Question

In a restaurant, the proportion of people who order coffee with their dinner is .9. a simple random sample of 144 patrons of the restaurant is taken.
a. what are the expected value, standard deviation, and shape of the sampling distribution of ?
b. what is the probability that the proportion of people who will order coffee with their meal is between 0.85 and 0.875?
c. what is the probability that the proportion of people who will order coffee with their meal is at least 0.945?

Answer 1

Let P be the proportion of people who order coffee with their dinner . Let n be the sample size

P=0.9 and n=144

P follows Normal distribution .

a. The expected value is E(p) = P =0.9

Standard deviation = $\sqrt{\frac{p*(1-p)}{n}}$

= $\sqrt{\frac{0.9* (1-0.9)}{144}}$

Standard deviation = 0.025

The shape of sampling distribution is bell shaped symmetric about mean.

b. The probability that the proportion of people who will order coffee with their meal is between 0.85 and 0.875

P(0.85 < p < 0.875) = $P(\frac{0.85 -0.9}{0.025} < \frac{p-mean}{standard deviation} < \frac{0.875-0.9}{0.025} )$

= P(-2 < Z < -1)

= P(Z < -1) - P(Z < -2)

= 0.1587 - 0.0228

P(0.85 < p < 0.875) = 0.1359

The probability that the proportion of people who will order coffee with their meal is between 0.85 and 0.875 is 0.1359

c. the probability that the proportion of people who will order coffee with their meal is at least 0.945

P(p > 0.945) = $P( \frac{p-mean}{standard deviation} > \frac{0.945 - 0.9}{0.025} )$

= P(z > 1.8)

= 1 - P(z < 1.8)

= 1 -0.9641

P(p > 0.945) = 0.0359

the probability that the proportion of people who will order coffee with their meal is at least 0.945 is 0.0359