Chapter 20, Problem 12E

(a)

To determine

To write appropriate hypotheses.

Answer to Problem 12E

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. Thus, the null hypothesis states that the rate is equal to $5\%$ . And the alternative hypothesis states that the rate is more than $5\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

(b)

To determine

To check the necessary assumptions.

Answer to Problem 12E

All the conditions are met.

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. Thus, the null hypothesis states that the rate is equal to $5\%$ . And the alternative hypothesis states that the rate is more than $5\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

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 with congenital abnormalitiesare independent from each other.

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

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

  $\begin{array}{l}np=0.05(384)>10\\ n(1-p)=0.05(384)>10\end{array}$

Thus, all the conditions are satisfied.

(c)

To determine

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

Answer to Problem 12E

The P-value is zero.

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. Thus, the null hypothesis states that the rate is equal to $5\%$ . And the alternative hypothesis states that the rate is more than $5\%$ . So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

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. Thus, we have,

  $\begin{array}{l}x=46\\ n=384\\ c=0.95=95\%\end{array}$

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,

  $\begin{array}{l}z=6.276\\ P-\text{value}\approx 0\end{array}$

(d)

To determine

To explain carefully what the P-value means in the context.

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

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

  $\begin{array}{l}z=6.276\\ P-\text{value}\approx 0\end{array}$

Thus, we can say that the p-value is the probability that the sample proportion was that far from $0.05$ by simple chance.

(e)

To determine

To find out what is your conclusion.

Answer to Problem 12E

We have sufficient evidence to claim that the risk has increased.

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

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

  $\begin{array}{l}z=6.276\\ P-\text{value}\approx 0\end{array}$

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

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

So, we conclude that we have sufficient evidence to claim that the risk has increased.

(f)

To determine

To explain do environmental chemicals cause congenital abnormalities.

Answer to Problem 12E

No, we do not have evidence that environmental chemicals cause congenital abnormalities.

Explanation of Solution

It is given in the question that it is believed that congenital abnormalities affected about $5\%$ of the nation’s children. So, we define hypothesis as:

  $\begin{array}{l}{H}_0:p=0.05\\ H_a:p>0.05\end{array}$

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

  $\begin{array}{l}z=6.276\\ P-\text{value}\approx 0\end{array}$

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

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

So, we conclude that, no, we do not have evidence that environmental chemicals cause congenital abnormalities.