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The following partial JMP regression output for the Fresh detergent data relates to predicting demand for future sales periods in which the price difference will be .10. SE Fit = .0647970, s = 0.316516
 
 	Predicted Demand	Lower 95% Mean
Demand	Upper 95% Mean
Demand
31	8.0806090242	7.9478783815	8.213339667
 	StdErr Indiv
Demand	Lower 95% Indiv
Demand	Upper 95% Mean
Demand
 	0.3231244931	7.4187185048	8.7424995436
 
   Click here for the Excel Data File
 
(a) Report a point estimate of and a 95 percent confidence interval for the mean demand for Fresh in all sales periods when the price difference is .10. (Round your CI answers to 3 decimal places and other answer to 4 decimal places.)
 
 
 
(b) Report a point prediction of and a 95 percent prediction interval for the actual demand for Fresh in an individual sales period when the price difference is .10. (Round your PI answers to 3 decimal places and other answer to 4 decimal places.)
 
 
 
(c) StdErr Indiv Demand on the JMP output equals s1+distance value ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√  . Using this information, find 99 percent confidence and prediction intervals for mean and individual demands when x = .10. (Round your answers to 4 decimal places.)
 
 

Let's go through the steps to answer each part of the question:

### Part (a)
**Point Estimate of Mean Demand and 95% Confidence Interval:**

The point estimate for the mean demand when the price difference is 0.10 is provided directly:
- **Point Estimate of Mean Demand:** 8.0806 (rounded to 4 decimal places)

The 95% confidence interval for the mean demand is also given:
- **Lower 95% CI for Mean Demand:** 7.9479 (rounded to 3 decimal places)
- **Upper 95% CI for Mean Demand:** 8.2133 (rounded to 3 decimal places)

### Part (b)
**Point Prediction of Actual Demand and 95% Prediction Interval:**

The point prediction for the actual demand in an individual sales period is the same as the point estimate for the mean demand:
- **Point Prediction of Actual Demand:** 8.0806 (rounded to 4 decimal places)

The 95% prediction interval for the actual demand in an individual sales period is:
- **Lower 95% PI for Actual Demand:** 7.4187 (rounded to 3 decimal places)
- **Upper 95% PI for Actual Demand:** 8.7425 (rounded to 3 decimal places)

### Part (c)
**99% Confidence and Prediction Intervals:**

To find the 99% confidence and prediction intervals, we need to use the standard error of the individual demand and the t-distribution. However, without additional details like the exact t-values, we can't directly compute these intervals here. But generally, the intervals can be found using the formula:

- **CI = Point Estimate ± (Critical t-value * StdErr Fit)**
- **PI = Point Estimate ± (Critical t-value * StdErr Indiv)**

For the 99% confidence and prediction intervals, the critical t-value would be higher than for the 95% intervals. However, to compute these intervals precisely, we would need the critical t-value for the specific degrees of freedom, which isn't provided here.

Would you like more details on how to compute these intervals, or do you have any other questions?

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### Questions:

1. How is the point estimate of mean demand determined in regression analysis?
2. What are the differences between confidence intervals and prediction intervals?
3. How do standard errors affect the width of confidence and prediction intervals?
4. Why is it important to specify the level of confidence (e.g., 95%, 99%) in statistical analysis?
5. What does a 95% confidence interval imply about the population parameter?
6. What does a 95% prediction interval imply about future observations?
7. How can the critical t-value be determined for different confidence levels?
8. How does the sample size affect the accuracy of point estimates and intervals?

**Tip:** Understanding the distinction between confidence and prediction intervals is crucial in statistical analysis, as they serve different purposes and have different implications for data interpretation.

Learn how to analyze JMP regression output for Fresh detergent data to predict demand and calculate confidence intervals and prediction intervals. Understand the point estimates, standard errors, and their implications in statistical analysis.

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