the average length of time for students to register for fall classes at a certain college has been 50 minutes. a new registration procedure using modern computing machine is being tried. if a random sample of 12 students had an average registration time of 42 minutes with a standard deviation of 11.90 minutes under the new system, test the hypothesis that the population mean is now less than 50 using 0.01 level of significance. Assume the population of times to be normal
To test the hypothesis that the population mean registration time is now less than 50 minutes using a 0.01 level of significance, we can perform a one-sample t-test. Here's how to do it:
1. State the null hypothesis (H₀) and the alternative hypothesis (H₁):
- Null hypothesis: The population mean registration time is equal to 50 minutes (µ = 50).
- Alternative hypothesis: The population mean registration time is less than 50 minutes (µ <50).
2. Determine the significance level (α):
- Given in the problem as 0.01 or 1%.
3. Compute the test statistic:
- The test statistic for a one-sample t-test is given by:
t = (x̄ - µ) / (s / √n)
where x̄ is the sample mean, µ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
In this case, x̄ = 42 minutes, µ = 50 minutes, s = 11.90 minutes, and n = 12 students.
t = (42 - 50) / (11.90 / √12) = -2.19 (rounded to two decimal places)
4. Determine the critical value:
- Since we are testing whether the population mean is less than 50, this is a left-tailed test.
- We need to find the critical t-value for a one-tailed test with a significance level of 0.01 and degrees of freedom (df) equal to n - 1.
- Since n = 12, df = 12 - 1 = 11.
- Using a t-distribution table or a statistical calculator, the critical t-value is approximately -2.718 (rounded to three decimal places).
5. Compare the test statistic and the critical value:
- Since the test statistic (-2.19) is not more extreme than the critical value (-2.718), we fail to reject the null hypothesis.
6. Make a conclusion:
- Based on the sample data and the statistical test, there is not enough evidence to support the claim that the population mean registration time is now less than 50 minutes. The new registration procedure does not significantly reduce the registration time compared to the previous average of 50 minutes.
Note: The p-value can also be computed for this test, but it is not necessary for reaching a conclusion in this case since the critical value approach was used.