Question

Investigators are trying to determine if the contamination of a town well led to significant increases in adverse health effects. During the period of time when water was consumed from this contaminated well, there were 16 birth defects among 414 births. After this well was shut off, there were 3 birth defects among 228 births. Investigators asked to determine if the rate of birth defects was higher when the contaminated well was in use.

(a) Estimate the probability of a birth defect when the contaminated water was consumed. Estimate the probability of a birth defect when the contaminated well was shut off.

(b) Calculate the 95% confidence interval for the true population proportion of birth defects when the contaminated water was consumed. Calculate the 95% confidence interval for the true population proportion of birth defects when the well was shut off.

(c) Test the claim that the contaminated well was not associated with an increase in the rate of birth defects in the community at the alpha = 0.05 level, by using the two-proportion z test.

(d) Calculate a 95% confidence interval for the true difference between the proportions of birth defects when the contaminated well was in use versus when the well was shut down.

Answer 1

a

The probability of a birth defect when the contaminated water was consumed =p1 =x1/n1 =16/414 =0.039

The probability of a birth defect when the contaminated water was shut off =p2 =x2/n2 =3/228 =0.013

b.

95% confidence interval for the true population proportion of birth defects when the contaminated water was consumed =P1 = =0.039 (1.96* ) =(0.020, 0.576)

95% confidence interval for the true population proportion of birth defects when the well was shut off =P2 = =0.013 (1.96* ​​​​​​) =(0.002, 0.028)

c.

H0: The difference between the population proportion is not significant. P1 - P2 =0

H1: The deference between the population proportions is significant. P1 - P2 0

Overall proportion, p =(x1+x2)/(n1+n2) =(16+3)/(414+228) =19/642 =0.03

Test statistic, Z =(p1 - p2)/ =(0.039 - 0.013)/ =0.026/0.01407 =1.85

Critical Z-value at 0.05 significance level for two-tailed test is Zcrit =1.96

Conclusion: Z < Zcrit (1.85 < 1.96) - No sufficient evidence to reject the null hypothesis(H0) and so, we failed to reject H0. Thus, we cannot claim that the difference between two population proportions (P1 - P2) is significant.

d.

Difference in sample proportions =p1 - p2 =0.039 - 0.013 =0.026

Standard error, SE =0.01407 (since from d. above)

95% confidence interval for the true difference between the proportions of birth defects when the contaminated well was in use versus when the well was shut down is:

P1 - P2 =(p1 - p2) (Z*SE) =0.026 (1.96*0.01407) =(-0.0016, 0.0536)

The above interval includes 0. So, the difference between true (population) proportions is not significant (same conclusion as in d. above).