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Chapter 10: Q2BSC (page 468)

Prediction Interval Using the heights and weights described in Exercise 1, a height of 180 cm is used to find that the predicted weight is 91.3 kg, and the 95% prediction interval is (59.0 kg, 123.6 kg). Write a statement that interprets that prediction interval. What is the major advantage of using a prediction interval instead of simply using the predicted weight of 91.3 kg? Why is the terminology of prediction interval used instead of confidence interval?

Short Answer

Expert verified

The 95% prediction interval stands for the fact that for thegiven value of height (x) at 180 cm, there is 95% confidence that the predicted value of weight (y) will fall between 59.0 kg and 123.6 kg.

The advantage of the prediction interval over the point estimate of y is that it tells the accuracy of the prediction. It considers the margin of error (likely to occur) when predicting the weight for a given height.

Since the interval estimate is used to define the predicted value of the response variable and not any of its parameters (mean, variance, proportion, etc.), the computed interval is called the prediction interval and not the confidence interval.

Step by step solution

01

Given information

The 95% prediction interval for the weight of males (y) for a height (x) of 180 cm is (59.0 kg, 123.6 kg).

02

Prediction interval

A prediction interval is the interval estimate of the predicted value of the response variable for a given value of the predictor variable.

Here, the 95% prediction interval of the weight of males when the height is 180 cm is obtained as (59.0 kg, 123.6 kg).

This means that there is a 95% confidence that the predicted value of the weight will fall within 59.0 kg and 123.6 kg when the height is 180 cm.

03

Advantage of using a prediction interval instead of using only the predicted weight of 91.3 kg

An interval estimate has the advantage of defining a range with a known probability of capturing the desired value.

Here, it is better to define a prediction interval than a simple estimate because the prediction interval provides some accuracy in the fact that the predicted value of the weight will lie between 59.0 kg and 123.6 kg and not pinpoint just a single value.

The prediction interval allows you to be more cautious in the interpretation of data and assists in maintaining the right perspective.

04

Prediction interval versus confidence interval

A confidence interval is used to define the range within which the value of a parameter is likely to fall.For example, a 95% confidence interval can be constructed to define a set of values that will include the slope coefficient.

On the other hand, a prediction interval is used to define the range within which the value of the response variable is likely to fall.

It is given that the 95% prediction interval for the value of the weight of males when the height is 180 cm is (59.0 kg, 123.6 kg). Thus, it would be incorrect to call it a confidence interval because it is not used to define the value of a parameter.

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Most popular questions from this chapter

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Use the shoe print lengths and heights to find the best predicted height of a male who has a shoe print length of 31.3 cm. Would the result be helpful to police crime scene investigators in trying to describe the male?

Outlier Refer to the accompanying Minitab-generated scatterplot. a. Examine the pattern of all 10 points and subjectively determine whether there appears to be a correlation between x and y. b. After identifying the 10 pairs of coordinates corresponding to the 10 points, find the value of the correlation coefficient r and determine whether there is a linear correlation. c. Now remove the point with coordinates (10, 10) and repeat parts (a) and (b). d. What do you conclude about the possible effect from a single pair of values?

Interpreting the Coefficient of Determination. In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Weight , Waist r = 0.885 (x = weight of male, y = waist size of male)

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Using the listed duration and interval after times, find the best predicted “interval after” time for an eruption with a duration of 253 seconds. How does it compare to an actual eruption with a duration of 253 seconds and an interval after time of 83 minutes?

Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2

1.3

1.5

1.6

8

3.4

3.5

2.8

2.6

2.2

3

7.1

2.3

2.1

3.4

6.4

5

4.2

2.8

3.9

5.2

6.9

6.9

5

5.5

6

5.5

8.6

9.4

10

7.6










Pain Intensity After Duragesic Treatment

0.4

1.4

1.8

2.9

6

1.4

0.7

3.9

0.9

1.8

0.9

9.3

8

6.8

2.3

0.4

0.7

1.2

4.5

2

1.6

2

2

6.8

6.6

4.1

4.6

2.9

5.4

4.8

4.1










Two Independent Samples The methods of Section 9-2 can be used to test the claim that two populations have the same mean. Identify the specific claim that the treatment is effective, then use the methods of Section 9-2 to test that claim. The methods of Section 9-2 are based on the requirement that the samples are independent. Are they independent in this case?
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