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Chapter 7: Q 128SE (page 443)

-Question:Consumers鈥� use of discount coupons. In 1894, druggist Asa Candler began distributing handwritten tickets to his customers for free glasses of Coca-Cola at his soda fountain. That was the genesis of the discount coupon. In 1975, it was estimated that 65% of U.S. consumers regularly used discount coupons when shopping. In a more recent consumer survey, 81% said they regularly redeem coupons (NCH Marketing Services 2015 Consumer Survey). Assume the recent survey consisted of a random sample of 1,000 shoppers.

a. Does the survey provide sufficient evidence that the percentage of shoppers using cents-off coupons exceeds 65%? Test using 伪 = 0.05.

b. Is the sample size large enough to use the inferential procedures presented in this section? Explain.

c. Find the observed significance level for the test you conducted in part a and interpret its value.

Short Answer

Expert verified
  1. Since the test statistic exceeds the critical value, reject the null hypothesis.
  2. Therefore, there is enough evidence that the percentage of shoppers using cents-off coupons exceeds 65%. The sample size is large enough to use the inferential procedures.
  3. The observed significance level is 0.00. It is the probability of getting the test statistic more than the observed value.

Step by step solution

01

Given Information

A random sample of size n = 1000 shoppers is surveyed. The sample proportion of shoppers that uses cents-off coupons is .

02

Define the claim

The researcher wants to test the claim that the percentage of shoppers using cents-off coupons exceeds 65%.

The null and alternative hypotheses are:

H0 : p = 0.65AgainstHa: p > 0.65

03

Computing the test statistic

a.

The test statistic is:

The test statistic is z = 10.596.

The z-critical value at a 5% significance level for the right-tailed test using the z-table is 1.65.

Since the test statistic exceeds the critical value, reject the null hypothesis.

Therefore, there is enough evidence that the percentage of shoppers using cents-off coupons exceeds 65%.

04

Validating the condition

(b)

The minimum sample size required to use the z-test for the significance of the population proportion is obtained by the following conditions:

and.

Here,

and

Since both conditions are satisfied, the sample size is large enough to use the inferential procedures.

05

Computing the value of the observed significance level

(c)

Referring to part a., the test statistic is z = 10.596. The significance level is 伪 = 0.05.

The observed significance level or the p-value for the right-tailed test is obtained as follows:

From the z-table, the p-value is approximately zero.

Therefore, the observed significance level is 0.00. It is the probability of getting the test statistic more than the observed value.

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Most popular questions from this chapter

a.Consider testing H0: m=80. Under what conditions should you use the t-distribution to conduct the test?

b.In what ways are the distributions of the z-statistic and t-test statistic alike? How do they differ?

Authorizing computer users with palm prints. Access to computers, email, and Facebook accounts is achieved via a password鈥攁 collection of symbols (usually letters and numbers) selected by the user. One problem with passwords is that persistent hackers can create programs that enter millions of combinations of symbols into a target system until the correct password is found. An article in IEEE Pervasive Computing (October-December 2007) investigated the effectiveness of using palm prints to identify authorized users. For example, a system developed by Palmguard, Inc. tests the hypothesis

\({H_0}\): The proposed user is authorized

\({H_a}\): The proposed user is unauthorized

by checking characteristics of the proposed user鈥檚 palm print against those stored in the authorized users鈥� data bank.

a. Define a Type I error and Type II error for this test. Which is the more serious error? Why?

Intrusion detection systems. The Journal of Research of the National Institute of Standards and Technology (November鈥� December 2003) published a study of a computer intrusion detection system (IDS). The IDS is designed to provide an alarm whenever unauthorized access (e.g., an intrusion) to a computer system occurs. The probability of the system giving a false alarm (i.e., providing a warning when no intrusion occurs) is defined by the symbol 伪, while the probability of a missed detection (i.e., no warning given when an intrusion occurs) is defined by the symbol 尾. These symbols are used to represent Type I and Type II error rates, respectively, in a hypothesis-testing scenario

a. What is the null hypothesis, H0?

b. What is the alternative hypothesis,Ha?

c. According to actual data collected by the Massachusetts Institute of Technology Lincoln Laboratory, only 1 in 1,000 computer sessions with no intrusions resulted in a false alarm. For the same system, the laboratory found that only 500 of 1,000 intrusions were actually detected. Use this information to estimate the values of 伪and 尾.

Packaging of a children鈥檚 health food. Can packaging of a healthy food product influence children鈥檚 desire to consume the product? This was the question of interest in an article published in the Journal of Consumer Behaviour (Vol. 10, 2011). A fictitious brand of a healthy food product鈥攕liced apples鈥攚as packaged to appeal to children (a smiling cartoon apple was on the front of the package). The researchers showed the packaging to a sample of 408 school children and asked each whether he or she was willing to eat the product. Willingness to eat was measured on a 5-point scale, with 1 = 鈥渘ot willing at all鈥� and 5 = 鈥渧ery willing.鈥� The data are summarized as follows: \(\bar x = 3.69\) , s = 2.44. Suppose the researchers knew that the mean willingness to eat an actual brand of sliced apples (which is not packaged for children) is \(\mu = 3\).

a. Conduct a test to determine whether the true mean willingness to eat the brand of sliced apples packaged for children exceeded 3. Use\(\alpha = 0.05\)

to make your conclusion.

b. The data (willingness to eat values) are not normally distributed. How does this impact (if at all) the validity of your conclusion in part a? Explain.

Customers who participate in a store鈥檚 free loyalty card program save money on their purchases but allow the store to keep track of the customer鈥檚 shopping habits and potentially sell these data to third parties. A Pew Internet & American Life Project Survey (January 2016) revealed that 225 of a random sample of 250 U.S. adults would agree to participate in a store loyalty card program, despite the potential for information sharing. Letp represent the true proportion of all customers who would participate in a store loyalty card program.

a. Compute a point estimate ofp

b. Consider a store owner who claims that more than 80% of all customers would participate in a loyalty card program. Set up the null and alternative hypotheses for testing whether the true proportion of all customers who would participate in a store loyalty card program exceeds .8

c. Compute the test statistic for part b.

d. Find the rejection region for the test if 伪=.01.

e. Find the p-value for the test.

f. Make the appropriate conclusion using the rejection region.

g. Make the appropriate conclusion using the p-value.
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