With more women in the workforce, it is believed that men are now performing mor

Question

With more women in the workforce, it is believed that men are now performing more of the household tasks. Consider grocery shopping. If the above belief is true then more advertising should be directed at men. Suppose that 87 people were interviewed at the checkouts at several major supermarkets. Those using the express lanes were excluded. The data collected at various times, including weekday mornings, evenings, and weekends, showed that 31 were male.

a) Construct a 98% confidence interval for the true proportion of grocery store customers who are male.

b) A similar study done ten years ago found that only 25% of shoppers were male. Do you have any strong reasons for concluding that the proportion has increased? Use your confidence interval in a) to justify your conclusions.

c) If you wanted to estimate the proportion to within +/-4% with 98% confidence, how large a sample would be necessary?

Explanation / Answer

a) P^ = x / n

     = 31/87

=0.3563

        Confidence Interval:

         p^ (-/+) E

E = zc * Ö[(p^*(1-p^))/n]

zc is the critical value for 98% confidence interval. We get this from Normal Distribution Table. This value is 2.326

n = 87

E = 2.326 * Ö((0.3563*(1-0.3563))/87) = 0.1194

0.3563 (-/+) 0.1194

= 0.237 and 0.476

Answer:

The 98% confidence interval is (0.237 and 0.476)

b) As 25% falls in this interval so we do not have any strong reasons for concluding that the proportion has increased.

c) n = (zc /E)^2*p*q

= (2.326/0.04)^2*0.3563*(1-0.3563)

              = 775.5 ~ 776

Answer: 776

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