Z TEST OF HYPOTHESIS ABOUT mu

TEST OF HYPOTHESIS ABOUT m

USING A Z-SCORE

A nutritionist believes that 12-ounce boxes of cereal should contain an average of 1.2 ounces of bran. She suspects that a popular cereal has a different mean bran content. She carefully analyzes the contents of a random sample of twenty 12-ounce boxes of the cereal and finds that the mean bran content is 1.16 ounces. It is known that the standard deviation of the bran content of all such boxes of cereal is 0.08.

A. Do the data provide sufficient evidence to conclude that the mean bran content of all 12-ounce boxes of this cereal differs from 1.2 ounces? Use a .05 level of significance.

B. State when this procedure is valid.

SOLUTION

A. The parameter of interest is m, the mean bran content of all 12-ounce boxes of this cereal. It is desired to test a hypothesis about m.

Decision Rule: Accept H_a if the calculated p-value < .05.

p-value = 2(.5 - .4875) = .0250

Interpretation: At a .05 level of significance it can be concluded that the mean bran content of all 12-ounce boxes of this cereal is less than 1.2 ounces.

A graph of the appropriate z-distribution and the resulting p-value is below.

B. Since a random sample of n = 20 boxes of cereal was observed, and n = 20 < 30, the population of all bran contents must be normally distributed for this procedure to be valid. If a large random sample had been obtained, the population would not need to have a normal distribution.

COMMENTS ABOUT THE SOLUTION

The value of the population standard deviation, s = 0.08, is given, so a "z-test" was used.

The following comments will apply to all the test of hypotheses problems you solve in this course.

Different instructors may require slightly different formats for presenting the solution to a test of hypothesis problem. All should require the following six components:

State the null hypothesis, H₀; State the alternative hypothesis, H_a; Give the decision rule; Give the formula for the test statistic; Calculate the value of the test statistic and the p-value; Interpret the results of your calculations in terms of the problem. This will include the variable, units, etc. of the problem.

Whether it is required or not it is a good idea to begin by defining the parameter of interest. Also summarize the information provided in the problem. This helps identify what type of problem is to be solved.

When asked, be able to state when the procedure is valid. Even if you are not asked, you should practice stating this. Again, this helps to identify what procedures are used to solve various problems.

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