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HYPOTHESIS TESTING: ONE SAMPLE Z AND T TESTS If the sample size is small and σ is not known, then the t-statistic is used instead of Z. How do I know what test to use? Use Z-statistic if: Use t-statistic if: If the population standard deviation σ is known or the sample is more than 30. If the population standard deviation ( σ )is unknown and the sample is less than 30. Comparison of Z-statistic and t-statistic formulas Z-statistic T-statistic Use the t-statistic for testing a sample mean against a population mean if σ is unknown and n<30. When using the t-statistic, you need to find the critical value that corresponds to n-1 degrees of freedom. Distributions of the t-statistic symmetrical and bell-shaped but are flatter and more spread out (greater variability). T-distribution is the same as normal distribution when sample size is infinite The critical value for the t-statistic uses the t-Distribution Table. 1. Decide whether you’re doing a one or two-tailed test. (directional vs. non-directional) 2. Select your alpha level. 3. Find the row that corresponds to n-1 degrees of freedom. Table C: Percentile points of t- Distribution (p.621) Example: 95% confidence, two-tailed test, n=14 Critical value is 2.160
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What are degrees of freedom? The number in a sample that are free to vary. The sample mean places a restriction on the value of one score in the sample. Therefore, there are n-1 values that are free to vary. We have n-1 degrees of freedom with the t statistic. Review Questions 1. As the value for df gets smaller, the t distribution resembles a normal distribution more and more. (true or false?) 2. As the value for df gets larger, s provides a better estimate of σ. (true or false?) 3. For df = 10, what t values are associated with: a. The top 1% of the t distribution b. The bottom 5% of the t distribution c. A two-tailed distribution for 99% confidence. Example 1 The average writing score on a test in the population is 368. A random sample of 25 students took the test with a sample mean of 372.5 with a sample standard deviation of 15. Do the students have a different mean than the population? Test at the 0.05 level. 1. State the null and alternative hypotheses. 2. Select a level of significance (i.e. 0.10, 0.05, 0.01) 3. Identify the test statistic 4. Formulate a decision rule 5. Take a sample and arrive at decision: a) Calculate test statistic b) Compare it to our critical value(s). c) Make a decision. Example 3: An insurance company reports the mean cost to process a claim is $60. An industry comparison showed this amount to be larger than most other insurance companies, so the company instituted cost-cutting measures. To evaluate the effect of the cost-cutting measures, the Supervisor of the Claims Department selected a random sample of 26 claims processed last month.
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