Economics 30330: Statistics for Economics
Problem Set 8 - Suggested Solutions
University of Notre Dame
Instructor: Julio Gar´ ın Spring 2012

Hypothesis Testing (80 Points)
1. Consider the following hypothesis test:
H0 : µ ≥ 20
HA : µ < 20
A sample of 40 provided a sample mean of 19.4. The population standard deviation is 2.
(a) Create a 95% confidence interval for the mean.
We know σ, therefore we should use the z − table. This is a one-tailed (lower tail) test, so the 95% confidence interval will be given then by σ x − z.05 √ , ∞
¯
n
2
19.4 − 1.65 √ , ∞
40
The 95% confidence interval is µ ∈ [18.878, ∞).
(b) What is the p-value?
The p-value is the area in the lower tail. First, we calculate the z-value: z= 19.4 − 20
√
= −1.9
2/ 40

Using the normal table with z = -1.9, p-value =.0287.
(c) At α = 0.01, what is your conclusion? p-value > .01, so we fail reject H0 at the 99% level.
(d) What is the rejection rule using the critical value? What is your conclusion? c Reject H0 at the 99% level if z ≤ zα =-2.33. In this example, -1.9 > -2.33, so we fail to reject H0 at the 99% level.
2. Consider the following hypothesis test:
H0 : µ = 15
HA : µ = 15
A sample of 50 provided a sample mean of 14.5. The population standard deviation is 3.
1

(a) Create a 95% confidence interval for the mean. We know σ, therefore we should use the z − table. This is a two-tailed test, so the 95% confidence interval will be given then by σ σ x − z.025 √ , x + z.025 √
¯
¯ n n
3
3
14.5 − 1.96 √ , 14.5 + 1.96 √
50
50
The 95% confidence interval is µ ∈ (13.67, 15.33).
(b) What is the p-value?
The p-value is the area in the upper and lower tails. First, we calculate the z-value: z= 14.5 − 15
√
= −1.18
3/ 50

Using the normal table with z = -1.18, p-value = 2(.1190) = 0.238.
(c) At α = 0.01, what is your conclusion? p-value > .01, so