Question

Does Mars, Incorporated use the same proportion of red candies in its plain and peanut varieties? A random sample of 56 plain M\&M'S contained 12 red candies, and another random sample of 32 peanut M\&M'S contained 8 red candies. a. Construct a \(95 \%\) confidence interval for the difference in the proportions of red candies for the plain and peanut varieties. b. Based on the confidence interval in part a, can you conclude that there is a difference in the proportions of red candies for the plain and peanut varieties? Explain.

Short answer

Short Answer: The 95% confidence interval for the difference in proportions of red candies in plain and peanut M&M's is calculated using the sample proportions, the standard error, and a critical value (z = 1.96). By analyzing the confidence interval, we can determine if there is a significant difference in the proportions of red candies for the two varieties.

Step-by-step solution

Calculate sample proportions

Calculate the sample proportions for the plain M&M's (d₁) and the peanut M&M's (d₂) by dividing the number of red candies by the total number of candies in each sample: \(d₁ = \frac{12}{56}\), \(d₂ = \frac{8}{32}\)

Calculate difference in proportions

Calculate the difference of the sample proportions: \(\overset{\ \ }{p} = d₁ - d₂ = \frac{12}{56} - \frac{8}{32}\)

Calculate the standard error of the difference in proportions

Calculate the standard error of the difference in proportions using the formula: \(\overset{\ \ }{SE}(d₁ - d₂) = \sqrt{\frac{d₁(1-d₁)}{n₁} + \frac{d₂(1-d₂)}{n₂}}\) Plug in the values: \(\overset{\ \ }{SE}(\overset{\ \ }{p})= \sqrt{\frac{\left( \frac{12}{56}\right) \left( 1 - \frac{12}{56}\right)}{56} + \frac{\left( \frac{8}{32}\right) \left( 1 - \frac{8}{32}\right)}{32}}\)

Construct the 95% confidence interval

Use the standard error and a critical value (z = 1.96 for a 95% confidence interval) to construct the confidence interval: \(\overset{\ \ }{p} \pm z * \overset{\ \ }{SE}(\overset{\ \ }{p})\) Plug in the values: \(\left( \frac{12}{56} - \frac{8}{32} \right) \pm 1.96 * \sqrt{\frac{\left( \frac{12}{56}\right) \left( 1 - \frac{12}{56}\right)}{56} + \frac{\left( \frac{8}{32}\right) \left( 1 - \frac{8}{32}\right)}{32}}\)

Draw conclusions based on the confidence interval

Calculate the confidence interval and observe if it includes zero. If it does, we cannot conclude that there's a significant difference in the proportions of red candies for the plain and peanut varieties. If it doesn't include zero, we can conclude that there's a significant difference. Based on the calculated confidence interval, we can determine whether there is a difference in the proportions of red candies for the plain and peanut varieties.