Question

-Question:Consumers’ 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 α = 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

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

Given Information

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

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 = 0.65AgainstH_a: p > 0.65

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%.

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.

Computing the value of the observed significance level

(c)

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