Question

Does Mars, Incorporated use the same proportion of red candies in its plain and peanut varieties? A random sample of 54 plain M&M'S contained 14 red candies, and another random sample of 40peanut M&M'S contained 6 red candies. (Use p₁ for the proportion of red candies in plain M&M'S and p₂ for the proportion of red candies in peanut M&M'S.)

(a) Construct a 95% confidence interval for the difference in the proportions of red candies for the plain and peanut varieties (p₁ − p₂). (Round your answers to three decimal places.)

______ to _______

(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.

A) Since the value p₁ − p₂ = 0 is in the confidence interval, it is possible that p₁ = p₂. We should not conclude that there is a difference in the proportion of red candies in plain and peanut M&M'S.

B) Since the value p₁ − p₂ = 0 is in the confidence interval, it is possible that p₁ = p₂. We should conclude that there is a difference in the proportion of red candies in plain and peanut M&M'S.

C) Since the value p₁ − p₂ = 0 is not in the confidence interval, it is possible that p₁ = p₂. We should conclude that there is a difference in the proportion of red candies in plain and peanut M&M'S.

D) Since the value p₁ − p₂ = 0 is not in the confidence interval, it is possible that p₁ = p₂. We should not conclude that there is a difference in the proportion of red candies in plain and peanut M&M'S.

Answer 1

Now for sample 1,

p1 = 14 / 54 = 0.2592

For sample 2,

p2 = 6 / 40 = 0.15

a.)

For 95% confidence interval, z = 1.96

Now let's go step by step:

(p1 * (1-p1)) / n1 = (0.2592 * 0.7408) / 54 = 0.00355

Similarly,

(p2 * (1-p2)) / n2 = (0.15 * 0.85) / 40 = 0.00318

So, Standard Error,

SE =

SE = 0.082

Lower Limit = (p1 - p2) - Z * SE

Lower Limit = (0.2592 - 0.15) - 1.96 * 0.082 = 0.1092 - 0.1607 = -0.0515

Upper Limit = (p1 - p2) + Z * SE

Lower Limit = (0.2592 - 0.15) + 1.96 * 0.082 = 0.1092 + 0.1607 = 0.2699 = 0.27

Hence Confidence Interval is -0.0515 to 0.27

b.)

If confidence interval contains 0, this means that there is no statistically significant between two groups.

Hence Option A is correct

Since the value p1 − p2 = 0 is in the confidence interval, it is possible that p1 = p2.

We should not conclude that there is a difference in the proportion of red candies in plain and peanut M&M'S