Consider the simple linear regression model \( y = 50 + 10x + \varepsilon \), where \( \varepsilon \) is \( \text{NID}(0, 16) \). Suppose that \( n = 20 \) pairs of observations are used to fit this model. Generate 500 samples of 20 observations, drawing one observation for each level of \( x = 1, 1.5, 2, \ldots, 10 \) for each sample.

a. For each sample, compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of these estimates. Discuss the shape of these histograms.

b. For each sample, compute an estimate of \( E(y|x = 5) \). Construct a histogram of the estimates you obtained. Discuss the shape of the histogram.

c. For each sample, compute a 95% confidence interval (CI) on the slope. How many of these intervals contain the true value \( \beta_1 = 10 \)? Is this what you would expect?

d. For each estimate of \( E(y|x = 5) \) in part b, compute the 95% CI. How many of these intervals contain the true value of \( E(y|x = 5) = 100 \)? Is this what you would expect?

Question

21-08-2024
Mathematics

High School

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Consider the simple linear regression model \( y = 50 + 10x + \varepsilon \), where \( \varepsilon \) is \( \text{NID}(0, 16) \). Suppose that \( n = 20 \) pairs of observations are used to fit this model. Generate 500 samples of 20 observations, drawing one observation for each level of \( x = 1, 1.5, 2, \ldots, 10 \) for each sample.

a. For each sample, compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of these estimates. Discuss the shape of these histograms.

b. For each sample, compute an estimate of \( E(y|x = 5) \). Construct a histogram of the estimates you obtained. Discuss the shape of the histogram.

c. For each sample, compute a 95% confidence interval (CI) on the slope. How many of these intervals contain the true value \( \beta_1 = 10 \)? Is this what you would expect?

d. For each estimate of \( E(y|x = 5) \) in part b, compute the 95% CI. How many of these intervals contain the true value of \( E(y|x = 5) = 100 \)? Is this what you would expect?

Answer :

Final answer:

For each sample, compute the least-squares estimates of the slope and intercept. Construct histograms of the sample values of slope and intercept. Compute an estimate of E(y|x = 5) for each sample and construct a histogram of the estimates. Compute a 95% CI on the slope for each sample and count how many contain the true value. Compute a 95% CI on the estimate of E(y|x = 5) for each sample and count how many contain the true value.

Explanation:

To generate the samples, for each level of x from 1 to 10, we draw one observation for each sample. We then compute the least-squares estimates of the slope and intercept for each sample. The histograms of the sample values of the slope and intercept can help us understand their distributions.

For part b, we can compute an estimate of E(y|x = 5) for each sample. Constructing a histogram of these estimates can give us an idea of their distribution.

In part c, we can compute a 95% confidence interval on the slope for each sample. We can then count how many of these intervals contain the true value of β1 = 10.

Similarly, in part d, we can compute a 95% confidence interval on the estimate of E(y|x = 5) for each sample and see how many of these intervals contain the true value of E(y|x = 5) = 100.

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