Chapter 23, Problem 29E

(a)

To determine

To write appropriate hypotheses.

Answer to Problem 29E

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

In which, the $H_0$ is the null hypothesis and the $H_1$ is the alternative hypothesis.

(b)

To determine

To explain do think the necessary assumption for inference are satisfied.

Answer to Problem 29E

The necessary assumption for inference are satisfied.

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

Now, we will check if the necessary assumption for inference are satisfied or not, as:

First is random condition: It is satisfied because the sample is given to be a random sample. The second is the independent condition: It is satisfied since the sample of $40$ men is less than $10\%$ of the population of all men.

And the third is the normal condition: It is satisfied since the sample size $40$ is $30$ or more and then the central limit theorem tells us that the sampling distribution of the sample mean is approximately Normal. Thus, all the conditions are met.

(c)

To determine

To describe the approximate sampling distribution model for the mean age in such samples.

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

Since the sample of size $40$ is $30$ or more, the central limit theorem tells us that the sampling distribution of the sampling distribution of the sample mean is approximately normal.

The degrees of freedom is the sample size decreased by one as:

  $\begin{array}{l}df=n-1\\ =40-1\\ =39\end{array}$

And the test statistic is as:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{\overline{x}-23.3}{\frac{s}{\sqrt{40}}}\end{array}$

Thus we know that $\frac{\overline{x}-23.3}{\frac{s}{\sqrt{40}}}$ has a student t -distribution with $39$ degrees of freedom.

(d)

To determine

To find out what is the P-value for this result.

Answer to Problem 29E

  $P>0.10$ .

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

And the test statistic is as:

  $\begin{array}{l}t=\frac{\overline{x}-{\mu }_0}{\frac{s}{\sqrt{n}}}\\ =\frac{24.2-23.3}{\frac{5.3}{\sqrt{40}}}\\ =1.07\end{array}$

The P-value is the number in the column title of table T in appendix F containing the t -value in the row $n-1=32-1=31$ is:

  $P>0.10$ .

(e)

To determine

To explain what this P-value means.

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

And the P-value is $P>0.10$ . The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, if the null hypothesis is true. It interprets that if a mean age of the first marriage of all Americans men is $23.3$ years then we have a more than $10\%$ chance of obtaining a t -statistic of $1.07$ or more extreme.

(f)

To determine

To explain what is your conclusion.

Explanation of Solution

In the question, it is given that census results indicated that the age at which American men first married had a mean of $23.3$ years. It is widely suspected that young people today are waiting longer to get married. In this, we want to find out if the mean age of first marriage has increased during the past $40$ years. Thus, the appropriate hypotheses for this context will be as:

  $\begin{array}{l}{H}_0:\mu =23.3\\ H_1:\mu >23.3\end{array}$

And the P-value is $P>0.10$ . It interprets that if a mean age of the first marriage of all Americans men is $23.3$ years then we have a more than $10\%$ chance of obtaining a t -statistic of $1.07$ or more extreme. Now if the P-value is smaller than the significance level then the null hypothesis is rejected, then,

  $P>0.05=5\%\Rightarrow \text{Fail to reject }{H}_0$

Thus, we conclude that there is no sufficient evidence to support the claim that young people today are waiting longer to get married.