Question

Isabella continues her hypothesis test, by finding the p-value to make a conclusion about the null hypothesis.

• H0:μ=48.4; Ha:μ≠48.4, which is a two-tailed test.
• α=0.1.
• z0=1.95

Which is the correct conclusion of Isabella's one-mean hypothesis test at the 10% significance level?

z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890

Use the above portion of the Standard Normal Table.

Answer 1

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =48.4 Ha: μ ≠48.4 This corresponds to a Two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2a) Critical Value Based on the information provided, the significance level is α=0.1, therefore the critical value for this Two-tailed test is Zc​=1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>1.6449 i.e. Z>1.6449 or Z<-1.6449 (3) Test Statistics The z-statistic is computed as follows: z = 1.95 (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 p =P(|Z|>1.95)=0.0512 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=1.95 > Zc​=1.6449, 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=0.0512, and since p=0.0512≤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 48.4, at the 0.1 significance level.