Suppose 40% of the employees in a very large corporation are women.
If a random sample of 30 employees is chosen from the corporation, let X be
the number of women in the sample.
a) For a specific x, the R function pbinom(x, 25, 0.3) computes P{X ≤ x}.
Use it to evaluate P{X ≤ 17}, P{X ≤ 6}, and hence P{7 ≤ X ≤ 17}.
b) Find µ = E(X) and σ = SD(X). Use the normal approximation to evaluate P{7 ≤ X ≤ 17}. That is, take Z = (X − µ)/σ to be approximately
standard normal. It is best to start with P{6.5 <><>
c) Now suppose the proportion π of women in the corporation is unknown.
A random sample of 30 employees has 20 women. Do you believe π is as
small as 0.4? Explain.
d) In the circumstances of part (c), use formula (1.2) to find an approximate
95% confidence interval for π.
Hints and comments: For (a) and (b), about 0.96; you should give 4-place accuracy.
The margin of error in (d) is about 0.17. Example 1.5 shows that the actual coverage
probability of the confidence interval in (d) may differ substantially from 95%; a
better confidence interval in this case is based on the Agresti-Coull adjustment of
Problem 1.16: (0.486, 0.808).