# Probability exam problem

1. Apr 16, 2009

### superwolf

a) Because of traffic, Fabian is lead to believe that the bike trip to the university takes longer time the later he starts in the morning. He collects data the for one week, keeping track of his starting time ($$t_i$$ minutes after 7:00) and the bike time ($$y_i$$), see Table 2.

Construct a simple linear regression model for the bike time. Specify your assumptions. Write down the least-squares estimator for the parameters of the model ( do not show how they appear by a mathematical proof). Use, without proof, that these estimators are unbiased. Assume, in the rest of the exercise, that the regression noise terms $$\epsilon$$ are normal distributed with expectation 0 and known variance 0.5^2. Formulate Fabians theory as a hypothesis test. Do the test at significance level 1%.

this one is OK.

b) One day Fabian starts at 8:30. Use the regression model in d) to predict Fabians bike time. Calculate a 95% prediction interval for Fabians bike time. Discuss the result.

$$Y_{estimate} = 5.31 + 0.037 t = 5.31 + 0.037 \cdot 90 = 8.64$$, which is correct.

Now, how do I proceed? The following formula is in my book:

Can I use this?

I know that the prediction error $$Y_{estimate} - Y$$ has mean value 0 and variance $$\frac{\sigma^2_{epsilon}}{n} + (t_0 - \bar{t})^2 \frac{\sigma^2_{epsilon}}{\Sigma^n_{i=n}(t_i - t_{mean})^2} + \sigma^2_{epsilon}$$

Should I use n = 5 here? If so, I get $$\sigma = \sqrt{0.7}$$ and, using the formula from my book, I get the prediction interval [6.84, 10.44]. Here I have used $$z_{\alpha / 2} = 1.96$$.

The correct answer is [6.51, 10.81]