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Chapter 8: Q76E (page 452)

Random samples of size n1 = 21 and n2 = 31 were drawn from populations 1 and 2, respectively. For each of the following cases, determine the test statistic and appropriate decision for a two-tailed test of the null hypothesis \({H_0}:\sigma _1^2 = \sigma _2^2\) against \({H_a}:\sigma _1^2 \ne \sigma _2^2\,at\,\alpha = 0.05\)

  1. \(s_1^2 = 500\,and\;s_2^2 = 900\)

Short Answer

Expert verified
  1. The test statistic is 1.8.
    We fail to reject the null hypothesis.

Step by step solution

01

Given Information

The hypothesis are given by

\(\begin{aligned}{H_0}:\sigma _1^2 = \sigma _2^2\\{H_0}:\sigma _1^2 \ne \sigma _2^2\end{aligned}\)

The level of significance is 0.05.

02

Compute the test statistic

The test statistic is calculated as

\(\begin{aligned}F &= \frac{{s_2^2}}{{s_1^2}}\\ &= \frac{{900}}{{500}}\\ &= \frac{9}{5}\\ &= 1.8\end{aligned}\)

Therefore, the test statistic is 1.8.

03

Decision-rule

The rejection region for two tailed test is

\(F > {F_{\frac{\alpha }{2}}}\)

For

\(\begin{aligned}{F_{\frac{\alpha }{2}}}\left( {{n_1} - 1,{n_2} - 1} \right)\\ = {F_{0.025}}\left( {20,30} \right)\end{aligned}\)

Critical value: 0.4258 and 2.1952

Hence, the statistic doesn鈥檛 fall in rejection region.

Therefore, we fail to reject the null hypothesis.

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Most popular questions from this chapter

Fingerprint expertise.A study published in PsychologicalScience(August 2011) tested the accuracy of experts andnovices in identifying fingerprints. Participants were presentedpairs of fingerprints and asked to judge whetherthe prints in each pair matched. The pairs were presentedunder three different conditions: prints from the same individual (match condition), non-matching but similar prints (similar distracter condition), and nonmatching and very dissimilar prints (non-similar distracter condition). The percentages of correct decisions made by the two groups under each of the three conditions are listed in the table.

Conditions

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Non-similar distracter

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Independent random samples from normal populations produced the results shown in the next table.

Sample 1


Sample 2

1.23.11.72.83.0

 4.22.73.63.9

a. Calculate the pooled estimate of 蟽2.

b. Do the data provide sufficient evidence to indicate that 渭2&驳迟;渭1? Test using 伪=.10.

c. Find a 90% confidence interval for (渭1鈭�渭2).

d. Which of the two inferential procedures, the test of hypothesis in part b or the confidence interval in part c, provides more information about (渭1鈭�渭2)?
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