Question

Here are the steps:
1). What are the hypotheses?
2). What is the rejection rule?
3). What is the test statistic?
4). What is your conclusion about the null hypothesis?
5). What is the conclusion in terms of the specific problem information?


The mean time using the present technology that Donald Co. uses to make widgets takes 10 minutes a widget. Donald Co. is experimenting with a new method, and conducted a sample by making 25 widgets with the new method. The sample using the new method obtained a mean of 9.2 minutes per widget, with a sample standard deviation of 1.5 minutes. At the 0.01 significance level, can we conclude that the mean time using the new method is less than 10 minutes?

Answer 1

One-Sample t-test

The sample mean is Xˉ=9.2, the sample standard deviation is s=1.5, and the sample size is n=25. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =10 Ha: μ <10 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=25-1=24. Therefore the critical value for this Left-tailed test is tc​=-2.4922. 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<-2.4922 (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 0.0067 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=-2.6667 < tc​=-2.4922, 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=0.0067, and since p=0.0067≤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 less than 10, at the 0.01 significance level. Therefore  at the 0.01 significance level, we can we conclude that the mean time using the new method is less than 10 minutes

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