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Hypothesis Testing Examples: (1) The average production of peanuts in the state of Virginia is 3000 pounds per acre. A new plant food has been developed and is tested on 60 individual plots of land. The mean yield with the new plant food is 3120 pounds of peanuts per acre. The population standard deviation is known to be 578 pounds. At α = 0.05, can one conclude that the average production has increased? H 0 : 3000 H a : 3000 (one-tailed: right) 0.05 3120 3000 1.608 578 60 CV : z0.05 1.645 z Compare : 1.608 1.645 ( Inside CV) Conclusion: cannot reject H 0 (2) The average salary for public school teachers for a specific year was reported to be $39,385. A random sample of 50 public school teachers in a particular state had a mean of $41,680. The known standard deviation is $5975. Is there sufficient evidence at the 0.05 level to conclude that the mean salary differs from $39,985? H 0 : 39,385 H a : 39,385 (two-tailed: need to divide by 2) 0.05, / 2 .025 41, 680 39,385 2.72 5975 50 CV : z.025 1.96 z Compare : 2.72 1.96 ( Outside CV) Conclusion: reject H 0 . This suggests that the average salary has indeed changed. (3) A survey of 15 large cities finds that the average commute time one way is 25.4 minutes. A chamber of commerce executive feels that the commute in his city is less and wants to publicize this. He randomly selects 25 commuters and finds the average is 22.1 minutes with a sample standard deviation of 5.3 minutes. At α = 0.10, is he correct? First, note that we are given the sample standard deviation, and that he population standard deviation is unknown. So we need to use the t-distribution. H 0 : 25.4 H a : 25.4 (one-tailed, left) 0.1, df n 1 24 22.1 25.4 3.11 5.3 25 CV : t0.1 1.38 t Compare : 3.11 1.38 ( Outside CV) Conclusion: reject H 0 . This suggests that the average commute is indeed shorter. (4) The U.S. Bureau of Labor and Statistics reported that a person between the ages of 18 and 34 has had an average of 9.2 jobs. To see if this average is correct, a researcher selected a sample of 8 workers between the ages of 18 and 34 and asked how many different places they had worked. The results were as follows: 8 12 15 6 1 9 13 2 At α = 0.05 can it be concluded that the mean is 9.2? Put the data into L1 : STAT 1 run 1-var stats: STAT right 1 ENTER H 0 : 9.2 H a : 9.2 (two-tailed,divide by 2) 0.05, / 2 .025, df 8 1 7 8.25 9.2 .53 5.06 8 CV : t0.025 2.365 t Compare : .53 2.365 ( Inside CV) Conclusion: Cannot reject H 0 . (5) The average 1-ounce chocolate chip cookie contains 110 calories. A random sample of 15 different brands of 1-ounce chocolate chip cookies resulted in the following calorie amounts. At the α = 0.01 level, is there sufficient evidence that the average calorie content is greater than 110 calories? 100 160 125 100 150 150 160 140 185 135 125 120 155 110 145