The average length of time for students to register for summer classes at a certain college has been 50 minutes. A new registration procedure using modern computing machines is being tried. If a random sample of 35 students had an average registration time of 42 minutes with a standard deviation of 11.9 minutes under the new system, test the hypothesis that the population mean is now less than 50 minutes, using a level of signficance of 0.01. (critical value = +/- 2.33)
To test the hypothesis that the population mean registration time is less than 50 minutes, we can use a one-sample t-test. Here's how you can conduct the hypothesis test:
Step 1: State the null and alternative hypotheses:
- Null Hypothesis (H₀): The population mean registration time is equal to 50 minutes.
- Alternative Hypothesis (H₁): The population mean registration time is less than 50 minutes.
Step 2: Set the significance level (alpha):
The level of significance is given as 0.01, which means we want to be 99% confident in our conclusion. Therefore, alpha (α) is 0.01.
Step 3: Calculate the test statistic:
The test statistic for a one-sample t-test is calculated using the formula:
t = (x̄ - μ) / (s / √n)
where:
- x̄ is the sample mean (42 minutes),
- μ is the population mean (50 minutes) stated in the null hypothesis,
- s is the sample standard deviation (11.9 minutes), and
- n is the sample size (35).
Plugging in the values:
t = (42 - 50) / (11.9 / √35)
Step 4: Determine the critical value:
The critical value for a one-tailed test with a significance level of 0.01 and 34 degrees of freedom (35 - 1) is approximately -2.33. Since we are testing if the population mean is less than 50 minutes, we only need to consider the left tail.
Step 5: Make a decision:
Compare the test statistic to the critical value. If the test statistic is less than the critical value, we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, if the test statistic is greater than or equal to the critical value, we fail to reject the null hypothesis.
Step 6: Calculate the p-value (optional):
If you want to calculate the p-value, you can use the t-distribution with the degrees of freedom to find the probability of obtaining a test statistic as extreme as the one observed (or more extreme).
That's how you can test the hypothesis that the population mean registration time is less than 50 minutes using a t-test with the given information and significance level.