Question

1. The data given below includes data from 42 ​candies, and 7 of them are red. The company that makes the candy claims that 33​% of its candies are red. Use the sample data to construct a 95​% confidence interval estimate of the percentage of red candies. What do you conclude about the claim of 33​%?
2. Construct a 95 % confidence interval estimate of the population percentage of candies that are red.
Weights (g) of a Sample Bag of Candy
Red Blue Brown Green Yellow
0.792 0.982 0.765 0.738 0.837
0.942 0.769 0.745 0.823 0.719
0.906 0.997 0.897 0.729 0.796
0.941 0.877 0.816 0.866 0.712
0.959 0.788 0.902 0.792 0.704
0.976 0.965 0.956 0.983
0.809 0.999 0.846
0.849 0.986 0.834
0.955 0.845
0.736
0.721

Answer 1

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:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}$

The information provided is:

n = 42

X = number of red candies = 7

Compute the sample proportion of candies that are red as follows:

$\hat p=\frac{X}{n}=\frac{7}{42}=0.167$

The critical value of z for 95% confidence level is:

$z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96$

*Use a z-table for the critical value.

Compute the 95% confidence interval for true proportion of red candies as follows:

$CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\=0.167\pm 1.96\times \sqrt{\frac{0.167(1-0.167)}{42}}\\=0.16\pm0.1137\\=(0.0463, 0.2737)\\\approx(0.05, 0.27)$

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%.