Chapter 20, Problem 14E

(a)

To determine

To write appropriate hypotheses.

Answer to Problem 14E

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

Explanation of Solution

It is given in the question that the National centre of education monitors many aspects of elementary and secondary education nationwide. So, $31\%$ of students reported that their mothers have graduated from college. Thus, the null hypothesis states that the rate is equal to $31\%$ . And the alternative hypothesis states that the rate is not equal to $31\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

(b)

To determine

To check the necessary assumptions.

Answer to Problem 14E

All the conditions are met.

Explanation of Solution

It is given in the question that the National centre of education monitors many aspects of elementary and secondary education nationwide. So, $31\%$ of students reported that their mothers have graduated from college. Thus, the null hypothesis states that the rate is equal to $31\%$ . And the alternative hypothesis states that the rate is not equal to $31\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

Thus, let us check the appropriate conditions and assumptions for inference to proceed further to test hypothesis as:

Random condition: It is satisfied as we assumed that the sample was randomly selected from the population.

Independent condition: It is satisfied as the children are independent from each other.

  $10\%$ condition: It is satisfied as the sample is less than $10\%$ of all children used in this sample.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=0.31(8368)>10\\ n(1-p)=0.69(8368)>10\end{array}$

Thus, all the conditions are satisfied.

(c)

To determine

To perform the test and find what is the P-value.

Answer to Problem 14E

The P-value is $0.0499$ .

Explanation of Solution

It is given in the question that the National centre of education monitors many aspects of elementary and secondary education nationwide. So, $31\%$ of students reported that their mothers have graduated from college. Thus, the null hypothesis states that the rate is equal to $31\%$ . And the alternative hypothesis states that the rate is not equal to $31\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result.

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=1.961\\ P-\text{value}\approx 0.0499\end{array}$

(d)

To determine

To state your conclusion.

Explanation of Solution

It is given in the question that the National centre of education monitors many aspects of elementary and secondary education nationwide. So, $31\%$ of students reported that their mothers have graduated from college. So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=1.961\\ P-\text{value}\approx 0.0499\end{array}$

Thus, as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that there is we have sufficient evidence to support the claim that there is change in education level among mothers.

(e)

To determine

To explain do you think this difference is meaningful.

Answer to Problem 14E

Yes, there is a significant difference.

Explanation of Solution

It is given in the question that the National centre of education monitors many aspects of elementary and secondary education nationwide. So, $31\%$ of students reported that their mothers have graduated from college. So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.31\\ H_a:p\ne 0.31\end{array}$

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=1.961\\ P-\text{value}\approx 0.0499\end{array}$

Thus, as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that there is we have sufficient evidence to support the claim that there is change in education level among mothers. Yes, there is a significant difference.