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Chapter 7: Q67E (page 424)

Refer to Exercise 6.44 (p. 356), in which 50 consumers taste-tested a new snack food. Their responses (where 0 = do not like; 1 = like; 2 = indifferent) are reproduced below

  1. Test \({H_0}:p = .5\) against \({H_0}:p > .5\), where p is the proportion of customers who do not like the snack food. Use \(\alpha = 0.10\).
    1 0 0 1 2 0 1 1 0 0 0 1 0 2 0 2 2 0 0 1 1 0 0 0 0 1 0 2 0 0 0 1 0 0 1 0 0 1 0 1 0 2 0 0 1 1 0 0 0 1

Short Answer

Expert verified
  1. The test statistic is 1.142

Step by step solution

01

Given Information

The number of sample size is 50.

The hypothesis are given by

\(\begin{aligned}{H_0}:p = .5\\{H_a}:p > .5\end{aligned}\)

02

Null hypothesis

A null hypothesis is a statistical supposition that claims there is no difference between specific features of a population as well as data-generating activity. The alternate hypothesis asserts that there is a distinction. Hypothesis test enables you to reject a null hypothesis with a particular confidence level.

03

Testing of hypothesis

The proportion of customers who do not like the snack food is given below

\(\begin{aligned}\hat p &= \frac{{29}}{{50}}\\ &= 0.58\end{aligned}\)

The test statistic is calculated as

\(\begin{aligned}z &= \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ &= \frac{{0.58 - 0.5}}{{\sqrt {\frac{{.5 \times .5}}{{50}}} }}\\ &= \frac{{0.08}}{{0.07}}\\ &= 1.142\end{aligned}\)

Therefore, the test statistic is 1.142.

Rejection region \(z > {z_{0.1}} = 1.282\)

Therefore, the calculated z-value does not fall in the rejection region. We do not reject the null hypothesis.

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Most popular questions from this chapter

Consider the test \({H_0}:\mu = 70\) versus \({H_a}:\mu \ne 70\) using a large sample of size n = 400. Assume\(\sigma = 20\).

a. Describe the sampling distribution of\(\bar x\).

b. Find the value of the test statistic if\(\bar x = 72.5\).

c. Refer to part b. Find the p-value of the test.

d. Find the rejection region of the test for\(\alpha = 0.01\).

e. Refer to parts c and d. Use the p-value approach to

make the appropriate conclusion.

f. Repeat part e, but use the rejection region approach.

g. Do the conclusions, parts e and f, agree?

For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.

a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)

b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)

c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)

d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)

e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)

f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)

g. For each rejection region specified in parts a鈥揻, state the probability notation in z and its respective Type I error value.

In a test of \({H_0}:\mu = 100\) against \({H_a}:\mu \ne 100\), the sample data yielded the test statistic z = 2.17. Find the p-value for the test.

Question: Point spreads of NFL games. Refer to the Chance (Fall 1998) study of point-spread errors in NFL games, Exercise 7.41 (p. 411). Recall that the difference between the actual game outcome and the point spread established by odds makers鈥攖he point-spread error鈥攚as calculated for 240 NFL games. The results are summarized as follows: . Suppose the researcher wants to know whether the true standard deviation of the point spread errors exceeds 15. Conduct the analysis using 伪 = 0.10.

Arresting shoplifters. Shoplifting in the United States costs retailers about $35 million a day. Despite the seriousness of the problem, the National Association of shoplifting Prevention (NASP) claims that only 50% of all shoplifters are turned over to police (www.shopliftingprevention.org). A random sample of 40 U.S. retailers were questioned concerning the disposition of the most recent shoplifter they apprehended. A total of 24 were turned over to police. Do these data provide sufficient evidence to contradict the NASP?

a. Conduct a hypothesis test to answer the question of interest. Use\(\alpha = 0.05\).

b. Is the sample size large enough to use the inferential procedure of part a?

c. Find the observed significance level of the hypothesis test in part a. Interpret the value.

d. For what values \(\alpha \) would the observed significance level be sufficient to reject the null hypothesis of the test you conducted in part b?
See all solutions

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