SMAM 314 Practice Final Examination Winter 2003

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1 SMAM 314 Practice Final Examination Winter 2003 You may use your textbook, one page of notes and a calculator. Please hand in the notes with your exam. 1. Mark the following statements True T or False F A The mean always divides the data in half. That is, half of the data values exceed the mean and half of the data values are less than the mean. B. Like the correlation coefficient the values of the coefficient of determination (R 2 ) are between 1 and +1. C. Events A and B are independent and events A and B are mutually exclusive means the same thing. D. When the correlation coefficient is negative the slope will be positive. E. In a control chart points outside the 3 sigma limits require action to monitor the process. F The p value of a test of hypothesis is The null hypothesis should be rejected at α =.03. G A statistical test rejects H 0 at α =.10. It is possible that H 0 will be rejected at α =.01. H One hundred 95% confidence intervals for the mean are found after taking 100 samples of size 36 from a population with unknown true mean. About 95 of these confidence intervals contain the true mean. It is possible to tell which 5 intervals do not contain the true mean. I Ten companies have their production line employees take a course about on the job safety. The number of accidents in each plant is recorded for the three month period before and after the course is taught. This would be an example of paired data and a t test on the before after differences might be appropriate to evaluate the effectiveness of the course for reduction of accidents. J. A test or confidence interval that uses the standard normal distribution is appropriate when the sample size is small and the standard deviation is unknown. 1

2 2.The data in the table below concerns twelve residential properties. The prediction variable x is number of square feet of living space. The response variable y is the selling price of the house in thousands of dollars. Row x y Consider the scatter plot below. 2

3 A. Base your answers to the questions that follow on the scatter plot only (1) Does the data appear to be positively or negatively correlated.?explain your answer.) (1 point) (2) Would a linear regression equation appear to be a reasonable model? Explain.(1 point) Regression Analysis: y versus x The regression equation is y = x Predictor Coef SE Coef T P Constant x S = R-Sq = 85.4% R-Sq(adj) = 83.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs x y Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual Use the Minitab output above to answer the questions below. B.. What price would the regression line predict for a house with 2000 sq ft.? (3 points) C What is the residual when x =1600? (4 points) D. Predictor Coef SE Coef T P Constant x The second line of the above tests the hypothesis that for an equation y = a +bx H 0 b =0 vs. H 1 b 0 Based on the p value would you reject or not reject H 0 at α =.001? Explain. (2 points) E. Based on the scatter plot, the correlation coefficient, the above test of hypothesis and the percentage of variation accounted for comment on the appropriateness of the regression model. Defend your statements. (4 points)

4 3. A 10 foot beam is inspected by dividing it up into disjoint segments and inspecting each segment one at a time. The probability of a defect for a segment is 0.1L where L is the length of the segment. A. Suppose the beam is divided into 10 1 ft. segments. What is the probability that exactly three of the segments have defects? (4 points) B. Suppose the beam is divided into 100 equal segments. What is the probability of at least one defect? (4 points) 4. The weight of a plated bracket is normally distributed with mean ounces and standard deviation.1 ounce. What is the probability that a randomly selected bracket will meet specifications of 1.5 ±.15 ounces?(7 points) 5.The ages of US commercial aircraft have a mean of 13.0 years and a standard deviation of 10.5 years. The federal Aviation administration randomly selects 49 commercial aircraft for special stress tests. Let x represent the mean age of the sample. A. What is the mean and the standard deviation of x? (2 points) B. Find P(x > 16) (4 points) C. Let A be the age of an aircraft where P(x < A) =.90. Find A. (4 points) 6. In Gotham city the proportion of highway sections that need repair so that Batman may navigate his Batmobile safely in any given year has the continuous probability density function f(x) = 12x 2 (1 x) 0 < x < 1 0 elsewhere A. What is the probability that in a given year at least half of the highway sections need repair? (4 points) B. What is the mean and the standard deviation of the proportion of highway that needs repair? (6 points) 7. A random sample of nine stainless steel bars was obtained. The strength in pounds per square inch was measured. The sample values were 45,950 43,080 47,440 42,330 41,950 45,020 42,230 41,950 45,020 40,950 47,090 43,480 ( x = s = 2327) A. Find a 95% confidence interval for (1) The true mean strength of the population of all such bars. (4 points) (2) The true standard deviation of the population of all such bars.(4 points)

5 B. Basing your answer only on Part A above (no computations needed) would you reject H 0 when performing the following tests of hypothesis at α =.05. Explain. (4 points-2 each) (1) (2) H 0 µ = 44,000 H 1 µ 44,000 H 0 σ = 5000 H 1 σ Consider a production process for which on Monday a sample of 100 devices contained 12 that did not conform to specifications. Some adjustments were made in the production process. On Tuesday after the adjustments were made a sample of size 200 yielded 16 nonconforming devices. At α =.05 can it be concluded that the adjustments improved the process? Perform an appropriate test of hypothesis. (10 points) What is the p value? 9. The following random samples are measurements of the heat-producing capacity in millions of calories per ton from two coal mines. Mine Mine A determination is to be made whether there is a difference between the mean heat producing capacity for the two mines. Use the Minitab data and your knowledge of Statistical Inference to Answer the following questions. No computations are needed. Two-sample T for C4 C3 N Mean StDev SE Mean Difference = mu (1) - mu (2) Estimate for difference: % CI for difference: (133.4, 446.6) T-Test of difference = 0 (vs not =): T-Value = 4.19 P-Value = DF = 9 Both use Pooled StDev = 114 A. What is the null and the alternative hypothesis? (2 points) B. Is this a one or a two tailed test? Explain. (2 points) C. Based on the test statistic and the p value would you reject H 0 and conclude at α =.01 that there is a significant difference in the heat producing capacity of the two mines? Explain your answer. (2 points)

6 D. Interpret the 95% confidence interval. (2 points) 6 E. Based on the graphs and information above (1) Does the assumption of equal standard deviations seem reasonable? Explain. (2 points) (2) Do the boxplots support the conclusion of the test of hypothesis? Explain.(2points)

7 F. Based on the normal probability plots above does the assumption that the data comes from normal populations seem reasonable? Explain. (2 points) 7

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SMAM 314 Exam 42 Name

SMAM 314 Exam 42 Name Mark the following statements True (T) or False (F) (10 points) 1. F A. The line that best fits points whose X and Y values are negatively correlated should have a positive slope.