Chapter 21, Problem 31E

(a)

To determine

To find out what is the alternative hypothesis.

Answer to Problem 31E

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

Explanation of Solution

It is given in the question that in a drawer are two coins and they look the same but one coin produces heads $90\%$ of the time when spun while the other one produces heads only $30\%$ of the time. Thus, we can say that the alternative hypothesis states that the null hypothesis is false, thus the alternative hypothesis states that the coin is not a $90\%$ head coin and then the coin has to be the $30\%$ head coin. So, we have,

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

(b)

To determine

To explain what would you decide given that the outcome of your spin is tail and what if it were heads.

Answer to Problem 31E

For tails: We reject the null hypothesis.

For heads: We fail to reject the null hypothesis.

Explanation of Solution

It is given in the question that in a drawer are two coins and they look the same but one coin produces heads $90\%$ of the time when spun while the other one produces heads only $30\%$ of the time. So, we have,

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

The $90\%$ head coin is more likely to flip heads than the $30\%$ head coin. The $30\%$ head coin is more likely to flip tails than the $90\%$ head coin. If we flip tails, then it is more likely that the tails originates from the $30\%$ head coin and thus we would reject the null hypothesis. If we flip heads then it is more likely that the heads originates from the $90\%$ head coin and thus we would fail to reject the null hypothesis.

(c)

To determine

To find out how large is $\alpha$ in this case.

Answer to Problem 31E

  $0.10$ .

Explanation of Solution

It is given in the question that in a drawer are two coins and they look the same but one coin produces heads $90\%$ of the time when spun while the other one produces heads only $30\%$ of the time. So, we have,

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

As we know that $\alpha$ is the probability of a Type I error that is, rejecting the null hypothesis when the null hypothesis is true. So, we have,

  $\begin{array}{l}\alpha =P(\text{Tails | }p=0.90)\\ =1-P(\text{Heads | }p=0.90)\\ =1-0.90\\ =0.10\end{array}$

(d)

To determine

To explain how large is the power of the test.

Answer to Problem 31E

  $0.70$ .

Explanation of Solution

It is given in the question that in a drawer are two coins and they look the same but one coin produces heads $90\%$ of the time when spun while the other one produces heads only $30\%$ of the time. So, we have,

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

As we know that the power is the probability of rejecting the null hypothesis when the null hypothesis is false. So, we have,

  $\begin{array}{l}\text{Power}=P(\text{Tails | }p=0.30)\\ =1-P(\text{Heads | }p=0.30)\\ =1-0.30\\ =0.70\end{array}$

(e)

To determine

To explain how could you lower the probability of a Type I error ad increase the power of the test at the same time.

Explanation of Solution

It is given in the question that in a drawer are two coins and they look the same but one coin produces heads $90\%$ of the time when spun while the other one produces heads only $30\%$ of the time. So, we have,

  $\begin{array}{l}{H}_0:p=0.90\\ H_a:p=0.30\end{array}$

Thus, we could lower the probability of a Type I error ad increase the power of the test at the same time as the probability of a type I error will decrease and the power will increase, if we spin the coin more than once and then base our decision (rejecting or failing to reject the null hypothesis) on the proportion of heads.