Question

A test of sobriety involves measuring the subject's motor skills. Twenty randomly selected sober subjects take the test and produce a mean score of 41.0 with a standard deviation of 3.7. At the 0.01 level of significance, test the claim that the true mean score for all sober subjects is equal to 35.0.  

The point estimate is:   (rounded to 1 decimal place)

The test statistic is:    (rounded to 2 decimal places)

The p-value is:   (rounded to 4 decimal places)

The Conclusion   is type R (Reject the Null) or F (Fail to Reject the Null)

Answer 1

One-Sample t-test

The sample mean is Xˉ=41, the sample standard deviation is s=3.7, and the sample size is n=20. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =35 Ha: μ ≠35 This corresponds to a Two-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=20-1=19. Therefore the critical value for this Two-tailed test is tc​=2.8609. This can be found by either using excel or the t distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |t|>2.8609 i.e. t>2.8609 or t<-2.8609 (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 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |t|=7.2521 > tc​=2.8609, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0, and since p=0≤0.01, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is different than 35, at the 0.01 significance level.

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