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Tests of Hypotheses: z-test and t-test 0801-HypothesisTests.doc

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TESTS OF HYPOTHESES

As was mentioned earlier, sometimes we cannot survey or test all persons or objects; therefore, we have to take a sample. From the results of analysis from the sample data, we can predict the results from the population. Some questions that one may want to answer are 1. Are unmarried workers more likely to be absent from work than married workers? 2. Are the sixth graders in a certain school significantly less skilled in their mathematical abilities than the average student in the district? 3. In Fall 1996, did students in Math 163-01 score the same on the exam as students in Math 163-02? ...view middle of the document...

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2. Select the appropriate test statistic and level of significance. When testing a hypothesis of a proportion, we use the z-statistic or z-test and the formula ˆ p− p z= pq n

When testing a hypothesis of a mean, we use the z-statistic or we use the t-statistic according to the following conditions.

If the population standard deviation, σ, is known and either the data is normally distributed or the sample size n > 30, we use the normal distribution (z-statistic). When the population standard deviation, σ, is unknown and either the data is normally distributed or the sample size is greater than 30 (n > 30), we use the t-distribution (t-statistic).

A traditional guideline for choosing the level of significance is as follows: (a) the 0.10 level for political polling, (b) the 0.05 level for consumer research projects, and (c) the 0.01 level for quality assurance work.

NOES

3. State the decision rules. The decision rules state the conditions under which the null hypothesis will be accepted or rejected. The critical value for the test-statistic is determined by the level of significance. The critical value is the value that divides the non-reject region from the reject region. 4. Compute the appropriate test statistic and make the decision. When we use the z-statistic, we use the formula x−µ z=

σ

n

When we use the t-statistic, we use the formula x−µ t= s n

Compare the computed test statistic with critical value. If the computed value is within the rejection region(s), we reject the null hypothesis; otherwise, we do not reject the null hypothesis. 5. Interpret the decision. Based on the decision in Step 4, we state a conclusion in the context of the original problem.

Tests of Hypotheses: z-test and t-test 0801-HypothesisTests.doc

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EXAMPLE: The average score of all sixth graders in school District A on a math aptitude exam is 75 with a standard deviation of 8.1. A random sample of 100 students in one school was taken. The mean score of these 100 students was 71. Does this indicate that the students of this school are significantly less skilled in their mathematical abilities than the average student in the district? (Use a 5% level of significance.) SOLUTION: In this problem, we know the mean and standard deviation for the population, µ = 75 and σ = 8.1 (all sixth graders in district A). We also know the mean for the sample of 100 students in a certain school in district A. Thus, we are testing the sample mean against the population mean with a population standard deviation (σ is known). This is a large sample because n > 30 which is usually used to determine whether to use large or small sample techniques. Since σ is known and n > 30, we use the z-test that is...

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