Question

1. in a random sample of 50 creamery customers, 30 preferred chocolate over vanilla. is there evidence at 0.01 level of significance that the percentage of creamery customers who prefer chocolate ice cream over vanilla less than 80%?

Explanation:

Step1: State the hypotheses

Let \(p\) be the proportion of customers who prefer chocolate over vanilla.
\(H_0:p = 0.8\) (null hypothesis)
\(H_1:p<0.8\) (alternative hypothesis)

Step2: Calculate the sample proportion

\(\hat{p}=\frac{30}{50}=0.6\)

Step3: Calculate the test - statistic

The formula for the test - statistic \(z\) in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.8\), \(\hat{p}=0.6\), \(n = 50\)

$$
LATEXBLOCK0
$$

Step4: Find the critical value

For a one - tailed test with \(\alpha=0.01\), the critical value \(z_{\alpha}\) is \(z_{0.01}=- 2.33\) (from standard normal distribution table)

Step5: Make a decision

Since the test - statistic \(z=-3.53<-2.33\) (critical value), we reject the null hypothesis.

Answer:

Yes, there is evidence at the \(0.01\) level of significance that the percentage of Creamery customers who prefer chocolate ice cream over vanilla is less than \(80\%\)