Solutions to Worksheet on Hypothesis Tests

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1 s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality control specialist responsible for the production line decides to do a hypothesis test at the 90 percent significance level to determine whether the production line is really producing rulers that are inches long or not. : µ = : µ z = x µ 0 s n. =.5 =.4 49 z > z α = z.05 =.645 Do not reject since.4 is not greater than.645. We cannot conclude that the mean length of the population of rulers from the production line is not equal to. We should continue with our working assumption that it is equal to.. A sample of 0 OU freshmen had a mean GPA of.8 over all their courses taken in their first semester at OU. This had a variance of.5. Perform a hypothesis test at the 95 percent level to determine if the first semester GPA of all OU freshmen is less than a B (3.0). : µ = 3.0 : µ < 3.0 (one-sided test, "less than" t = x µ 0 s n = (small sample) t < t n,α = t 9,.05 =.79

2 Reject since -.79 < We must reject our working hypothesis that the mean GPA is 3.0 or more and conclude that it is less than A researcher decides to look at the variance of the production line in Problem She decides to do a hypothesis test at the 90 percent significance level to determine if the variance is actually less than.5. : σ =.5 : σ <.5 χ = ( n )s = ( 49 )(.5) = 48 σ 0.5 χ < χ n, α = χ 48, We cannot reject since 48 is not less than 9.05 We must continue with our working assumption that the true variance is.5 or more and cannot assume that it is less than.5.

3 4. A math teacher for Math 4753 wants to determine if a new book proposed for the course is better than the old book that was used before. The math teacher decides that the two classes might have different math ability coming into the course. He gives a test at the beginning of the semester to measure the math ability the students have when they come into the course. He uses this test to create a matched pair design for 30 students from each class. The scores he gets are as follows: Student Pair New Book Class Old Book Class Perform a hypothesis test at the 95 percent level for the matched pairs design in Problem 8 to see if the scores for the new book are higher than those for the old book. : µ D = µ µ = 0 ( µ is the mean for the new book) : µ D = µ µ > 0

4 First we need to find the differences, their mean and their variance: Student Pair New Book Class Old Book Class Differences d = -0. s d = The value of the test statistic is then t = d δ 0 s d n t > t n,α = t 9,.05 =.699 We cannot reject since is not greater than.699. =

5 We must continue to use the working assumption that the means are not different and cannot conclude that the means for the new book are higher than for the old book. 5. Five engineering students and 5 computer science students are arranged in a matched pair design on the basis of their GPA prior to the last semester of their senior year. Their GPAs for their last semester are as follows: Pair Number Engineering Major CS Major Perform a 90 percent hypothesis test to determine whether the means for CS majors are less than the means for Engineering majors. : µ D = µ µ = 0 (This is Engineers minus CS majors) : µ D = µ µ > 0 (Since this is Engineering majors minus CS majors and we are interested in whether the means for CS majors are less, we can answer the question by seeing if the Engineering majors are more and that's what we are doing since we have already found the differences as Engineering majors minus CS majors and not the other way around.) First we need to find the differences, their mean and their variance: So t = d δ 0 s d n Pair Number Engineering CS Major Differences Major d = 0.34 s d =.0443 = t > t n,α = t 4,. =.533

6 We cannot reject since.709 is not greater than.533. We must continue to assume that there is no difference between the grades for the two majors and cannot conclude that CS majors have lower grades. 6. The teacher in Problem 5 is still not sure about the differences in Problem 5 between CS and Engineering majors. He takes independent samples of size 0 from each major at random. He does not use their previous GPA to make matched pairs. The 0 Engineering majors have a mean last-semester GPA of.89 with a variance of.5. The 0 CS majors have a mean last semester GPA of.66 with a variance of.3. He decides to do a hypothesis test to see if the GPAs are different for the two populations. He does the test at the 90 percent level. : µ µ = 0 : µ µ 0 t = x y δ s + = (.7) n n where s = ( n )s + ( n )s = ( 0 )(.5) + ( 0 ) (.3) n + n 0 +0 t > t n +n,α = t 8,.05 =.734 We cannot reject because.47 is not greater than.734. = = 0.7 We cannot conclude that the two means are different since we cannot reject the null hypothesis. 7. Suppose in the situation in Problem 6, the teacher had the GPAs for a sample of 0 Engineering majors but only 8 CS majors but otherwise the data were the same. Suppose he decided to do a 90 percent hypothesis test for the difference in GPAs for the two populations of majors. : µ µ = 0

7 : µ µ 0 t = x y δ s + s = (.5) n n 0 + ( ) t > t ν, α = t 5,.05 =.753 s + s n where ν = n = s s n n + n n (.5) ( ) (.5) (.3) We cannot reject because.08 is not greater than.753. ( ).0365 = We cannot conclude that the means are different and must continue to assume they are equal. 8. The math teacher for Math 4753 wants to determine if the new book proposed for the course is better than the old book that was used before. He teaches two classes that are each a random sample of 4 people. The class using the new book got an average of 88 on the final exam given by the math teacher and had a standard deviation of 3 points. The class using the old book had an average of 90 with a standard deviation of points. Suppose the teacher decides to do a hypothesis test at the 98 percent level to see if the variances for Engineering majors is lower than the variances for CS majors. : σ σ : σ >σ F = s s = ( 3) ( ) =.5

8 F > f = f n, n,α 4,4,.0. using the values for 40,40 rather than for 4,4. We can reject since.5 is greater than.. We conclude that the variance for the new books is higher than the variance for the old books. 9. A survey of 4000 people in the US finds that 856 of them believe that daily weather reports are totally useless because meteorology is not really a science. Given this data perform a 95 percent hypothesis test to see if more than half of the people in the US believe that weather reports are useless. : p.5 : p >.5 p z = ˆ p =.4 p 0 q n 4000 z > z α = z We can reject since 7.09 >.645. We can conclude that it is likely (at the 95% confidence level) that more than half of the people in the US believe that weather reports are useless. 0. A survey of 00 regular viewers of Channel 5 in Oklahoma City show that 68 believe that Gary England is God. A survey of 00 regular viewers of Channel 9 in Oklahoma City show that 68 of them also believe that Gary England is God. Given these data perform a 90 percent hypothesis to determine if the proportion of Channel 9 viewers who believe that Gary England is God is greater than the proportion of Channel 5 viewers who believe it. : p p δ 0 = 0 (Population are the regular Channel 9 viewers) : p p > δ 0 = 0

9 p ˆ z = p ˆ p ˆ q ˆ + = n n where p ˆ = x + y = and we use this form of the test statistic because n + n δ 0 = 0 (see Equation 9.). z > z α = z..85 Reject because 5.58 >.85. This means that we can conclude that H is true and that p p > 0 or, in other words, p > p (a greater proportion of regular Channel 9 viewers believe that Gary England is God than the proportion of regular Channel 5 viewers that believe that Gary England is God. [Oh, please, Gary, forgive the Channel 5 viewers for they know not what they do]).

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