Question

1. The Gallup Poll asked a probability sample of 2500 U.S. adults whether they used a credit card to pay for groceries that week. Suppose that 25% of the adult population did. We would like to know the probability that an SRS of size 2500 would come within plus or minus 2 percentage points of this true value.
   1. If p-hat is the proportion of the sample who did use a credit card to pay for groceries, what is the mean of p-hat? (2 points)
      
      
   2. What is the standard deviation of p-hat? (2 points)
      
      
   3. Explain why you can use the formula for the standard deviation of p-hat in this setting.(check that the population is at least 10 times larger than the sample) (2 points)
      
   4. Check that you can use the normal approximation for the distribution of p-hat.(2 points)
      
      
   5. Find the probability that p-hat takes a value between 0.23 and 0.27. (4 points)

Answer 1

a) = p = 0.25

b) = sqrt(p(1 - p)/n)

          = sqrt(0.25 * 0.75/2500)

         = 0.009

c) When the population size is at least 10 times larger than the sample size, then the standard deviation can be approximated by the formula     = sqrt(p(1 - p)/n)

d) np = 2500 * 0.25 = 625

   n(1 - p) = 2500 * (1 - 0.25) = 1875

Since np > 5 and n(1 - p) > 5, so we can use normal approximation.

e) P(0.23 < < 0.27)

= P((0.23 - )/ < ( - )/ < (0.27 - )/)

= P((0.23 - 0.25)/0.009 < Z < (0.27 - 0.25)/0.009)

= P(-2.22 < Z < 2.22)

= P(Z < 2.22) - P(Z < -2.22)

= 0.9868 - 0.0132

= 0.9736