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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 σ12=σ22=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 areσ12=σ22=14.

Using the standard normal table, the critical value at a 95% confidence level isZα2=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=Zα22σ12+σ22SE2=1.96214+141.82=1.962281.82=33.2≈34

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

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Most popular questions from this chapter

Patron amenability to supply biomass. Relate to the Biomass and Energy (Vol. 36, 2012) study of the amenability of directors to supply biomass products similar to fat hay, Exercise8.20 (p. 469). Recall that independent samples of Missouri directors and Illinois directors were surveyed. Another aspect of the study concentrated on the service directors who were willing to supply. One essential service involves windrowing (mowing and piling) hay. Of the 558 Missouri directors surveyed, 187 were willing to offer windrowing. Of the 940 Illinois directors surveyed, 380 were willing to offer windrowing services. The experimenters want to know if the proportion of directors willing to offer windrowing services to the biomass request differs for the two areas, Missouri and Illinois.

a. Specify the parameter of interest to the experimenters.

b. Set up the null and indispensable suppositions for testing whether the proportion of directors willing to offer windrowing services differs in Missouri and Illinois.

c. A Minitab analysis of the data is given below. Detect the test statistic on the printout.

d. provide the rejection region for the test using a = .01.

e. Detect the p- the value of the test on the printout.

f. Make the applicable conclusion using both the p-value and rejection region approach. Your conclusions should agree.

Given that xis a hypergeometric random variable, computep(x)for each of the following cases:

a. N= 8, n= 5, r= 3, x= 2

b. N= 6, n= 2, r= 2, x= 2

c. N= 5, n= 4, r= 4, x= 3

4.132 Suppose xis a random variable best described by a uniform

probability distribution with c= 3 and d= 7.

a. Find f(x)

b. Find the mean and standard deviation of x.

c. FindP(μ-σ≤x≤μ+σ)

Question: Refer to the Bulletin of Marine Science (April 2010) study of lobster trap placement, Exercise 6.29 (p. 348). Recall that the variable of interest was the average distance separating traps—called trap-spacing—deployed by teams of fishermen. The trap-spacing measurements (in meters) for a sample of seven teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap-spacing measurements for eight teams from the Punta Abreojos (PA) fishing cooperative are listed. For this problem, we are interested in comparing the mean trap-spacing measurements of the two fishing cooperatives.

BT Cooperative

93

99

105

94

82

70

86

PA Cooperative

118

94

106

72

90

66

98


Source: Based on G. G. Chester, “Explaining Catch Variation Among Baja California Lobster Fishers Through Spatial Analysis of Trap-Placement Decisions,” Bulletin of Marine Science, Vol. 86, No. 2, April 2010 (Table 1).

a. Identify the target parameter for this study.b. Compute a point estimate of the target parameter.c. What is the problem with using the normal (z) statistic to find a confidence interval for the target parameter?d. Find aconfidence interval for the target parameter.e. Use the interval, part d, to make a statement about the difference in mean trap-spacing measurements of the two fishing cooperatives.f. What conditions must be satisfied for the inference, part e, to be valid?

Let t0 be a particular value of t. Use Table III in Appendix D to find t0 values such that the following statements are true.

a.=P(-t0<t<t0).95wheredf=10b.P(t≤-t0ort≥t0)wheredf=10c.P(t≤t0)=.05wheredf=10d.P(t≤-t0ort≥t0)=.10wheredf=20e.P(t≤-t0ort≥t0)=.01wheredf=5
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