Question

The data given to the right includes data from 32 ​candies, and 6 of them are red. The company that makes the candy claims that ​31% of its candies are red. Use the sample data to construct a ​90% confidence interval estimate of the percentage of red candies. What do you conclude about the claim of ​31%?

Red Blue Brown Green Yellow 0.959 0.759 0.901 0.829 0.882 0.989 0.913 0.918 0.767 0.777 0.857 0.904 0.734 0.838 0.984 0.911 0.961 0.882 0.824 0.848 0.737 0.803 0.864 0.874 0.994 0.752 0.911 0.915 0.918 0.882 0.936 0.833 0.946 0.774 0.802 0.765 0.927 0.836 0.895 0.849

Answer 1

To construct a confidence interval for the percentage of red candies, we can use the sample proportion (the proportion of red candies in the sample) and apply the formula for a confidence interval for a population proportion. The formula is:

Confidence Interval= p^ ±Z √p^(1− p^ ) ÷ n

​where:

p^ is the sample proportion (proportion of red candies in the sample),

Z is the z-score corresponding to the desired confidence level,

n is the sample size.

Given that 6 out of 32 candies are red, the sample proportion p^ is 6/32=0.1875.

For a 90% confidence interval, the critical z-value is approximately 1.645.

Let's calculate the confidence interval:

Confidence Interval= 0.1875±1.645 √ 0.1875(1−0.1875) ÷ 32

​Calculating this will give you the interval estimate. Please note that this is a one-sample z-interval for a population proportion.

Confidence Interval≈(0.066,0.309)

This means that we are 90% confident that the true percentage of red candies is between 6.6% and 30.9%.

Now, comparing this interval with the company's claim of 31%, we can see that the entire 90% confidence interval does not include 31%. Therefore, based on this sample data, there is evidence to suggest that the true percentage of red candies may be different from the company's claim of 31%.

Explanation: