Question

A study was conducted to test the effectiveness of a software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in the table below. The "before" value is matched to an "after" value, and the differences are calculated. The differences have a normal distribution. Test at the 1% significance level. Installation A B C D E F G H

Before 3 6 4 2 5 8 2 6

After 1 5 2 0 1 0 2 2

A) State the null and alternative hypotheses.

H₀: μ_d < 0
H_a: μ_d ≥ 0

H₀: μ_d = 0
H_a: μ_d ≠ 0    

H₀: μ_d ≤ 0
H_a: μ_d > 0

H₀: μ_d ≠ 0
H_a: μ_d = 0

H₀: μ_d ≥ 0
H_a: μ_d < 0

B) Draw the graph of the p-value.

Answer 1

Part A

State the null and alternative hypotheses.

H0: μd ≤ 0
Ha: μd > 0

This is an upper tailed test.

We take difference as before minus after.

Part B

Test statistic for paired t test is given as below:

t = (Dbar - µd)/[Sd/sqrt(n)]

From given data, we have

Dbar = 2.8750

Sd = 2.4749

n = 8

df = n – 1 = 7

α = 0.01

t = (Dbar - µd)/[Sd/sqrt(n)]

t = (2.8750 – 0)/[ 2.4749/sqrt(8)]

t = 3.2857

The p-value by using t-table is given as below:

P-value = 0.0067

P-value < α = 0.01

So, we reject the null hypothesis

There is sufficient evidence to conclude that the software patch is effective in reducing system failures over a six-month period.