three examples of testing a single mean

Three Examples of Testing a Single Mean

Module 17

Three Examples of Significance Tests

• Example 1: Arsenic– “A factory that discharges waste water into the sewage system is

required to monitor the arsenic levels in its waste water and report the results to the Environmental Protection Agency (EPA) at regular intervals. Sixty beakers of waste water from the discharge are obtained at randomly chosen times during a certain month. The measurement of arsenic is in nanograms per liter for each beaker of water obtained.” (from Graybill, Iyer and Burdick, Applied Statistics, 1998).

– Suppose the EPA wants to test if the average arsenic level exceeds 30 nanograms per liter at the 0.05 level of significance.


• Arsenic Example– Information given: 37.6 56.7 5.1 3.7 3.5 15.7

20.7 81.3 37.5 15.4 10.6 8.3

23.2 9.5 7.9 21.1 40.6 35

19.4 38.8 20.9 8.6 59.2 6.2

24 33.8 21.6 15.3 6.6 87.7

4.8 10.7 182.2 17.6 15.3 37.6

152 63.5 46.9 17.4 17.4 26.1

21.5 3.2 45.2 12 128.5 23.5

24.1 36.2 48.9 16.5 24.1 33.2

25.6 33.6 12.2 9.9 14.5 30

83.31x

Assume it is known that = 34.

Sample size: n = 60.


• Arsenic Example– 1. State the null hypothesis:

– 2. State the alternative hypothesis:

– 3. State the level of significance


• Arsenic Example– 4. Calculate the test statistic.

– 5. Find the P-value.

n

xz

0


• Arsenic Example– 6. Do we reject or fail to reject H0 based on the P-

value?

– 7. State the conclusion.


• Example 2: Cereal– “An operator of cereal-packaging machine monitors the

net weights of the packaged boxes by periodically weighing random samples of boxes…One condition required for the proper operation of the machine is that the mean is 453 grams.” (from Graybill, Iyer and Burdick, Applied Statistics, 1998).

– The operator wishes to see if a random sample of 50 boxes gives evidence that the mean is different than 453 grams.


• Cereal Example– Information given:

8.449 ofmean a yields 50 of sample random The x


• Cereal Example– 1. State the null hypothesis:




• Cereal Example– 4. Calculate the test statistic.


n

xz

0


• Cereal Example– 6. Do we reject or fail to reject H0 based on the P-

value?



• Example 3: Salary– A state representative wants to know the mean salary for the

faculty at a state university in her district. She recently read a newspaper article that stated the mean salary for faculty in state universities is $51,000. At present, the representative does not plan to support any bill that would increase faculty salaries at the university. However, if she can be convinced that the mean salary of the faculty is less than $51,000, she will support such a bill. The representative asked an intern to select a simple random sample of 25 faculty from the list of faculty in the university catalog and use these salaries to conduct a statistical test using = 0.025 (adapted from Graybill, Iyer and Burdick, Applied Statistics, 1998).


• Salary Example– Information given:

741,48$ ofmean a yields 25 of sample random The x


• Salary Example– 1. State the null hypothesis:




• Salary Example– 4. Calculate the test statistic.


n

xz

0


• Salary Example– 6. Do we reject or fail to reject H0 based on the P-

value?


three examples of testing a single mean

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