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Chapter 10: Q. 10.102 (page 431)

Suppose that the sample sizes, $n_{4}$ and $n_{2}$ , are equal for independent simple random samples from two populations.
a. Show that the values of the pooled and nonpooled r-statistics will be identical. (Hint: Refer to Exercise 10.61 on page 417.)
b. Explain why part (a) does not imply that the two t-tests are Equivalent (i.e., will necessarily lead to the same conclusion) when the sample sizes are equal.

Short Answer

Expert verified

a) The standard deviations for both are different.

b) The standard deviations for both are different.

Step by step solution

01

Part (a) Step 1: Given Information

To discover the proof that pooled and non-pooled t-test values are identical.

02

Part (a) Step 2: Explanation

When the tests are run, it is discovered that the values for the pooled and non-pooled tests are not identical.

03

Part (b) Step 1: Given Information

To ascertain the reason for the lack of equivalence.

04

Part (b) Step 2: Explanation

Because the standard deviations for both tests differ, the tests do not have the same values for the same sample sizes.

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Most popular questions from this chapter

$H_{2} : μ_{1} > μ_{2}$

V. Tangpricha et al. did a study to determine whether fortifying orange juice with Vitamin D would result in changes in the blood levels of five biochemical variables. One of those variables was the concentration of parathyroid hormone (PTH), measured in picograms/milliliter ( pg/ml ). The researchers published their results in the paper "Fortification of Orange Juice with Vitamin D: A Novel Approach for Enhancing Vitamin D Nutritional Health" (American Journal of Clinical Nutrition, Vol. 77, pp. 1478-1483). Concentration levels were recorded at the beginning of the experiment and again at the end of $12$ weeks. The following data, based on the results of the study, provide the decrease (negative values indicate an increase) in PTH levels, in pg/ml, for those drinking the fortified juice and for those drinking the unfortified juice.

At the $5 %$ significance level, do the data provide sufficient evidence to conclude that drinking fortified orange juice reduces PTH level more than drinking unfortified orange juice? (Note: The mean and standard deviation for the data on fortified juice are $9.0 pg / mL$ and $37.4 pg / mL$ , respectively, and for the data on unfortified juice, they are $1.6 pg / mL$ and $34.6 pg / mL$ , respectively.)

In this Exercise, we have provided summary statistics for independent simple random samples from two populations. In each case, use the pooled $t$ -test and the pooled $t$ -interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.

${\bar{x}}_{1} = 20, s_{1} = 4, n_{1} = 10, {\bar{x}}_{2} = 23, s_{2} = 5, n_{2} = 15$

a. Left-tailed test, $α = 0.05$

b. $90 %$ confidence interval

Data on household vehicle miles of travel (VMT) are compiled annually by the Federal Highway Administration and are published in the National Household Travel Survey, Summary of Travel Trends. Independent random samples of $15$ midwestern households and $14$ southern households provided the following data on last year's VMT, in thousands of miles.

At the $5 %$ significance level, does there appear to be a difference in last year's mean VMT for midwestern and southern households? (Note: ${\bar{x}}_{1} = 16.23, s_{1} = 4.06, {\bar{x}}_{2} = 17.69$ , and $s_{2} = 4.42$ .)

Consider the quantities $μ_{1}, σ_{1}, {\bar{x}}_{1}, s_{1}, μ_{2}, σ_{2}, {\hat{x}}_{2}$ , and $s_{2}$ .

a. Which quantities represent parameters and which represent statistics?

b. Which quantities are fixed numbers and which are variables?

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