Question

Suppose a random sample of n measurements is selected from a binomial population with probability of success p = .38. Given n = 300, describe the shape, and find the mean and the standard deviation of the sampling distribution of the sample proportion,  .

Answer 1

The Sampling Distribution of the Sample Proportion. For large samples, the sample proportion is approximately normally distributed, with mean μˆP=p. and standard deviation σˆP=√pqn.

1) for large samples (n=300) the sample proportion is approximately normally distributed,

Hence,

The Shape of  sampling distribution of the sample proportion is symmetrical.( The shape of normal distribution is symmetrical)

Given : p = 0.38 , n = 300

2) Mean = p = probability of success

Mean of the sampling distribution of the sample proportion is p = 0.38

Hence mean is 0.38

3) Standard deviation = √pqn

p= 0.38

q=1-p =1-0.38 = 0.62

n =300

Variance is npq = 0.38*0.62*300 = 70.68

Standard deviation of the sampling distribution of the sample proportion is √pqn =√0.38*0.62*300 =8.407139822793480

Hence, Standard deviation is ✓variance = 8.407139822793480

Standard deviation is 8.4071 (to 4 decimal places)