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).
