Chapter 8, Problem 19CAP

(1)

To determine

Identify the decision for each value of p if $\alpha =0.05$

1. a) $p=0.0250$
2. b) $p=0.1000$
3. c) $p=0.005$
4. d) $p=0.0001$

Answer to Problem 19CAP

1. a) The decision for $p=0.0250$ if $\alpha =0.05$ is rejecting the null hypothesis.
2. b) The decision for $p=0.1000$ if $\alpha =0.05$ is retaining the null hypothesis.
3. c) The decision for $p=0.005$ if $\alpha =0.05$ is rejecting the null hypothesis.
4. d) The decision for $p=0.0001$ if $\alpha =0.05$ is rejecting the null hypothesis.

Explanation of Solution

Decision Rule:

If the value of p is less than $\alpha$, then reject the null hypothesis.

a)

The data gives the value of p is 0.0250 and $\alpha =0.05$.

The p-value is less than the value of $\alpha$. That is, $p\left(=0.0250\right)<\alpha \left(=0.05\right)$. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $p=0.0250$ if $\alpha =0.05$ is rejecting the null hypothesis.

b)

The data gives the value of p is 0.1000 and $\alpha =0.05$.

The p-value is greater than the value of $\alpha$. That is, $p\left(=0.1000\right)<\alpha \left(=0.05\right)$. From the decision rule, fail to reject the null hypothesis.

Hence, the decision for $p=0.1000$ if $\alpha =0.05$ is retaining the null hypothesis.

c)

The data gives the value of p is 0.005 and $\alpha =0.05$.

The p-value is lesser than the value of $\alpha$. That is, $p\left(=0.005\right)<\alpha \left(=0.05\right)$. From the decision rule, reject the null hypothesis.

Hence, the decision for $p=0.005$ if $\alpha =0.05$ is rejecting the null hypothesis.

d)

The data gives the value of p is 0.0001 and $\alpha =0.05$.

The p-value is lesser than the value of $\alpha$. That is, $p\left(=0.0001\right)<\alpha \left(=0.05\right)$. From the decision rule, reject the null hypothesis.

Hence, the decision for $p=0.0001$ if $\alpha =0.05$ is rejecting the null hypothesis.

(2)

To determine

Identify the decision for each value of p if $\alpha =0.01$

1. a) $p=0.0250$
2. b) $p=0.1000$
3. c) $p=0.0050$
4. d) $p=0.0001$

Answer to Problem 19CAP

1. a) The decision for $p=0.0250$ if $\alpha =0.01$ is retaining the null hypothesis.
2. b) The decision for $p=0.1000$ if $\alpha =0.01$ is retaining the null hypothesis.
3. c) The decision for $p=0.0050$ if $\alpha =0.01$ is rejecting the null hypothesis.
4. d) The decision for $p=0.0001$ if $\alpha =0.01$ is rejecting the null hypothesis.

Explanation of Solution

a)

The data gives the value of p is 0.0250 and $\alpha =0.01$.

The p-value is greater than the value of $\alpha$. That is, $p\left(=0.0250\right)>\alpha \left(=0.01\right)$. From the decision rule, fail to reject the null hypothesis.

Hence, the decision for $p=0.0250$ if $\alpha =0.01$ is retaining the null hypothesis.

b)

The data gives the value of p is 0.1000 and $\alpha =0.01$.

The p-value is greater than the value of $\alpha$. That is, $p\left(=0.1000\right)>\alpha \left(=0.01\right)$. From the decision rule, fail to reject the null hypothesis.

Hence, the decision for $p=0.1000$ if $\alpha =0.01$ is retaining the null hypothesis.

c)

The data gives the value of p is 0.0050 and $\alpha =0.01$.

The p-value is lesser than the value of $\alpha$. That is, $p\left(=0.0050\right)<\alpha \left(=0.01\right)$. From the decision rule, reject the null hypothesis.

Hence, the decision for $p=0.0050$ if $\alpha =0.01$ is rejecting the null hypothesis.

d)

The data gives the value of p is 0.0001 and $\alpha =0.01$.

The p-value is lesser than the value of $\alpha$. That is, $p\left(=0.0001\right)<\alpha \left(=0.01\right)$. From the decision rule, reject the null hypothesis.

Hence, the decision for $p=0.0001$ if $\alpha =0.01$ is rejecting the null hypothesis.