Name__Answers___________ STA 6166 Exam #2 Spring 2000 1. In 1992, 100 potato plants were randomly chosen from a field and 25 were found to be infected with a certain virus. In 1993, another 100 plants were obtained from the field and 20 were infected. Construct a statistical test of whether the infection rate decreased from 1992 to 1993. Define all terms and symbols. (5) a. Ho: Ha: (5) b. Test statistic: (5) c. Conclusion: 2. In an experiment to evaluate the effectiveness of a device that is supposed to increase gasoline mileage in automobiles, 10 cars were run over a prescribed course both with and without the device, in random order. Miles per gallon (mpg) were: Car 1 2 3 4 5 6 7 8 9 10 mean W 21.1 30.6 29.8 27.3 27.7 33.1 20.8 28.2 29.1 18.8 26.65 W/O 20.2 29.5 30.3 26.1 26.3 32.4 21.3 27.8 28.5 19.1 26.15 (5) Construct a 95% confidence interval for the increase in mpg due to using the device: 0.50, 0.71, / 0.22yssn== = t.025,9=2.26 95% CI = 0.50 ± 2.26(0.22) = 0.50 ± 0.50 (5) Interpret the confidence interval: We are 95% confident that μd is in the interval (.50 - .50, .50 + .50)3. Clay content was measured in ten soil samples from one area and in twelve samples from another area. Data produced means and variances as follows: Area 1 Area 2 n 10 12 mean 9.9 8.4 variance 3.1 3.6 Perform a statistical test of whether the clay content in Area 1 is greater than that of Area 2. (5) a: Ho: μ1 = μ2 Ha: μ1 > μ2 (5) b. Test statistic: 212( ) / (1/10 1/12) (9.9 8.4) / 0.79 1.9ptyy s=− + = − = (5) c. You should have a value of the test statistic in b. approximately equal to 1.9. What is the approximate level of significance (p-value)? Df =20 t.05 = 1.724, t.025 = 2.131 .025 < p < .05 (5) d. Conclusion: Statistical evidence that μ1 > μ2 (p < .05) (5) e. What assumptions of the data must be true in order for your inference to be valid? Normally distributed data Independent samples Equal variances4. Lawn fertilizer is prepared by mixing sand with sources of nitrogen (N) and other elements and minerals. In preparing a batch of fertilizer which will be labeled as containing 10% N, specifications require that the standard deviation σ of the per cent N be no larger than 1%. Nine samples from a batch yielded the following % N measurements: 8.5 11.2 9.4 9.8 10.7 10.8 9.5 11.3 8.6 (mean=10.0, std. dev.=1.07) (5) a. Construct a 95% confidence interval for the percent nitrogen content of the batch. .025,8/ 1.07 / 3 0.36±t / = 10.0 ±2.32(0.36)snysn== (5) b. Discuss whether you believe the batch is in compliance: The data indicate (but not prove) out of compliance because std is too large. (5) c. State Ho and Ha that would be appropriate for this problem: Ho: H211σ≤a: 21 1σ> 5. Conduct a statistical test for the equality of variances in problem 3. (5) a: Ho: 2122σσ= Ha: 2212σσ≠ (5) b. Test statistic: F’ = 3.6/3.1 =1.1 F.25,11,9 = 1.59 inplies p>.2 (5) c. Conclusion: No statistical evidence that 2122σσ≠ (p>.2) 6. You are going to run another experiment like the one in problem 2, but with a different device. How many cars would be needed to obtain an estimate of the improvement in miles per gallon with a margin of error of 0.2 mpg? Assume σd = 0.7. (5) 224(.7) /(.2) 12.25n ==6. Short answer and true-false (2 pts each): a. Define Type I error _probability of rejecting true null hypothesis b. Define Type II error _probability of not rejecting false null hypothesis c. Increasing the sample size would increase the value of β (T-F) __F__. d. Power of a test is equal to 1-α, where α=probability of type I error (T-F)__F__. e. The p-value of a statistical test is the probability that H0 is true (T-F)__F__. f. The critical value z.025 for a standard normal distribution is less than the critical value t.025 for a t distribution, regardless of the degrees of freedom (T-F)
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