Chapter 6: Q13 (page 264)

In Exercises 11–14, use the population of {34, 36, 41, 51} of the amounts of caffeine (mg/12oz) in Coca-Cola Zero, Diet Pepsi, Dr Pepper, and Mellow Yello Zero.
Assume that random samples of size n = 2 are selected with replacement.
Sampling Distribution of the Range Repeat Exercise 11 using ranges instead of means.

Short Answer

Expert verified

a. The following table represents the sampling distribution of the sample ranges:

Sample	Sample Range
(34,34)	0
(34,36)	2
(34,41)	7
(34,51)	17
(36,34)	2
(36,36)	0
(36,41)	5
(36,51)	15
(41,34)	7
(41,36)	5
(41,41)	0
(41,51)	10
(51,34)	17
(51,36)	15
(51,41)	10
(51,51)	0

Combining all the same values of ranges, the following table is obtained:

Sample Range	Probability
0	$\frac{4}{16}$
2	$\frac{2}{16}$
5	$\frac{2}{16}$
7	$\frac{2}{16}$
10	$\frac{2}{16}$
15	$\frac{2}{16}$
17	$\frac{2}{16}$

b. The population range is not equal to the mean of the sampling distribution of the sample range.

c. Since the population range is not equal to the mean of the sample ranges, it can be said that the sample ranges do not target the value of the population range.

Since the mean value of the sampling distribution of the sample range is not equal to the population range, the sample range cannot be considered as a good estimator of the population range.

Step by step solution

Given information

A population of the amounts of caffeine in 3 different drink brands is provided.

Samples of size equal to 2 are extracted from this population with replacement.

Sampling distribution of sample ranges

All possible samples of size 2 selected with replacement are tabulated below:

(34,34)	(36,34)	(41,34)	(51,34)
(34,36)	(36,36)	(41,36)	(51,36)
(34,41)	(36,41)	(41,41)	(51,41)
(34,51)	(36,51)	(41,51)	(51,51)

The range has the following formula:

$R a n g e = Maximum Value - Minimum Value$

The following table shows all possible samples of size equal to 2 and the corresponding sample ranges:

Sample	Sample Range
(34,34)	$Sample {Range}_{1} = 34 - 34 = 0$
(34,36)	$Sample {Range}_{2} = 36 - 34 = 2$
(34,41)	$Sample {Range}_{3} = 41 - 34 = 7$
(34,51)	$Sample {Range}_{4} = 51 - 34 = 17$
(36,34)	$Sample {Range}_{5} = 36 - 34 = 2$
(36,36)	$Sample {Range}_{6} = 36 - 36 = 0$
(36,41)	$Sample {Range}_{7} = 41 - 36 = 5$
(36,51)	$Sample {Range}_{8} = 51 - 36 = 15$
(41,34)	$Sample {Range}_{9} = 41 - 34 = 7$
(41,36)	$Sample {Range}_{10} = 41 - 36 = 5$
(41,41)	$Sample {Range}_{11} = 41 - 41 = 0$
(41,51)	$Sample {Range}_{12} = 51 - 41 = 10$
(51,34)	$Sample {Range}_{13} = 51 - 34 = 17$
(51,36)	$Sample {Range}_{14} = 51 - 36 = 15$
(51,41)	$Sample {Range}_{15} = 51 - 41 = 10$
(51,51)	$Sample {Range}_{16} = 51 - 51 = 0$

Combining the values of ranges that are the same, the following probability values are obtained:

Sample Range	Probability
0	$\frac{4}{16}$
2	$\frac{2}{16}$
5	$\frac{2}{16}$
7	$\frac{2}{16}$
10	$\frac{2}{16}$
15	$\frac{2}{16}$
17	$\frac{2}{16}$

Population range and mean of the sample ranges

The population range is computed as shown below:

$Population Range = Maximum Value - Minimum Value = 51 - 34 = 17$

Thus, the population range is equal to 17.

The mean of the sample ranges is computed below:

$Mean of Sample Ranges = \frac{Sample {Range}_{1} + Sample {Range}_{2} + . . . . . + Sample {Range}_{16}}{16} = \frac{0 + 2 + . . . . + 0}{16} = 7$

Thus, the mean of the sampling distribution of the sample range is equal to 7.

Here, the population range (17) is not equal to the mean of the sampling distribution of the sample range (7).

Good estimator

Since the population range is not equal to the mean of the sample ranges, it can be said that the sample rangesdo not target the value of the population range.

A good estimator is a sample statistic whose sampling distribution has a mean value equal to the population parameter.

The mean value of the sampling distribution of the sample range (7) is not equal to the population range (17).

Thus, the sample rangecannot be considered as a good estimator of the population range.

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Short Answer

Step by step solution

Given information

Sampling distribution of sample ranges

Population range and mean of the sample ranges

Good estimator

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