Question

In the 1990s, it was generally believed that genetic abnormalities affected about 8% 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, which could have important implications for health insurance companies. A recent study examined 369 randomly selected children and found that 40 of them showed signs of a genetic abnormality.

(a) Which hypotheses should be used to test if the proportion of genetic abnormalities has increased in recent years?

H₀: p = 0.08 vs. H_a: p > 0.08

H₀: p = 0.1084 vs. H_a: p > 0.1084  

H₀: p = 0.08 vs. H_a: p < 0.08

H₀: p = 0.1084 vs. H_a: p ≠ 0.1084

H₀: p = 0.08 vs. H_a: p ≠ 0.08

H₀: p = 0.1084 vs. H_a: p < 0.1084

(b) Are the conditions met for doing the hypothesis test?

None of the conditions are met.

The children taking part in the study were independent of each other.

The sample was randomly chosen.

The 15 successes and failures condition is met.

(c) What is the p-value? (Use 3 decimals.)  

(d) What does this p-value mean?

The p-value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 369 children, if the true proportion of children with genetic abnormalities is 10.84%.

The p-value gives the actual percentage of children who have genetic abnormalities.   

The p-value is the chance of observing 8% of children with genetic abnormalities.

The p-value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 369 children, if the true proportion of children with genetic abnormalities is 8%.

(e) What is the conclusion of the hypothesis test, for α = 0.05?

Reject H₀. There is sufficient evidence that more than 8% of this nation's children have genetic abnormalities.

Do not reject H₀. There is insufficient evidence that more than 8% of this nation's children have genetic abnormalities.    

Reject H₀. There is insufficient evidence that more than 8% of this nation's children have genetic abnormalities.

Do not reject H₀. There is sufficient evidence that more than 8% of this nation's children have genetic abnormalities.

(f) Do this study show that environmental chemicals cause congenital abnormalities?

No, the hypothesis test shows that environmental chemicals do not cause the genetic abnormality.

Yes, the hypothesis test shows that environmental chemicals cause the genetic abnormality.    

This study does not address what causes the genetic abnormality.

Answer 1

Given : n=369 , X=40 , , ,

The sample proportion is , p=X/n=40/369=0.1084

(a) Hypothesis : Vs  

(b) Condition :

The children taking part in the study were independent of each other.

The sample was randomly chosen.

The 15 successes and failures condition is met. (np=369*0.08=29.52>15 and nq=369*0.92=339.48>15)

(c) The test statistic is ,

The p-value is ,

p-value=

; From standard normal probability table

(d) The p-value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 369 children, if the true proportion of children with genetic abnormalities is 8%.

(e) Decision : Here , p-value=0.022<0.05

Therefore , reject Ho.

Conclusion : There is sufficient evidence that more than 8% of this nation's children have genetic abnormalities.

(f) Yes, the hypothesis test shows that environmental chemicals cause the genetic abnormality.    

Because  it was generally believed that genetic abnormalities affected about 8% of a large nation's children.