Question

Question 3:

Suppose that 1000 customers are surveyed and 850 are satisfied or very satisfied with a corporation’s products and services.

1. Test the hypothesis H₀: p = 0.9 against H₁: p ≠ 0.9 at α = 0.05. Find the p-value.
2. Explain how the question in part (a) could be answered by constructing a 95% two-sided confidence interval for p.

Answer 1

a)

H0: p = 0.9

Ha: p 0.9

Sample proportion = 850 / 1000 = 0.85

Test statistics

z = - p / sqrt [ p( 1 - p) / n ]

= 0.85 - 0.90 / sqrt [ 0.90 * 0.10 / 1000 ]

= -5.27

This is test statistics value,

p-value = 2 * P(Z < z)

= 2 * P(Z < -5.27)

= 2 * 0

= 0

Since p-value < 0.05 significance level, reject H0.

b)

95% confidence interval for p is

- Z/2 * sqrt [ ( 1 - ) / n ] < p < + Z/2 * sqrt [ ( 1 - ) / n ]

0.85 - 1.96 * sqrt [ 0.85 * ( 1 - 0.85) / 1000] < p < 0.85 + 1.96 * sqrt [ 0.85 * ( 1 - 0.85) / 1000]

0.828 < p < 0.872

Since claimed proportion 0.90 does not contained in the confidence interval, reject H0.