Question

M&M'S MILK CHOCOLATE: 24% cyan blue, 20% orange, 16% green, 14% bright yellow, 13% red, 13% brown.

Each large production batch is blended to those ratios and mixed thoroughly. However, since the individual packages are filled by weight on high-speed equipment, and not by count, it is possible to have an unusual color distribution.

Part 1: Confidence Interval for Small n

Milk Chocolate M&M’s come in 6 colors; blue, orange, green, yellow, red, and brown.

Choose the color of M&M’s you will be working with for this project

Using the collected data below from a single fun-sized bag, provide the frequency and proportion of M&M’s in your color of choice.

Red	Orange	Yellow	Green	Blue	Brown
2	1	2	3	5	1

Number of M&M's in your color:

Total number of M&M's:

Proportion of M&M's in your color:

Construct a 95% confidence interval for the proportion of M&M’s one can expect to find in the color of your choice.

Check the requirements for constructing a confidence interval for the proportion are satisfied. Show your work.

The conditions might not be satisfied, depending on how many candies were in your bag. If the conditions are not met, what could you do?

Part 2: Confidence Interval for Larger n

Now, use the data collected below from a collection of fun-sized bags to provide the frequency and proportion of M&M’s in your original color of choice.

Red	Orange	Yellow	Green	Blue	Brown
54	49	52	51	84	109

Number of M&M's in your color:

Total number of M&M's:

Proportion of M&M's in your color:

Construct a 95% confidence interval for the proportion of M&M’s one can expect to find in the color of your choice.

Write an interpretation of your confidence interval specific to your color.

Check the requirements for constructing a confidence interval for the proportion are satisfied. Show your work.

Part 3: Answer Some Questions

How would are confidence intervals affected by sample size?

How does the margin of error for the confidence interval in Part 1 compare to the margin of error for your confidence interval in Part 2? (compute both and compare)

Does the confidence interval you constructed in Part 2 contain the claimed proportion given by Mars Inc?

Do you believe the claims given by Mars Inc?

Answer 1

Choose RED. The population proportion of RED is

i) Here in the sample RED occurs 2 times out of 14 times. The sample proportion of RED is

The sample size is only . Hence normality assumption is not possible. Use T-distribution.

We have to test

The two sided CI is .  When
The 95% CI is

Since the 95% CI includes 0.13 we conclude that the sample proportion is not different from the hypothesized proportion 0.13. Accept null hypothesis.

ii) When the sample size is large , we can use normal distribution.

Here in the sample RED occurs 54 times out of 399 times. The sample proportion of RED is

The two sided CI is .  When
The 95% CI is

Since the 95% CI includes 0.13 we conclude that the sample proportion is not different from the hypothesized proportion 0.13 . Accept null hypothesis.

iii) The width of the confidence interval reduces as the sample size increases. The margin of error is lesser for larger sample size.

The claims are right.