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Chapter 8: Q63E (page 452)

Suppose you want to estimate the difference between two population means correct to within 1.8 with a 95% confidence interval. If prior information suggests that the population variances are approximately equal to 蟽12=蟽22=14 and you want to select independent random samples of equal size from the populations, how large should the sample sizes n1, and n2, be?

Short Answer

Expert verified

The sample sizes andshould be at least 34.

Step by step solution

01

Given information

The sampling error isSE=1.8

The population variances are蟽12=蟽22=14.

Using the standard normal table, the critical value at a 95% confidence level isZ伪2=1.96

02

Sample size

The sample sizes are a crucial part of any statistical experiment. Considering a sufficient sample size per population is crucial to getting significant results. It can be calculated using the sampling error, population variances, and confidence level.

03

Calculating the sample size

Assume

n1=n2=n

The required sample size is

n=Z伪22蟽12+蟽22SE2=1.96214+141.82=1.962281.82=33.2鈮�34

Hence the required sample size is n1=n2=34.

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Most popular questions from this chapter

Traffic sign maintenance. Refer to the Journal of Transportation Engineering (June 2013) study of traffic sign maintenance in North Carolina, Exercise 8.54 (p. 489). Recall that the proportion of signs on NCDOT-maintained roads that fail minimum requirements was compared to the corresponding proportion for signs on county-owned roads. How many signs should be sampled from each maintainer to estimate the difference between the proportions to within .03 using a 90% confidence interval? Assume the same number of signs will be sampled from NCDOT-maintained roads and county-owned roads

Question: Deferred tax allowance study. A study was conducted to identify accounting choice variables that influence a manager鈥檚 decision to change the level of the deferred tax asset allowance at the firm (The Engineering Economist, January/February 2004). Data were collected for a sample of 329 firms that reported deferred tax assets in 2000. The dependent variable of interest (DTVA) is measured as the change in the deferred tax asset valuation allowance divided by the deferred tax asset. The independent variables used as predictors of DTVA are listed as follows:

LEVERAGE: x1= ratio of debt book value to shareholder鈥檚 equity

BONUS: x2 = 1 if firm maintains a management bonus plan,

0 if not

MVALUE: x3 = market value of common stock

BBATH: x4 = 1 if operating earnings negative and lower than last year,

0 if not

EARN: x5 = change in operating earnings divided by total assets

A first-order model was fit to the data with the following results (p-values in parentheses):

Ra2 = .280

y^=0.044+0.006x1-0.035x2-0.001x3+0.296x4+0.010x5

(.070) (.228) (.157) (.678) (.001) (.869)

  1. Interpret the estimate of the 尾 coefficient for x4.
  2. The 鈥淏ig Bath鈥� theory proposed by the researchers鈥� states that the mean DTVA for firms with negative earnings and earnings lower than last year will exceed the mean DTVA of other firms. Is there evidence to support this theory? Test using 伪 = .05.
  3. Interpret the value of Ra2.

Question: Summer weight-loss camp. Camp Jump Start is an 8-week summer camp for overweight and obese adolescents. Counselors develop a weight-management program for each camper that centers on nutrition education and physical activity. To justify the cost of the camp, counselors must provide empirical evidence that the weight-management program is effective. In a study published in Paediatrics (April 2010), the body mass index (BMI) was measured for each of 76 campers both at the start and end of camp. Summary statistics on BMI measurements are shown in the table.

Source: Based on J. Huelsing, N. Kanafani, J. Mao, and N. H. White, "Camp Jump Start: Effects of a Residential Summer Weight-Loss Camp for Older Children and Adolescents," Pediatrics, Vol. 125, No. 4, April 2010 (Table 3).

a. Give the null and alternative hypotheses for determining whether the mean BMI at the end of camp is less than the mean BMI at the start of camp.

b. How should the data be analyzed, as an independent samples test or as a paired difference test? Explain.

c. Calculate the test statistic using the formula for an independent samples test. (Note: This is not how the test should be conducted.)

d. Calculate the test statistic using the formula for a paired difference test.

e. Compare the test statistics, parts c and d. Which test statistic provides more evidence in support of the alternative hypothesis?

f. The p-value of the test, part d, was reported as p 6 .0001. Interpret this result, assuming a = .01.

g. Do the differences in BMI values need to be normally distributed in order for the inference, part f, to be valid? Explain.

h. Find a 99% confidence interval for the true mean change in BMI for Camp Jump Start campers. Interpret the result.

Question: Independent random samples selected from two normal populations produced the sample means and standard deviations shown below.

Sample 1

Sample 2

n1= 17x炉1= 5.4s1= 3.4

role="math" localid="1660287338175" n2= 12x炉2=7.9s2=4.8

a. Conduct the testH0:(渭1-渭2)>10against Ha:(渭1-渭2)鈮�10. Interpret the results.

b. Estimate渭1-渭2 using a 95% confidence interval

Question: Two independent random samples have been selected鈥�100 observations from population 1 and 100 from population 2. Sample means x炉1=26.6,x炉2= 15.5 were obtained. From previous experience with these populations, it is known that the variances are蟽12=9and蟽22=16 .

a. Find 蟽(x炉1-x炉2).

b. Sketch the approximate sampling distribution for (x炉1-x炉2), assuming (渭1-渭2)=10.

c. Locate the observed value of (x炉1-x炉2)the graph you drew in part

b. Does it appear that this value contradicts the null hypothesis H0:(渭1-渭2)=10?

d. Use the z-table to determine the rejection region for the test againstH0:(渭1-渭2)鈮�10. Use伪=0.5.

e. Conduct the hypothesis test of part d and interpret your result.

f. Construct a confidence interval for 渭1-渭2. Interpret the interval.

g. Which inference provides more information about the value of 渭1-渭2鈥� the test of hypothesis in part e or the confidence interval in part f?
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