Question

In the 1980s, it was generally believed that congenital abnormalities affected about 7% of a large nation's children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of abnormalities. A recent study examined 352 randomly selected children and found that 30 of them showed signs of an abnormality. Is this strong evidence that the risk has increased? (We consider a P-value of around 5% to represent reasonable evidence.) Complete parts a through f. Assume the independence assumption is met.

a) Write appropriate hypotheses. Let p be the proportion of children with genetic abnormalities. Choose the correct answer below.

A. H₀: p = 0.0852 vs H_A: p < 0.0852

B. H₀: p = 0.07 vs H_A: p ≠ 0.07

C. H₀: p = 0.0852 vs H_A: p ≠ 0.0852

D. H₀: p = 0.0852 vs H_A: p > 0.0852

E. H₀: p = 0.07 vs H_A: p > 0.07

F. H₀: p = 0.07 vs H_A: p < 0.07

b) Check the necessary assumptions. Which of the following are satisfied? Select all that apply.

A. Less than​ 10% of the population was sampled.

B. The sample is random.

C. There are more than 10 successes and 10 failures.

D. The independence assumption is satisfied.

c) Perform the mechanics of the test. What is the P-value?

P-value = ___ (Round to three decimal places as needed.)
d) Explain carefully what the P-value means in this context. Choose the correct answer below.

A. The​ P-value is the chance of observing 30 or more children with genetic abnormalities in a random sample of 352

children if 7​% of children actually have genetic abnormalities.

B. The​ P-value is the chance of observing 7​% of children with genetic abnormalities.

C. The​ P-value is the actual percentage of children who have genetic abnormalities.

D. The​ P-value is the chance of observing 30 or more children with genetic abnormalities in a random sample of 352

children.

e) What's your conclusion?

A. Fail to reject H₀. There is sufficient evidence that more than 7​% of the​ nation's children have genetic abnormalities.

B. Reject H₀. There Is sufficient evidence that more than 7​% of the​ nation's children have genetic abnormalities.

C. Reject H₀. There is not sufficient evidence that more than 7​% of the​ nation's children have genetic abnormalities.

D. Fail to reject H₀. There is not sufficient evidence that more than 7​% of the​ nation's children have genetic abnormalities.

f) Do environmental chemicals cause congenital abnormalities?

A. ​No, the conclusion of the hypothesis test shows that environmental chemicals do not cause genetic abnormalities.

B. It is unknown if environmental chemicals cause genetic​ abnormalities, because the hypothesis test does not indicate the cause of any changes.

C. ​Yes, the conclusion of the hypothesis test shows that environmental chemicals cause genetic abnormalities.

Answer 1

Solution:

a) Write appropriate hypotheses. Let p be the proportion of children with genetic abnormalities. Choose the correct answer below.

Answer: $\small \boldsymbol{E. \ H_{0}: p = 0.07 \ vs \ H_{A}: p > 0.07}$

b) Check the necessary assumptions. Which of the following are satisfied? Select all that apply.

Answer: All assumptions are satisfied:

A. Less than​ 10% of the population was sampled.

B. The sample is random.

C. There are more than 10 successes and 10 failures.

D. The independence assumption is satisfied.

c) Perform the mechanics of the test. What is the P-value?

Answer: $\small \boldsymbol{P-value = 0.131}$

Explanation:

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Where:

$\small \hat{p}=\frac{30}{352}=0.08523$

0.08523 -0.07 ..p- vlaue = P > 0.07(1-0.07)

$\small =P\left ( z>1.1197\right )$

Now using the standard normal table, we have:

$\small p-value=P\left ( z>1.1197\right )=0.131$
d) Explain carefully what the P-value means in this context. Choose the correct answer below.

Answer: A. The​ P-value is the chance of observing 30 or more children with genetic abnormalities in a random sample of 352 children if 7​% of children actually have genetic abnormalities.

e) What's your conclusion?

Answer: D. Fail to reject H₀. There is not sufficient evidence that more than 7​% of the​ nation's children have genetic abnormalities.

f) Do environmental chemicals cause congenital abnormalities?

Answer: A. ​No, the conclusion of the hypothesis test shows that environmental chemicals do not cause genetic abnormalities.