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Don't be confused by π. It just represents the proportion (percentage or fraction).

From the sample population of 300 people, the proportion who are parents is 57/300=19/100 or 19%.

We're not given the sample standard deviation (SSD) so we need to calculate it. If 19% are parents, then 81% are not. The variance is 300(0.19)(0.81)=46.17, so the SSD is √46.17=6.79% approx. But this is for the sample, not the whole population, so we make a standard adjustment for the sample: 6.79/√300=0.3923 approx.

To bring in the CI we work on the Z score for the normal distribution. Z=|X-19|/0.3923 where X is a measure of the proportion in lots of different samples. The CI tells us how far X can be on either side of the mean proportion of 19% to lie within 95% of the results of measuring many samples. The significance level for a CI of 95% is (100-95)/2=2.5% because we have to remember that the normal distribution is symmetrical about the mean proportion, so that 5% is distributed equally on both sides of the mean in the tails of the distribution. The Z value is about ±1.96 for 95%, so |X-19|/0.3923=1.96, |X-19|=0.77, X=19.77 or 18.23.

Therefore 18.23%<π<19.77%, that is, the proportion can be expected to lie between 18.23% and 19.77% in 95% of the cases.

In decimals this is 0.182<π<0.198 (3 decimal places).

by Top Rated User (1.0m points)

If this method is correct you can use it to answer your other question on CI.

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