Question

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?

Short answer

The null and the alternative hypotheses are\({H_0}:P = 0.50\)and\({H_a}:P > 0.50\)

Step-by-step solution

Given information

The information is regarding the National Association of Shoplifting Prevention (NASP) U.S retailers.

Random samples of 40 U.S retailers were questioned concerning the deposition of the most recent shoplifter they apprehended.

A total of 24 were turned over to police. Here NASP claims that only 50% of all shoplifting are turned over to police.

From the above information,

\(\begin{aligned}{p_0} &=0.50\\{q_0} &= 1 - {p_0}\\ &= 0.50\end{aligned}\)

Concept of the null and the alternative hypothesis

The null hypothesis of a test always predicts no effect or no association between variables, while the alternative hypothesis states your research's prediction of an effect or relationship.

Setting up the null and the alternative hypothesis

a.

From this information,

Null hypothesis:

\({H_0}:P = 0.50\)

Only 50% of U.S. retailers are shoplifting is turned over to police

Alternative hypothesis:

\({H_a}:P > 0.50\)

Not only 50% of U.S. retailers are shoplifting is turned over to police.