Question

Our environment is very sensitive to the amount of ozone in the upper atmosphere. The level of ozone normally found is 4.8 parts/million (ppm). A researcher believes that the current ozone level is at an excess level. The mean of 26 samples is 5.0 ppm with a standard deviation of 1.2. Does the data support the claim at the 0.025 level? Assume the population distribution is approximately normal.

Step 1 of 5:

State the null and alternative hypotheses.

Step 2 of 5:

Find the value of the test statistic. Round your answer to three decimal places.

Step 3 of 5:

Specify if the test is one-tailed or two-tailed.

Step 4 of 5:

Determine the decision rule for rejecting the null hypothesis. Round your answer to three decimal places.

Step 5 of 5:

Make the decision to reject or fail to reject the null hypothesis.

Answer 1

One-Sample t-test

The sample mean is Xˉ=5, the sample standard deviation is s=1.2, and the sample size is n=26. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =4.8 Ha: μ >4.8 This corresponds to a Right-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: 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 0.2017 (3) One-tailed (4) Critical Value Based on the information provided, the significance level is α=0.025, and the degree of freedom is n-1=26-1=25. Therefore the critical value for this Right-tailed test is tc​=2.0595. This can be found by either using excel or the t distribution table. Rejection Region The rejection region for this Right-tailed test is t>2.0595 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=0.8498 < tc​=2.0595, 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=0.2017, and since p=0.2017>0.025, it is concluded that the null hypothesis is Not rejected. 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 4.8, at the 0.025 significance level.