Question

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

A shipping firm suspects that the mean life of a certain brand of tire used by its trucks is less than 33,000 miles. To check the claim, the firm randomly selects and tests 18 of these tires and gets a mean lifetime of 32,250 miles with a standard deviation of 1200 miles. At α = 0.05, test the shipping firm's claim.


a) State the Null and alternative hypotheses   


b) Find the test statistic . (1 mark)  


c) Fine the critical value. ( 1 mark)


d) Draw a final conclusion that addresses the claim ( 2 marks)

Answer 1

One-Sample t-test

The sample mean is Xˉ=32250, the sample standard deviation is s=1200, and the sample size is n=18. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =33000 Ha: μ <33000 This corresponds to a Left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. (2)Test Statistics The t-statistic is computed as follows: (3) Critical Value Based on the information provided, the significance level is α=0.05, and the degree of freedom is n-1=18-1=17. Therefore the critical value for this Left-tailed test is tc​=-1.7396. This can be found by either using excel or the t distribution table. Rejection Region The rejection region for this Left-tailed test is t<-1.7396 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 0.0084 (4) The Decision about the null hypothesis (a) Using the traditional method Since it is observed that t=-2.6517 < tc​=-1.7396, 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=0.0084, and since p=0.0084≤0.05, it is concluded that the null hypothesis is rejected. Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 33000, at the 0.05 significance level.

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