Chapter 18, Problem 5E

(a)

To determine

To explain what shape would you expect this histogram to be and why.

Answer to Problem 5E

The histogram will be symmetric.

Explanation of Solution

It is given in the question, the professor in the class has each person toss a coin $16$ times and calculate the proportion of his or her tosses that were heads. And the results of the students are plotted by a histogram by the professor of these several proportions. Since we know that the probabilities of the toss to be head and to be tail is equal in proportions as half. So, the histogram of these several proportions will be symmetric and in bell shaped as the probability of coming head and tail is equal.

(b)

To determine

To explain where do you expect the histogram to be centered.

Answer to Problem 5E

The histogram will be then centered at $0.5$ .

Explanation of Solution

It is given in the question, the professor in the class has each person toss a coin $16$ times and calculate the proportion of his or her tosses that were heads. And the results of the students are plotted by a histogram by the professor of these several proportions. Since we know that the probabilities of the toss to be head and to be tail is equal in proportions as half. So, the histogram of these several proportions will be symmetric. Thus, if the histogram will be symmetric then the data will follow the normal conditions and thus, the histogram will be then centered at $0.5$ .

(c)

To determine

To explain how much variability would you expect among these proportions.

Answer to Problem 5E

The variability we will expect among these proportions to be $0.125$ .

Explanation of Solution

It is given in the question, the professor in the class has each person toss a coin $16$ times and calculate the proportion of his or her tosses that were heads. And the results of the students are plotted by a histogram by the professor of these several proportions. Since we know that the probabilities of the toss to be head and to be tail is equal in proportions as half. So, the histogram of these several proportions will be symmetric. Thus, if the histogram will be symmetric then the data will follow the normal conditions and thus, the variability we will expect among these proportions to be $0.125$ .

(d)

To determine

To explain why a normal model should not be used here.

Explanation of Solution

It is given in the question, the professor in the class has each person toss a coin $16$ times and calculate the proportion of his or her tosses that were heads. And the results of the students are plotted by a histogram by the professor of these several proportions. Since we know that the probabilities of the toss to be head and to be tail is equal in proportions as half. So, the histogram of these several proportions will be symmetric. Thus, if the histogram will be symmetric then the data will follow the normal conditions and thus the normal model should not be used here because

  $\begin{array}{l}np=16\times 0.5\\ =8<10\end{array}$

Thus, the condition of the normal condition to be at least ten is not satisfied in this case.