Test Concerning Means:Small Sample

Test on Small Sample (t-test)

          When the population standard deviation is unknown and the sample size is small, that is, less than 30, the t test is appropriate for testing hypothesis involving means. The formula is

Example 2: In order to increase customer service, a muffler repair shop claims its mechanics can replace a muffler in 12 minutes. A time management specialist selected six (6) repair jobs and found their mean time to be 11.6 minutes. The standard deviation of the sample was 2.1 minutes. At 0.025 level of significance, is there enough evidence to conclude that the mean time in changing a muffler is less than 12 minutes?

Solution: Follow the steps in hypothesis testing.

1. State the null and alternative hypothesis. Mathematically,

Ho: μ = 12 minutes

Ha: μ < 12 minutes 

2. Level of significance α = 0.025.

3. Select an appropriate test statistic.

The test statistic is the t– test, the sample size is less than 30 and the formula is

4. Determine the critical value and critical region

Since the level of significance is 0.025 and df = n – 1 = 6 – 1 = 5, and the alternative hypothesis is left – tailed test, then the critical value ( = -2.571.

Reject Ho, if t computed is less than -2.571.

5. Compute the value of the test statistics:

Given:

Sample mean = 11.6 minutes

Population mean = 12 minutes

Standard deviation = 2.1 minutes

Sample size = 6 samples

6. Decision: Since the computed t = -0.47 is in the acceptance region, thus, we fail to reject Ho.

7. Conclusion:

          Therefore, the data does not provide enough evidence to conclude that that the mean time in changing a muffler is less than 12 minutes.