Question

An organizational psychologist was asked to test a claim by a franchise swimming instruction program. Specifically, the franchisers claimed that they could teach the average 7-year-old to swim across an Olympic-size pool in less than 2 hours. The psychologist arranged for eight randomly selected 7-year-old children to take instruction at this school and kept a record of how long it took each child to swim across the pool. The times (in minutes) were 60, 120, 110, 80, 70, 90, 100, and 130. Following a t test for a single sample (120 minutes is the “known” population mean) using the .01 significance level, what should the psychologist conclude?

Use the five steps of hypothesis testing!

Answer 1

One-Sample t-test

The sample mean is Xˉ=95, the sample standard deviation is s=24.4949, and the sample size is n=8. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =120 Ha: μ <120 This corresponds to a Left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. (2a) Critical Value Based on the information provided, the significance level is α=0.01, and the degree of freedom is n-1=8-1=7. Therefore the critical value for this Left-tailed test is tc​=-2.998. This can be found by either using excel or the t distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is t<-2.998 (3)Test Statistics The t-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is 0.0117 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=-2.8868 > tc​=-2.998, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0117, and since p=0.0117>0.01, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population mean μ is less than 120, at the 0.01 significance level.

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