Im having trouble with a question finding an estimate for $\displaystyle \beta_1$ as well as showing its unbiased and finding its variance. There is a set of data with the points

$\displaystyle (1,Y_1) , (2,Y_2) , (3,Y_3) , (4,Y_4), (5,Y_5) $

We want to fit a simple linear regression of the model:

$\displaystyle Y_i = \beta_0 + \beta_1i + \epsilon_i $

Where: $\displaystyle \epsilon_i ~ N(0,\sigma^2) $

The question asks me to find an estimate for $\displaystyle \beta_1$ by joining each set of points and writing the slope of each line formed. Thus, using the average of all the individual slopes to calculate $\displaystyle \beta_1$

$\displaystyle \hat\beta_1 = \frac{\sum\gamma_i}{4}$

Where: $\displaystyle \gamma_i$ is the slope between the points $\displaystyle (i, Y_i) \mbox{and} (i+1,Y_{i+1})$

My attempt:

The slope of any $\displaystyle \gamma_i $ is:
$\displaystyle \gamma_i = \frac{Y_{i+1} - Y_i}{i+1-i} = Y_{i+1} - Y_i $


$\displaystyle \sum\gamma_i = (Y_2 - Y_1) + (Y_3 - Y_2) + (Y_4 - Y_3) + (Y_5 - Y_4) = (Y_5 - Y_1)

Which leads me to the estimate $\displaystyle \hat\beta_1 = \frac{Y_5 - Y_1}{4}$

Can someone tell me if I am on the right track, and how can I determine the bias of this estimator? Aren't the values just constant...? Thanks