h'(2)$ equals
A. -2
B. -1
C. 0
D. 1
| B: The correct answer is (B). Since $h(x) = f(g(x))$, $h'(x) = f'(g(x)) \cdot g'(x)$ and $h'(2) = f'(g(2)) \cdot g'(2) = f'(3) \cdot 1 = -1. | Math/91 | Math | null |
The graph of $f$ is shown above, and $f$ is twice differentiable. Which of the following has the largest value?
I. $f(0)$
II. $f'(0)$
III. $f''(0)$
A. I
B. II
C. III
D. I and II
| C: Since $f$ is decreasing, $f' < 0$ and since $f$ is concave up, $f'' > 0$. The graph also shows that $f(0) < 0$. Thus $f''(0)$ has the largest value. | Math/92 | Math | null |
A slope field for a differential equation is shown above. Which of the following could be the differential equation?
A. $\frac{dx}{dy} = 2x$
B. $\frac{dx}{dy} = -2x$
C. $\frac{dx}{dy} = y$
D. $\frac{dx}{dy} = x - y$
| B: Note that for each column, all the tangents have the same slope. For example, when $x = 0$ all tangents are horizontal, which is to say their slopes are all zero. This implies that the slope of the tangents depends solely on the $x$-coordinate of the point and is independent of the $y$-coordinate. Also note that when $x > 0$ slopes are negative, and when $x < 0$ slopes are positive. Thus, the only differential equation that satisfies these conditions is $\frac{dy}{dx} = -2x$.
| Math/93 | Math | null |
The graph of $f''$, the second derivative of $f$ is shown above. The graph of $f''$ has horizontal tangents at $x = -2$ and $x = 2$. For what values of $x$ does the graph of the function $f$ have a point of inflection?
A. -4, 0 and 4
B. -2, 0 and 2
C. -4 and 4 only
D. 0 only | A: Note that $f'' > 0$ on the intervals $(-\infty, -4)$ and $(0, 4)$. Thus the graph of the function $f$ is concave up on these intervals. Similarly, $f'' < 0$ and concave down on the intervals $(-4, 0)$ and $(4, \infty)$. Therefore $f$ has a point of inflection at $x = -4, 0$, and $4$.
| Math/94 | Math | null |
The figure below shows the portions of the graphs of the ray $\theta = \frac{\pi}{4}$ and the curve $r = 2 \cos \theta + \sin \theta$ that lie in the first quadrant. Which of the following integrals expresses the area of the shaded region $R$?
A. $\int_{0}^{\frac{\pi}{4}} (\frac{\pi}{4} - r) d\theta$
B. $\int_{0}^{\frac{\pi}{4}} r^2 d\theta$
C. $\int_{0}^{\frac{\pi}{4}} \frac{r^2}{2} d\theta$
D. $\int_{0}^{1.6} (\frac{\pi}{4} - \frac{r^2}{2}) d\theta$
| C: The area of a simple region enclosed by a polar curve $r$ between two rays $\theta = \alpha$ and $\theta = \beta$ is $\int_{\alpha}^{\beta} \frac{r^2}{2} d\theta$.
In our case, the simple polar region $R$ is enclosed by $r$ and the rays $\theta = 0$ and $\theta = \frac{\pi}{4}$; therefore, the integral in answer (C) expresses the area of $R$.
| Math/95 | Math | null |
The graphs of the polar curves $r = 2 + \cos \theta$ and $r = -3 \cos \theta$ are shown on the graph below. The curves intersect when $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. Region $R$ is in the second quadrant, bordered by each curve and the y-axis.
Set up but do not evaluate a formula to find the area of $R$.
A. $\frac{1}{2} \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} ((2 + \cos \theta)^2 - (-3 \cos \theta)^2) d\theta$
B. $\int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} ((2 + \cos \theta)^2 - (-3 \cos \theta)^2) d\theta$
C. $\frac{1}{2} \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} ((-3 \cos \theta)^2 - (2 + \cos \theta)^2) d\theta$
D. $\frac{3}{4} \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} ((-3 \cos \theta)^2 - (2 + \cos \theta)^2) d\theta$ | C: When you see sine and cosine, most of the time it will involve the Pythagorean theorem, which states $\sin^2 x + \cos^2 x = 1$. You will want to solve for sine and cosine: $\frac{x}{2} = \sin t$ and $\frac{y}{3} = \cos t$. Now square both to get $\frac{x^2}{4} = \sin^2 t$ and $\frac{y^2}{9} = \cos^2 t$. Add the equations and simplify.
$\frac{x^2}{4} + \frac{y^2}{9} = \sin^2 t + \cos^2 t = 1$
Hence, (C) is correct. | Math/96 | Math | null |
The graph of the function $f$ is shown above. Which of the following statements is false?
A. $\lim_{x \to 1} f(x) = \infty$
B. $\lim_{x \to 3} f(x) = 1$
C. $\lim_{x \to 4} f(x) = 6$
D. $\lim_{x \to 5} f(x) = 4$ | D: The limit of a function exists at $x = a$ if both one-sided limits approach the same value. The limit does not have to equal the actual value of the function at that point. For this function, the limit from the left and from the right of $x = 1$ is $+\infty$ (because the graph increases without bound on both sides of the vertical asymptote), so the limit equals $\infty$ and you can eliminate (A). The limit from the left and from the right of $x = 3$ is 1 (even though $f(3)$ is 0), so eliminate (B). The limit from the left and from the right of $x = 4$ is 6, so eliminate (C). This means (D) must be correct. (The limit from the left of $x = 5$ is 4, while the limit from the right is 3, so the limit does not exist.)
| Math/97 | Math | null |
Let $h(x)$ be continuous on $[-5, -2]$ with some of the values shown in the following table:
If $h(x) = -1$ has no solutions on the interval $[-5, -2]$, which of the following values are possible for $a$?
I. -4
II. -3
III. 0
A. I only
B. II only
C. III only
D. I and II only | D: Since the function is continuous and never attains a value of $-1$ on the interval, neither will it attain any value greater than $-1$. Therefore, only $-4$ and $-3$ are possible values for $a$. See the discussion of the intermediate value theorem in 20.
| Math/98 | Math | null |
The function $f$ is continuous and differentiable on $0 < x < 10$. Use the table of values to determine an interval for which according to the mean value theorem $f'(c) = 0$ for some $c$ on the interval.
A. $6 < x < 7$
B. $5 < x < 7$
C. $1 < x < 7$
D. $5 < x < 6$
| C: The mean value theorem requires two points whose secant line has a slope equal to the slope of the tangent line. The slope of the tangent line is $f'(1, 5)$ and $(7, 5)$ are the only two points which qualify. The interval is $a < x < b$ or $1 < x < 7$.
| Math/99 | Math | null |
The table below includes all critical points of the continuous function $g(x)$. Use the table to determine where the function $g(x)$ is increasing within the interval $-2 < x < 20$.A. $0 < x < 9.08$ and $14 < x < 20$
B. $-2 < x < 0$
C. $-2 < x < -0.94$
D. $-2 < x < -0.94$ and $14 < x < 20$ | A: Notice that the information given is about $g(x)$, not $g'(x)$. A function can only change from increasing to decreasing or from decreasing to increasing at a critical point. $g(x)$ increases from $-0.33$ to $0$ on the interval $0 < x < 9.08$ and from $-7.3$ to $-4.4$ on the interval $14 < x < 20$. $g(x)$ is increasing on $0 < x < 9.08$ and $14 < x < 20$.
| Math/100 | Math | null |
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