Hypothesis Testing-Z-Test
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Here are the steps to solve this problem: 1. State the hypotheses: H0: μ = 100 H1: μ ≠ 100 2. The critical values are ±1.96 (two-tailed test, α=0.05) 3. Compute the test statistic: z = (140 - 100)/15/√40 = 20/15/2 = 4 4. The test statistic is in the critical region, so reject the null hypothesis. 5. There is strong evidence that the medication affected intelligence since the sample mean is much higher than the population mean.
Hypothesis Testing-Z-Test
- 1. OPENER Please find the critical value(s) for each situation and draw the appropriate figure, showing the critical region: a. Two-Tailed Test α=0.05 b. Right-Tailed Test α=0.02 c. Left-Tailed Test α=0.01
- 2. Homework Learning Gains Lesson 1 2 3 4 5 6 7 8 9 Gain 81.6 - - 92.0 77.5 Opener Closer Opener Closer Opener Closer Opener Closer Opener Closer Opener Closer Opener 81.4 94.0 81.9 90.0 - 65.0 - 90.0
- 3. Speaking of Homework / Grades • Check Edbox • Not everyone has handed in their homework calendars for Ch. 8 • Barring any discussed changes to your grade, what you see on Edbox is an accurate reflection of what will go home with your quarter grades.
- 4. Review • A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. • The numerical value obtained from a statistical test is called the test value.
- 5. Review • The critical value separates the critical region from the noncritical region. • The critical region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected.
- 6. Two Tailed v. Left/Right-Tailed Test Two-Tailed α=0.10 Non Critical Region Critical Region Critical Region α/2=0.05 α/2=0.05 CV=-1.65 CV=1.65 Critical Value Critical Value Left-Tailed Right-Tailed α=0.05 Non α=0.10 Non Critical Critical Region Region Critical Region Critical Region α=0.05 α=0.10 CV=-1.65 CV=1.28 Critical Value Critical Value
- 7. Five Steps in Hypothesis Testing 1. State the hypotheses and identify the claim. Make sure to use proper symbols. Please include proper units when given. 2. Find the critical value(s). Include a diagram that displays all the pertinent information. 3. Compute the test value. Include the proper formula. Round properly. Locate and place test value on the diagram. 4. Make the decision to reject or not to reject the null hypothesis. Use a complete sentence. 5. Summarize the result. Use complete sentence(s) that states the final conclusion clearly in the context of the problem. Use the proper vocabulary and statistical terms. Include the proper units if they are given.
- 8. Section 9-3 Z-Test for a Mean
- 9. When to Use the z Test • The z test is a statistical test for the mean of a population. It can be used when n> 30 or when the population is normally distributed and σ is known.
- 10. Formula for the z Test X z n
- 12. Example 9-3 • A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43, 260. At = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230. Non Rejection Region H0: µ≤$42,000 H1: µ>$42,000 (Claim)
- 13. Example 9-3 • A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43, 260. At = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230. Step 3: X 4 3, 2 6 0 4 2, 0 0 0 z z z 1 .3 2 5230 n 30 Step 4: Right-Tailed Do Not Reject the null hypothesis. Non Rejection Step 5: Region There is not enough evidence to support the claim that assistant Critical Region professors earn more on average than $42,000 a year CV=1.65 z 1 .3 2
- 14. Example 9-4 • A researcher claims that the average cost of men’s athletic shoes is less than $80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following cost (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researchers claim at = 0.10? Reference Page 351 for costs. Non Rejection Region H0: µ≥$80 H1: µ<$80 (Claim)
- 15. Example 9-4 • A researcher claims that the average cost of men’s athletic shoes is less than $80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following cost (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researchers claim at = 0.10? Step 3: X 75 80 z z z 1 .5 6 19.2 n 36 Step 4: Left-Tailed Reject the null hypothesis. Non Rejection Step 5: Region There is enough evidence to support the claim that the average cost of Critical men’s athletic shoes is less than $80 Region CV=-1.28 z 1 .5 6
- 16. Example 9-5 • The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24,672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is $25,226. At = 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $24,672? Assume SD is 3251.
- 17. Example 9-5 • The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is $24,672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is $25,226. At = 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from $24,672? Assume SD is 3251. Step 3: X 2 5, 2 2 6 2 4, 6 7 2 z z z 1 .0 1 3251 n 35 Step 4: Two-Tailed Do Not Reject the null hypothesis. Non Rejection Step 5: Region There is not enough evidence to support the claim that the average Critical Critical cost of rehabilitation at the Region Region particular hospital is different from CV=-2.58 CV=2.58 $24,672 z 1.01
- 18. Five Steps in Hypothesis Testing 1. State the hypotheses and identify the claim. Make sure to use proper symbols. Please include proper units when given. 2. Find the critical value(s). Include a diagram that displays all the pertinent information. 3. Compute the test value. Include the proper formula. Round properly. Locate and place test value on the diagram. 4. Make the decision to reject or not to reject the null hypothesis. Use a complete sentence. 5. Summarize the result. Use complete sentence(s) that states the final conclusion clearly in the context of the problem. Use the proper vocabulary and statistical terms. Include the proper units if they are given.
- 19. Your Moment of Zen
- 20. HOMEWORK Read Section 9-3 Pg. 358: 14-20all Assigned: 2012.04.11 Due: 2012.04.16 Please write this on your Calendar
- 21. CLOSER In the population, the average IQ is 100 with a standard deviation of 15. A team of scientists wants to test a new medication to see if it has an effect on intelligence (as measured by IQ). A sample of 40 participants who have taken the medication has a mean of 140. Did the medication affect intelligence? Use α=0.05.