Respuesta :
Answer:
The 95% confidence interval for true proportion of red candies is (5%, 27%).
The true proportion of red candies made by the company is different from 33%.
Step-by-step explanation:
The hypothesis to determine whether the claim made by the candy company is correct or not is:
H₀: The true proportion of red candies made by the company is 33%, i.e. p = 0.33.
Hₐ: The true proportion of red candies made by the company is different from 33%, i.e. p ≠ 0.33.
A (1 - α)% confidence interval can be constructed to check this claim.
The decision rule is:
If the confidence interval consists the null value then the null hypothesis will not be rejected. Otherwise it will be rejected.
The (1 - α)% confidence interval for population proportion is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The information provided is:
n = 42
X = number of red candies = 7
Compute the sample proportion of candies that are red as follows:
[tex]\hat p=\frac{X}{n}=\frac{7}{42}=0.167[/tex]
The critical value of z for 95% confidence level is:
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use a z-table for the critical value.
Compute the 95% confidence interval for true proportion of red candies as follows:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\=0.167\pm 1.96\times \sqrt{\frac{0.167(1-0.167)}{42}}\\=0.16\pm0.1137\\=(0.0463, 0.2737)\\\approx(0.05, 0.27)[/tex]
The 95% confidence interval for true proportion of red candies is (5%, 27%).
The confidence interval does not contains the null value.
Thus, the null hypothesis will be rejected at 5% level of significance.
Conclusion:
The true proportion of red candies made by the company is different from 33%.
