Question

Problem 4 (20 pts) A sociologist is interested in surveving workers in computer-related jobs to estimate the proportion of such workers who have changed the jobs within the past year. (a) (10 pts) In the absence of preliminary data, how large a sample must be taken to ensure that a 99% confidence interval will specify the proportion to within 20.05? (b) (5 pts) is the sample size obtnined in (a) large enough to ensure that confidence interval will specify the proportion to within +0.05? Explain les con borgo (0) 15 pts) is the sample size obtained in (a) large enough to ensure that a 99% confidence interval will specify the proportion to within 20.1? Explain.

Answer 1

(a)The ME = Zcritical * SQRT(p * q/n).

Squaring and solving, n = (Zc/ME)² * p * q

Here ME = 0.05, p = 0.5, q = 0.5

The Zc at $\alpha$ = 0.01 is 2.576

Therefore n = (2.576 / 0.05)² * 0.5 * 0.5 = 663.58

Therefore n = 664 (Rounding to the nearest Integer)

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(b) Yes, it is large enough.

The critical value is in the numerator, and as level of confidence decreases from 99% to 95%, the critical value decreases.

When critical value decreases, the sample size required also decreases, and therefore the sample obtained in (a) is large enough.

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(c) Yes, It is large enough

The ME is in the denominator, and therefore as ME increases from 0.05 to 0.1, the sample size required will decrease.

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