Question: Peanut Butter. American shoppers have a great variety of products from which to choose. Many people turn to information in consumer surveys and product comparisons to help make decisions. A consumer magazine rated 35 varieties of peanut butter. Each peanut butter was assigned a quality rating from 1 (low) to 100 (high) points. In addition, price (in cents) per 3-tablespoon serving was recorded. The goal of the survey was to find evidence whether higher rated products tend to be pricier, i.e. does better quality tend to cost more?

1. Refer to Figure 2 and describe the relationship between the quality rating and the price of the peanut butter. Make sure to focus on the 4 features discussed in class when describing the relationship between two variables. (b) Propose a theoretical model (i.e., a population model) that would be appropriate to model the linear relationship between the quality rating and the price of the peanut butter. Provide a brief description (for example, "explanatory variable") for any symbols that you use in the model.
2. Propose a theoretical model (i.e., a population model) that would be appropriate to model the linear relationship between the quality rating and the price of the peanut butter. Provide a brief description (for example, "explanatory variable") for any symbols that you use in the model.
3. Referring to the JMP output in Figure 2 what is the value of \(r\), measuring the strength of the linear relationship between quality rating and the price? Explain how you can determine the value of \(r\) based on the given output.
4. Is there sufficient evidence to conclude that peanut butter that get higher quality rating scores also tend to be pricier? Make sure to conduct a complete hypothesis test. That is, include an appropriate null and alternative hypothesis, test statistic, p-value, decision and conclusion within the context of the problem.
5. Provide an interpretation of the estimated slope within the context of the problem.
6. Assume we had evidence that the value of the population slope were 0.15. How would an interpretation of the population slope differ from the interpretation of the sample slope?
7. How would you interpret the estimated intercept? If you feel an interpretation would not be meaningful, explain what the general mathematical interpretation of the intercept is instead.
8. The consumer magazine is interested in summarizing their findings of their survey on the quality score and the price of the peanut butter by providing readers with a range for the average price per 3 -tablespoon serving for peanut butter products obtaining a quality score of 30, 60 , and 90 points, respectively. Construct the most appropriate \(99 \%\) interval for a quality rating of 30 points. The needed critical value for a \(99 \%\) level of confidence is 2.75. Provide an interpretation of the interval within the context of the problem. Refer to Figure 3 of the JMP output that may be helpful.
9. A \(99 \%\) confidence interval for a quality rating of 60 points would be
   \(\begin{array}{lll}\text { NARROWER } & \text { BE OF THE SAME WIDTH } & \text { WIDER }\end{array}\)
10. You decide to buy a jar of peanut butter that has a quality rating of 30 points. Do you feel \(99 \%\) confident that the price per 3 -tablespoon serving will fall in the interval that you obtained in part (8h)? Carefully explain why or why not.
11. Take a close look at the scatterplot in Figure 1 below and the LS regression line fitted through the data by JMP. For all the values of \(x\) check whether the line roughly falls where we would expect it to fall. Do you think the line describes the data well? Explain your answer.

If you think that the line does not describe the overall relationship as adequately as one might expect, point to what might be causing the line to deviate from what one would expect and draw a more appropriate line. If the line does provide an adequate fit, explain to someone not familiar with the principle of Least Squares regression why the line is adequate.

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