Question

suppose that in a random selection of 100 colored candies, 30% of them are blue. the candy company claims that the percentage of blue candies is equal to 24%. use a 0.01 significance level to test that claim. (round to three decimal places as needed.) identify the conclusion for this hypothesis test. a. fail to reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that the percentage of blue candies is equal to 24% b. reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that the percentage of blue candies is equal to 24% c. reject ( h_0 ). there is sufficient evidence to warrant rejection of the claim that the percentage of blue candies is equal to 24% d. fail to reject ( h_0 ). there is not sufficient evidence to warrant rejection of the claim that the percentage of blue candies is equal to 24%

Explanation:

Step1: State the hypotheses

Let \(p\) be the proportion of blue candies.
\(H_0:p = 0.24\) (null hypothesis, the company's claim)
\(H_1:p
eq0.24\) (alternative hypothesis)

Step2: Calculate the test statistic

The sample proportion \(\hat{p}=0.30\), \(n = 100\)
The formula for the test statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute the values: \(z=\frac{0.30 - 0.24}{\sqrt{\frac{0.24\times(1 - 0.24)}{100}}}=\frac{0.06}{\sqrt{\frac{0.24\times0.76}{100}}}=\frac{0.06}{\sqrt{0.001824}}\approx1.41\)

Step3: Find the critical values

For a two - tailed test with \(\alpha = 0.01\), the critical values are \(z=\pm2.58\)

Step4: Make a decision

Since \(|z| = 1.41<2.58\), we fail to reject \(H_0\)

Answer:

D. Fail to reject \(H_0\). There is not sufficient evidence to warrant rejection of the claim that the percentage of blue candies is equal to \(24\%\)