Question

A study was conducted to investigate the relationship between maternal smoking during pregnancy and congenital malformation. Consider children born with an oral cleft. In a random sample of 27 such infants, 15 have mothers who smoked during pregnancy.

1. What is a point estimate of population proportion, ?? Construct a 95% confidence interval for ?.

2. You would like to know whether the proportion of mothers who smoked during pregnancy for children with an oral cleft is identical to the proportion of mothers who smoked for children with other types of malformations, 32.8%. What are the null and alternative hypotheses of the appropriated test?

3. Conduct the test at the 0.01 level of significance.

4. What do you conclude?

5. If the true population proportion of children with an oral cleft whose mothers smoked is as low as 0.25, you want to risk only a 10% chance of failing to reject the null hypothesis. If you are conducting a two-sided test at the 0.01 level of significance, how large a sample would be required?

Answer 1

1)

sample size, n=27 point estimate of population proportion = sample proportion, p = 15 / 27 = 0.5556 we use normal approximation, for this we check that both np and n(1-p) > 5. Since n*p = 15.0012 > 5 and n*(1-p) = 11.9988 > 5, we can take binomial random variable as normally distributed, with mean = p = 0.5556 and std error = root( p * (1-p) /n ) = 0.095628

For constructing Confidence interval, Margin of Error (ME) = z x SD = 0.1874 95% confidence interval is given by: Sample Mean +/- (Margin of Error) 0.5556 +/- 0.1874 = (0.3682 , 0.743)

2)

We have n = 27, sample proportion p = 15 / 27 = 0.5556 and P = 0.328 we want to test whether hypothesis: H0: true P = 0.328 (null hypothesis) Ha: true P not equal to 0.328 (alternative hypothesis) now, calculating statistics, sigma s = sqrt(P*(1-P)/n) = 0.0904 (test statistic) z-score = (p-P)/s = 2.5177 this is two tailed test, hence p-value = P( z < -2.5177) + P( z > 2.5177) = 0.0117

since p-value is not less than 0.01 (alpha), we fail to reject null hypothesis, and hence

4) we conclude the true proportion, p is 32.8%

5)

here we note that:

$2014-11-18_12-03-09.png$

solving for n, we get:

$2014-11-18_11-59-44.png$

where alpha = 0.01, beta = 0.1, p0 = current probability = sample proportion = 0.5556 , p1 = probability in question = 0.25

n = $gif&s=45&w=330.&h=45.$

n = 42.4945 = 43 (taking ceiling)

Hence we have to take N = 43