Solutions to Practice Test Math 4753 Summer 005 This test is worth 00 points. Questions 5 are worth 4 points each. Circle the letter of the correct answer. Each question in Question 6 9 is worth the same number of points but different parts within the questions are not necessarily worth equal amounts of points. Any work for partial credit must be shown on this sheet. Mark answers in the spaces provided. No credit will be given if work is not shown. There is no partial credit for the multiple choice questions.. A confidence interval for a proportion indicates that the proportion is likely greater than one half if: a..5 is in the confidence interval. b..5 is below the lower confidence limit. c..5 is above the upper confidence limit. Comment: For a, it is possible for the proportion to equal.5 since it is in the interval. For c, the proportion is likely less than.5 since both limits of the confidence interval are below.5. So only b is correct.. A confidence interval for a mean is wider than another confidence interval for the same mean if: a. The n is larger for the first confidence interval. b. The α for the first confidence interval is larger. c. The y for the first confidence interval is larger than that for the second confidence interval but both have the same standard deviation s (and the same α and n). Comment: For a, a larger n gives a narrow confidence interval since it gives a smaller standard deviation. For c, the width of the confidence interval depends on α, the sample size and the standard deviation s. All these are the same for the two confidence intervals in c so the width is the same ( y doesn t affect the width of the confidence interval. For b, a larger α (say.05 instead of.0) means we are looking for a less precise confidence interval (say 95% instead of 99%) and since it is less precise it will be larger. So only b is correct.
3. A confidence interval for the ratio of two variances shows that one of the variances is greater than the other if. a. is in the confidence interval. b. Both limits of the confidence interval are less than. c. Both limits of the confidence interval are greater than. Comment: Fir a, if ins in the confidence interval then it is possible for the ratio of variances to equal and thus for the variances to be equal. For b, if both limits are less than then the ratio of variances is less than and the variance on bottom is larger. For C, if both limits are greater than then the ratio of variances is greater than and the variance on top is larger. So both b and c are correct which makes e the correct answer. 4. For a χ -distribution, we know: a. The distribution is symmetrical. b. χ α = χ α for the same degrees of freedom. c. The area under the χ pdf above χ α is the same as the area under the χ pdf below χ α for the same degree of freedom. Comments: All χ -distributions are asymmetrical so a is false. There is no relationship between the size of the number for χ α and χ α, even with the same degree of freedom, so b is false. On the other hand, for the same degree of freedom, ( ) = α = P χ > χ ( α ) so P χ > χ ( ) = α α = P χ < χ ( α ) P χ < χ α statement. So only c is correct. and c is a true 5. If the test statistic of a hypothesis test does not fall in the rejection region for that hypothesis test then: a. We likely have made a mistake in calculating the value of the test statistic. b. We accept H 0. c. We must continue to use H 0 as a working hypothesis. Comment: The fact that the test statistic is not in the rejection region tells us nothing about whether we have made a mistake so a is not true. We can never accept H 0 as a true statement no matter what we calculation so b is not true. However, if the test statistic is not in the rejection region we must continue to use H 0. So c is true.
6. Suppose that a random sample of 8 OU mechanical engineering majors had an average GPA of 3.54 with a standard deviation of.5 while a random sample of 9 Texas A&M mechanical engineering majors had an average GPA of 3.0 with a standard deviation of.40. a. Find a 98% confidence interval that will allow you to decide whether the variances are the same and, if not, which is greater. We want to compare variances and the only confidence interval that allows us to do that is the one for σ. So we want a confidence interval. We want it for σ. σ σ There is no difference for large or small samples or any other factor. So the form of the confidence interval is s s g F α,( ν, ν ) σ σ s s gf α,( ν, ν ) where ν = n and ν = n.we have s = (.5) = 0.065, s = (.40) = 0.6, n = 8, n = 9, ν = 8 = 7, ν = 9 = 8, and α =.0 since we want a 98% confidence interval. We need to find F α,( ν, ν ) = F.0,( 7,8) = F.0,( 7,8) and F α,( ν, ν ) = F.0,( 8,7) = F.0,(8,7). We need to turn to Table (for.0). We see there that F.0,( 7,8) = 6.8 and F.0,(8,7) = 6.84.
This gives us s s g F α,( ν, ν ) σ σ s s gf α,( ν, ν ).065.6 g 6.8 σ σ.065.6 g6.84 ( 0.39065) 6.8 σ σ ( 0.39065)6.84 0.063079 σ σ.6788 ( 0.063079,.6788) So the 98% confidence interval for the ratio of variances is about ( 0.06,.67). b. Based on the confidence interval in a, can we assume that the variance of the GPAs of mechanical engineering students at Texas A&M is greater than that of similar students at OU? Tell how you know. Notice that in the interval ( 0.06,.67), the lower confidence limit, 0.06, is less than and the upper confidence interval,.67, is greater than. That means that is in the interval, so we could have σ <, σ = or σ >. This means that we σ σ σ could have σ < σ, σ = σ or σ > σ. Therefore, from these data and at the 98% confidence level we cannot tell whether one variance is greater than the other. 7. 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. a. Find a 95 percent confidence interval to estimate the actual mean length of the population of rulers made on the production line. This is a CI for a population mean. The form for this CI is: s x t α,ν n < µ < x + t s α where ν = n.,ν n We have y =., s =.5, n = 49, and α =.05. This gives.000 = t.05,60 < t = t < t α.ν.05,48.05,40 =.0. We use the larger value to make the interval wider (and therefore take a more conservative approach. So we have:..0.5 49 < µ <. +.0.5 49. 0.44 < µ <. + 0.44.956 < µ <.44 b. Given this confidence interval, is it likely that the population mean is inches as it is supposed to be? Tell how you know. We cannot tell if the population mean is inches since is in the confidence interval.
8. 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). a. What is the null hypothesis? H 0 : µ = 3.0 b. What is the alternative hypothesis? H a : µ < 3.0 (one-sided test, "less than" c. What is the value of the test statistic? t = x µ 0 s n =.8 3.0.5 0.79 (small sample) d. What is the rejection region (with its numerical value)? t < t n,α = t 9,.05 =.79 e. What conclusion do you draw? Reject H 0 since -.79 < -.79. f. What does this mean in terms of the problem situation? We must reject our working hypothesis that the mean GPA is 3.0 or more and conclude that it is less than 3.0. 9. 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. a. What is the null hypothesis? H 0 : p p δ 0 = 0 (Population are the regular Channel 9 viewers) b. What is the alternative hypothesis? H a : p p > δ 0 = 0 c. What is the value of the test statistic? ˆp t = ˆp.68.34 = ˆp ˆq +.4533.5467 n n 00 + 00.34.06097 5.58
where p ˆ = x + y n + n = because δ 0 = 0 (see Equation 9.). 68 + 68 0.4533 and we use this form of the test statistic 00 +00 d. What is the rejection region (with its numerical value)? t > t α = t..85 with degrees of freedom e. What conclusion do you draw? Reject H 0 because 5.58 >.85. f. What does this mean in terms of the problem situation? This means that we can conclude that H a 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]).