A company believes that it controls more than 30% of the total market share for one of its products. To prove this belief, a random sample of 144 purchases of this product is contacted. It is found that 50 of the 144 purchased this company's brand of the product. If a researcher wants to conduct a statistical test for this problem, the test statistic would be ___.
A) 0.05
B) 0.103
C) 0.35
D) 1.24
E) 1.67

Question

Quizplus · Accepted Answer

To test if the company controls more than 30% of the total market share, we can use a one-tailed hypothesis test with the following null and alternative hypotheses:
H0: p <= 0.30
Ha: p > 0.30
where p is the population proportion of customers who purchased the company's brand.
The test statistic for this problem is given by:
z = (phat - p) / sqrt(p*(1-p)/n)
where phat is the sample proportion (phat = 50/144), n is the sample size (n = 144), and p = 0.30.
Substituting the values, we get:
z = (0.3472 - 0.30) / sqrt(0.30*0.70/144) = 1.24 (rounded to two decimal places)
Since the alternative hypothesis is that p is greater than 0.30, we look up the critical value from the standard normal table for a one-tailed test at 0.05 significance level (z = 1.645). Since the calculated z-value (1.24) is less than the critical value (1.645), we fail to reject the null hypothesis. This means that there is not enough evidence to support the claim that the company controls more than 30% of the total market share.

A Company Believes That It Controls More Than 30% of the Total