Questions: Suppose that in a random selection of 100 colored candies, 22% of them are blue. The candy company claims that the percentage of blue candies is equal to 20%. Use a 0.05 significance level to test that claim. Identify the null and alternative hypotheses for this test. Choose the correct answer below. A. H0: p ≠ 0.2 H1: p=0.2 B. H0: p=0.2 H1: p>0.2 C. H0: P=0.2 H1: p ≠ 0.2 D. H0: P=0.2 H1: p<0.2

Solution Steps

We are testing the claim made by the candy company regarding the proportion of blue candies. The null and alternative hypotheses are defined as follows:

\[ H_0: p = 0.2 \] \[ H_1: p \neq 0.2 \]

The test statistic \(Z\) is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Where:

• \(\hat{p} = 0.22\) (sample proportion)
• \(p_0 = 0.2\) (hypothesized population proportion)
• \(n = 100\) (sample size)

Substituting the values, we find:

\[ Z = \frac{0.22 - 0.2}{\sqrt{\frac{0.2(1 - 0.2)}{100}}} = 0.5 \]

The P-value associated with the test statistic \(Z = 0.5\) is calculated to be:

\[ \text{P-value} = 0.6171 \]

For a significance level of \(\alpha = 0.05\) in a two-tailed test, the critical region is defined as:

\[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

We compare the P-value with the significance level:

• Since \(0.6171 > 0.05\), we fail to reject the null hypothesis \(H_0\).

Final Answer