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# 1) Question: The NetWt (y) and Count (x) for a sample of n=3 bags of Almond Mamp;M's is shown below. Assume these were drawn at random from a...

**1) **

**Question:**The NetWt (y) and Count (x) for a sample of n=3 bags of Almond M&M's is shown below. Assume these were drawn at random from a production run of 5000 bags. Treat that production run as a population. A table of t-critical values is given at the end of the exam. The questions in this problem focus on numerical work, so save your deeper thoughts for page 3.

Count NtWt

13 40.6

14 41.3

12 36.8

(a) By hand and pocket calculator only where absolutely needed, compute the sample slope and intercept. Show your work - convince me that you haven't simply used built-in statistics functions on a fancy calculator. I recommend working in columns to the right of the given values. That will facilitate answering (b).

(b) Compute the y-hat values, the residuals, and the MSE. (You can show your work above.)

(c) Compute the sample standard error of .

(d) For the slope t-test, state the null and alternate hypotheses in symbols.

(e) Compute the t-statistic for the above t-test.

(f) State the rejection region for the test.

(g) Is the null hypothesis rejected or not? On what grounds?

(h) Sketch the relevant t-distribution. Show and label the t-critical values, the t-statistic, the rejection region, the alpha regions, the p-value regions.

(i) Define "p-value" as it is applied to this setting. Include the positive and the negative values of the t-statistic in your definition.

Consider the same setting as in the first problem, where we have a sample of size n=3 from a population of 5000 bags of Almond M&M's from a single production run. Now, we will focus on the simple linear regression model, itself:

Yx = + x + ε,

(aa) In this Almond M&M setting, what is meant by Yx on the left-hand side of that equation?

(bb) What is meant by x on the right-side in this Almond M&M setting?

(cc) What is the role of ε?

(dd) In Problem 1, you computed the MSE. What parameter of the regression model is estimated by the MSE? Write out the explanation in words, referencing Almond M&M's.

(ee) Given x=15, what parameter is estimated by y-hat? State your answer in words, not symbols.

(ff) Consider the population of 5000 bags of Almond M&M's with sample size n=3. Explain what it means to say that the sample slope is an unbiased estimator of . Your answer should include computing the size of the sample space and a clear description of the construction of the relevant daughter population.