We see that the test statistic is in the tail and thus we move to step 4 and reach a conclusion. Step 4 has us compare the test statistic and the critical value and mark these on the graph. We find that the calculated test statistic is 2.08, meaning that the sample mean is 2.08 standard deviations away from the hypothesized mean of 16.43. Step 3 is the calculation of the test statistic using the formula we have selected. This is the critical value and we can put this on our graph. Looking up 14 degrees of freedom at the 0.05 column of the t-table we find 1.761. For this problem the degrees of freedom are n-1, or 14.
We can now draw the graph of the t-distribution and mark the critical value. We have concluded that this is a t-test on the basis of the sample size and that we are interested in a population mean. To find the critical value we need to select the appropriate test statistic. (Reject the null hypothesis when the null hypothesis is true.) For this case the only concern with a Type I error would seem to be that Jeffery’s dad may fail to bet on his son’s victory because he does not have appropriate confidence in the effect of the goggles. The Type I error is to conclude that Jeffrey swims the 25-yard freestyle, on average, in less than 16.43 seconds when, in fact, he actually swims the 25-yard freestyle, on average, in 16.43 seconds. It is worth thinking about the meaning of this choice. Our step 2, setting the level of significance, has already been determined by the problem. Μ 0 = 16.43 comes from H 0 and not the data. The sample size is less than 30 and we do not know the population standard deviation so this is a t-test. Random variable: = the mean time to swim the 25-yard freestyle. The null and alternative hypotheses are thus:įor Jeffrey to swim faster, his time will be less than 16.43 seconds. Remember that the status quo must be defeated with a high degree of confidence, in this case 95 % confidence. The claim will always be in the alternative hypothesis because the burden of proof always lies with the alternative. The effect of this is to set the hypothesis as a one-tailed test. This is that the goggles will reduce the swimming time.
In this case there is an implied challenge or claim. Since the problem is about a mean, this is a test of a single population mean.
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