Question

Suppose we consider a bus "old" if it has been in service more than 8 years. At the 0.01 significance level, can we conclude that less than 40% of the district's buses are old? Report the p-value.

Age

5

2

10

2

2

5

10

8

7

9

10

10

4

6

6

6

3

3

8

8

3

9

7

3

9

8

4

6

9

6

9

9

1

6

8

9

9

11

9

7

6

9

7

8

8

8

10

14

4

6

4

10

9

8

7

4

6

9

10

4

2

4

10

4

12

7

2

7

10

7

10

5

6

6

11

7

10

5

8

9

Answer 1

As we are testing here whether the proportion of old buses is less than 0.4, therefore the null and the alternative hypothesis here are given as:

From the given sample values, we have here:
n = 80 as the sample size.
x = 28 as the number of buses with age greater than 8 years

The sample proportion here is computed as:
p = x/n = 28/80 = 0.35

The test statistic now is computed here as:

As this is a one tailed test, specifically a left tailed test, the p-value here is computed from the standard normal tables as:
p = P(Z < -0.9129) = 0.1807

Therefore 0.1807 is the required p-value here.