A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. Both populations have a normal distribution.
College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes.
College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class.
The community group believes that a student who graduates from College A has taken more math classes, on average. Is the community group correct?
To prove or disprove the community group’s beliefs, we need to conduct a hypothesis test or utilize a confidence interval to make an inference.
Using a hypothesis test, does the data suggest that College A has taken more math classes than College B?
Step 1: Write out the null and alternative hypotheses.
[latex]H_0: \mu_A - \mu_B = 0[/latex]
[latex]H_A: \mu_A - \mu_B \gt 0[/latex]
Step 2: Check the conditions for the hypothesis test.
Because the populations have a normal distribution, the conditions are met.
Step 3: Calculate the [latex]t[/latex]-test statistic.
Using the statistical tool above, [latex]t=0.890[/latex].
Step 4: Calculate a P-value.
Using the statistical tool above, P-value [latex]=0.1928[/latex]. Note that we are conducting a one-tailed test because our alternative hypothesis is testing “greater than”.
Step 5: Compare the P-value to the significance level, [latex]\alpha[/latex], to make a decision.
The P-value [latex]=0.1928 > \alpha = 0.05[/latex]. Therefore, there is not enough evidence to reject the null hypothesis.
Step 6: Write a conclusion in context.
At the [latex]5\%[/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.
Therefore, the community college’s beliefs are not correct.
Using a confidence interval, does the data suggest that College A has taken more math classes than College B?
Using the statistical tool, we find that the [latex]95\%[/latex] confidence interval for this data set is [latex](-0.683, 1.683)[/latex].
Interpretation: We are [latex]95\%[/latex] confident that the true difference in the mean math classes between College A and College B is [latex]-0.683[/latex] and [latex]1.683[/latex] classes.
The values in the [latex]95\%[/latex] confidence interval include some negative values, some positive values, and [latex]0[/latex]. This implies that we cannot conclude that a student who graduates from College A has taken more math classes, on average, than a student who graduates from College B.
We achieved the same conclusion as the hypothesis test.