Suppose that in a random selection of 100 colored candies, 25% of them are blue. The candy company claims that the percentage of blue candies is equal to 20%. Use a 0.10 significance level to test that claim.

1. Identify the null and alternative hypotheses for this test. Choose the correct answer below:
A. [tex]\(H_0: p = 0.2\)[/tex] [tex]\(H_1: p \ \textgreater \ 0.2\)[/tex]
B. [tex]\(H_0: p = 0.2\)[/tex] [tex]\(H_1: p \neq 0.2\)[/tex]
C. [tex]\(H_0: p \neq 0.2\)[/tex] [tex]\(H_1: p = 0.2\)[/tex]
D. [tex]\(H_0: p = 0.2\)[/tex] [tex]\(H_1: p \ \textless \ 0.2\)[/tex]

2. Identify the test statistic for this hypothesis test.
The test statistic for this hypothesis test is [tex]\(\square\)[/tex].
(Round to two decimal places as needed.)



Answer :

To address the question, we first need to set up our null and alternative hypotheses and then determine the test statistic.

### Hypotheses
The candy company claims that the proportion of blue candies is 20%. We want to test if our observed sample proportion of 25% deviates significantly from this claimed proportion using a significance level of 0.10.

- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of blue candies is 20%.
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The proportion of blue candies is not equal to 20%.

So, we have:
[tex]\[ H_0: p = 0.2 \][/tex]
[tex]\[ H_1: p \neq 0.2 \][/tex]

This describes a two-tailed test because we are checking for any significant difference from 20%, whether it be higher or lower.

Given this, the correct answer for the hypotheses:
Answer B:
[tex]\[ H_0: p = 0.2 \][/tex]
[tex]\[ H_1: p \neq 0.2 \][/tex]

### Test Statistic
The test statistic for a proportion is calculated using the formula:
[tex]\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the observed sample proportion (0.25 in this case),
- [tex]\(p_0\)[/tex] is the claimed proportion in the null hypothesis (0.20),
- [tex]\(n\)[/tex] is the sample size (100).

Given the values:
- [tex]\(\hat{p} = 0.25\)[/tex]
- [tex]\(p_0 = 0.20\)[/tex]
- [tex]\(n = 100\)[/tex]

First, we calculate the standard error (SE):
[tex]\[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.20 \times (1 - 0.20)}{100}} \][/tex]

Next, we calculate the test statistic ([tex]\(z\)[/tex]):
[tex]\[ z = \frac{0.25 - 0.20}{SE} \][/tex]

From our previous calculation, the test statistic is found to be:
[tex]\[ z = 1.25 \][/tex]

So, we fill in the blank with:
The test statistic for this hypothesis test is 1.25 (rounded to two decimal places as needed).