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Ch. 9 examples

Ch. 9 examples. Summary of Hypothesis test steps. Null hypothesis H 0 , alternative hypothesis H 1 , and preset α Test statistic and sampling distribution P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H 0 and say that the data are significant at level α

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Ch. 9 examples

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  1. Ch. 9 examples

  2. Summary of Hypothesis test steps • Null hypothesis H0, alternative hypothesis H1, and preset α • Test statistic and sampling distribution • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  3. Should you use a 2 tail, or a right, or left tail test? • Test whether the average in the bag of numbers is or isn’t 100. • Test if a drug had any effect on heartrate. • Test if a tutor helped the class do better on the next test. • Test if a drug improved elevated cholesterol.

  4. Type I and Type II error

  5. Probabilities associated with error

  6. Example #1- numbers in a bag • Recall that I claimed that my bag of numbers had a mean µ = 100 and a standard deviation =21.9. Test this hypothesis if your sample size n= 20 and your sample mean x-bar was 90.

  7. Ex #1- Hypothesis Test for numbers in a bag • H0: µ = 100 H1: µ ≠ 100 α = 0.05 • Z = = • P-value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  8. Ex #2– new sample mean for numbers in a bag • If the sample mean is 95, redo the test: • H0: µ = 100 H1: µ ≠ 100 α = 0.05 • Z = = • P-value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  9. Ex #3: Left tail test- cholesterol • A group has a mean cholesterol of 220. The data is normally distributed with σ= 15 • After a new drug is used, test the claim that it lowers cholesterol. • Data: n=30, sample mean= 214.

  10. Ex #3- cholesterol- test • H0: µ 220 (fill in the correct hypotheses here) H1: µ 220 α = 0.05 • Z = = • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  11. Ex #4- right tail- tutor • Scores in a MATH117 class have been normally distributed, with a mean of 60 all semester. The teacher believes that a tutor would help. After a few weeks with the tutor, a sample of 35 students’ scores is taken. The sample mean is now 62. Assume a population standard deviation of 5. Has the tutor had a positive effect?

  12. Ex #4: tutor • H0: µ 60 (fill in the correct hypotheses here) H1: µ 60 α = 0.05 • Z = = • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  13. 9.2– t tests • Just like with confidence intervals, if we do not know the population standard deviation, we • substitute it with s (the sample standard deviation) and • Run a t test instead of a z test

  14. Ex #5– t test – placement scores • The placement director states that the average placement score is 75. Based on the following data, test this claim. • Data: 42 88 99 51 57 78 92 46 57

  15. Ex #5 t test – placement scores • H0: µ 75 fill in the correct hypothesis here H1: µ 75 α = 0.05 • t = = • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  16. Ex #6- placement scores • The head of the tutoring department claims that the average placement score is below 80. Based on the following data, test this claim. • Data: 42 88 99 51 57 78 92 46 57

  17. Ex #6– t example • H0: µ 80 (fill in the correct hypotheses here) H1: µ 80 α = 0.05 • t = = • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  18. Ex #7- salaries– t • A national study shows that nurses earn $40,000. A career director claims that salaries in her town are higher than the national average. A sample provides the following data: • 41,000 42,500 39,000 39,999 • 43,000 43,550 44,200

  19. Ex #7- salaries • H0: µ 40000 (fill in the correct hypotheses here) H1: µ 40000 α = 0.05 • t = = • P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  20. Traditional Critical Value Approach • Redo Example #1 • Recall that I claimed that my bag of numbers had a mean µ = 100 and a standard deviation =21.9. Test this hypothesis if your sample size n= 20 and your sample mean x-bar was 90.

  21. Ex#1 redone with CV • H0: µ = 100 H1: µ ≠ 100 α = 0.05 • Z = = • CV 4. Test conclusion If p-value ≤ α, then test value is in RR, and we reject H0 and say that the data are significant at level α If p-value > α, then test value is not in RR, and we do not reject H0 5. Interpretation of test results

  22. Ex #3 redone with CV • A group has a mean cholesterol of 220. The data is normally distributed with σ= 15 • After a new drug is used, test the claim that it lowers cholesterol. • Data: n=30, sample mean= 214.

  23. Ex#3- 5 steps- done with CV • H0: µ = 220 H1: µ 220 (fill in) α = 0.05 • Z = = • CV 4. Test conclusion If p-value ≤ α, then test value is in RR, and we reject H0 and say that the data are significant at level α If p-value > α, then test value is not in RR, and we do not reject H0 5. Interpretation of test results

  24. 9.3 Testing Proportion p • Recall confidence intervals for p: • ± z

  25. Hypothesis tests for proportions • Null hypothesis H0, alternative hypothesis H1, and preset α 2. Test statistic and sampling distribution • P-value and/or critical value z= = 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  26. Ex #8- proportion who like job The HR director at a large corporation estimates that 75% of employees enjoy their jobs. From a sample of 200 people, 142 answer that they do. Test the HR director’s claim.

  27. Ex #8 • Null hypothesis H0, alternative hypothesis H1, and preset α H0: p=.75 (fill in hypothesis) H1: p α = • Test statistic and sampling distribution Z = = 3. P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

  28. Ex #9 Previous studies show that 29% of eligible voters vote in the mid-terms. News pundits estimate that turnout will be lower than usual. A random sample of 800 adults reveals that 200 planned to vote in the mid-term elections. At the 1% level, test the news pundits’ predictions.

  29. Ex #9 • Null hypothesis H0, alternative hypothesis H1, and preset α H0: p (fill in hypothesis) H1: p α = • Test statistic and sampling distribution Z = = 3. P-value and/or critical value 4. Test conclusion If p-value ≤ α, we reject H0 and say that the data are significant at level α If p-value > α, we do not reject H0 5. Interpretation of test results

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