Chapter 18, Problem 6E

(a)

To determine

To explain what shape would you expect to have if we plot a histogram showing the proportions of green candies in the various bags.

Answer to Problem 6E

The histogram of the proportion of green candies in the bags would be skewed slightly to the right.

Explanation of Solution

In the question, the candy company claims that $10\%$ of the M&M’s it produces are green. Suppose that the candies are packed at random in small bags. Then the class of elementary school opens the bags and count the proportion that are green candies. Thus, the histogram of the proportion of green candies in the bags would be skewed slightly to the right since the proportion of green M&Ms would never fall below zero on the left but the potential to be higher on the right.

(b)

To determine

To explain can that histogram be approximated by a normal model.

Answer to Problem 6E

The histogram cannot be approximated by a normal model.

Explanation of Solution

In the question, the candy company claims that $10\%$ of the M&M’s it produces are green. Suppose that the candies are packed at random in small bags. Then the class of elementary school opens the bags and count the proportion that are green candies. Thus, the histogram of the proportion of green candies in the bags would be skewed slightly to the right. Thus, the normal model cannot be used to approximate the histogram since the expected number of green M&Ms is:

  $\begin{array}{l}np=50(0.10)\\ =5<10\end{array}$

Therefore, the success or failure condition is not met in this case.

(c)

To determine

To find out where should the center of the histogram be.

Answer to Problem 6E

The histogram should be centered around $0.10$ .

Explanation of Solution

In the question, the candy company claims that $10\%$ of the M&M’s it produces are green. Suppose that the candies are packed at random in small bags. Then the class of elementary school opens the bags and count the proportion that are green candies. Thus, the histogram of the proportion of green candies in the bags would be skewed slightly to the right. Thus, the histogram be centered around the expected proportion of green M&Ms at around $0.10$ .

(d)

To determine

To find out what should the standard deviation of the proportion be.

Answer to Problem 6E

The standard deviation of the proportion is $0.042$ .

Explanation of Solution

In the question, the candy company claims that $10\%$ of the M&M’s it produces are green. Suppose that the candies are packed at random in small bags. Then the class of elementary school opens the bags and count the proportion that are green candies. Thus, the histogram of the proportion of green candies in the bags would be skewed slightly to the right. Thus, the standard deviation of the proportion be as:

  $\begin{array}{l}S.D.=\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.1(1-0.1)}{50}}\\ =0.042\end{array}$