Question

Consider a population proportion p = 0.37. [You may find it useful to reference the z table.]

a. Calculate the standard error for the sampling distribution of the sample proportion when n = 10 and n = 75? (Round your final answer to 4 decimal places.)

b. Is the sampling distribution of the sample proportion approximately normal with n = 10 and n = 75?

c. Calculate the probability that the sample proportion is between 0.35 and 0.37 for n = 75. (Round "z-value" to 2 decimal places and final answer to 4 decimal places.)

Answer 1

a) For n = 10

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

      = sqrt(0.37 * (1 - 0.37)/10)

      = 0.1527

n = 75

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

      = sqrt(0.75 * (1 - 0.75)/10)

      = 0.1369

b) For n = 10

np = 10 * 0.37 = 3.7

n(1 - p) = 10 * (1 - 0.37) = 6.3

Since np < 5, so the sampling distribution of the sample proportion is not approximately normal.

np = 75 * 0.37 = 27.75

n(1 - p) = 75 * (1 - 0.37) = 47.25

Since np > 5 and n(1 - p) > 5, so the sampling distribution of the sample proportion is approximately normal.

c) P(0.35 < < 0.37)

= P((0.35 - )/ < ( - )/ < (0.37 - )/)

= P((0.35 - 0.37)/0.1369 < Z < (0.37 - 0.37)/0.1369)

= P(-0.15 < Z < 0)

= P(Z < 0) - P(Z < -0.15)

= 0.5 - 0.4404

= 0.0596