Consider a population proportion p = 0.78.

Question

a-1. Calculate the expected value and the standard error of  P−
  with n = 40. (Round "expected value" to 2 decimal places and "standard deviation" to 4 decimal places.)

a-2. Is it appropriate to use the normal distribution approximation for  P−
  ?

multiple choice 1
Yes
No

b-1. Calculate the expected value and the standard error of  P−
  with n = 40. (Round "expected value" to 2 decimal places and "standard deviation" to 4 decimal places.)

b-2. Is it appropriate to use the normal distribution approximation for  P−
  ?

multiple choice 2
Yes
No

GPT 3.5 · Answer

a-1. To calculate the expected value and standard error of P−, we need to use the formula:

Expected value of P− = p = 0.78

Standard error of P− = sqrt(pq/n) where q = 1 - p

Standard error of P− = sqrt(0.78 * (1 - 0.78)/40) ≈ 0.0513 (rounded to 4 decimal places)

Therefore, the expected value of P− is 0.78 and the standard error is 0.0513.

a-2. No, it is not appropriate to use the normal distribution approximation for P− when n = 40. The normal approximation is generally considered appropriate when both np and n(1-p) are greater than or equal to 5, where n is the sample size. In this case, np = 31.2 (40 * 0.78) and n(1-p) = 8.8 (40 * 0.22), which are both less than 5. Therefore, the normal distribution approximation is not appropriate.

b-1. The expected value and standard error of P− with n = 40 are the same as in part a-1:

Expected value of P− = p = 0.78

Standard error of P− = sqrt(pq/n) where q = 1 - p

Standard error of P− = sqrt(0.78 * (1 - 0.78)/40) ≈ 0.0513 (rounded to 4 decimal places)

Therefore, the expected value of P− is 0.78 and the standard error is 0.0513.

b-2. No, it is not appropriate to use the normal distribution approximation for P− when n = 40. The normal approximation is generally considered appropriate when both np and n(1-p) are greater than or equal to 5, where n is the sample size. In this case, np = 31.2 (40 * 0.78) and n(1-p) = 8.8 (40 * 0.22), which are both less than 5. Therefore, the normal distribution approximation is not appropriate.