Question

5. A researcher suspects that the mean pollutant level in a region is higher than the state average of 4.3 grams/cubic liter. Data are collected; from 56 samples, the mean is found to be 5.4, with a standard deviation of 5.0. Test the null hypothesis that the true mean in the region is 4.3, against the alternative that it is higher. State the null and alternative hypotheses, give the critical value, find the test statistic, make a decision, and give the p-value. (5)

Answer 1

Assuming the level of significance to be 5%

The sample mean is Xˉ=5.4, the sample standard deviation is s=5, and the sample size is n=56. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =4.3 Ha: μ >4.3 This corresponds to a Right-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=0.05, and the degree of freedom is n-1=56-1=55. Therefore the critical value for this Right-tailed test is tc​=1.673. This can be found by either using excel or the t distribution table. (2b) Rejection Region The rejection region for this Right-tailed test is t>1.673 (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 0.0527 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=1.6463 < tc​=1.673, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0527, and since p=0.0527>0.05, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population mean μ is greater than 4.3, at the 0.05 significance level.