Question

You wish to test the following claim (Ha) at a significance level of α=0.10.

      Ho:μ=71.4
      Ha:μ≠71.4

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=74 with mean M=69.6 and a standard deviation of SD=6.5.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

Answer 1

Test Statistics
The t-statistic is computed as follows:

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 0.0198

The complete test:

One-Sample t-test

The sample mean is Xˉ=69.6, the sample standard deviation is s=6.5, and the sample size is n=74. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =71.4 Ha: μ ≠71.4 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=0.1, and the degree of freedom is n-1=74-1=73. Therefore the critical value for this Two-tailed test is tc​=1.666. 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|>1.666 i.e. t>1.666 or t<-1.666 (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 0.0198 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |t|=2.3822 > tc​=1.666, 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=0.0198, and since p=0.0198≤0.1, 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 71.4, at the 0.1 significance level.

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