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Introduction to Managerial StatisticsThis assignment consists

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Introduction to Managerial Statistics This assignment consists of four datasets 1. Clouds were randomly seeded or not with silver nitrate. Rainfall amounts were recorded from the clouds. The purpose of the experiment was to determine if cloud seeding increases rainfall. The following data is obtained: Seeded: 2745.6, 1697.8, 1656, 978, 703.4, 489.1, 430, 334.1, 302.8, 274.7, 274.7, 255, 242.5, 200.7, 198.6, 129.6, 119, 118.3, 115.3, 92.4, 40.6, 32.7, 31.4, 17.5, 7.7, 4.1 Unseeded: 1202.6, 830.1, 372.4, 345.5, 321.2, 244.3, 163, 147.8, 95, 87, 81.2, 68.5, 47.3, 41.1, 36.6, 29, 28.6, 26.3, 26.1, 24.4, 21.7, 17.3, 11.5, 4.9, 4.9, 1 a) For each data set, find the mean and standard deviation. b) While we have a before and after scenario, the treatments were not performed on the same subjects (clouds) so we could use a pooled (not paired) t-test to determine if the cloud seeding was effective. A t-test requires certain assumptions, including normality and constant variance. Construct a box-plot for the unseeded and seeded data. Based on the box-plots, does the data appear symmetric? Do the variances appear to be equal? c) Transformations are a common method of “fixing” datasets that do not meet required assumptions. Transforming a data set is applying an invertible, one to one transformation to the data, replacing the data point y with z=f(y). One of the most common transformations is the log transform. Apply the log transform to the data, replacing each data point y with z=ln(y). d) Construct a box-plot for the transformed data. Does the transformed data appear to be symmetric? Do the variances appear to be equal with the transformed data? e) Perform a pooled t-test on the transformed data to determine the effectiveness of cloud seeding. Choose the level of significance, and find the critical value for this level of significance. State the forma null and the alternative hypothesis. What is the p-value? Based on the t-test, what is your conclusion about the effect of cloud seeding on rainfall? 2. This dataset contains the labor force participation rate (LFPR) of women in 19 cities in the United States in each of two years (1968 and 1972). The data help to measure the growing presence of women in the labor force over this period. City 1972 1968 N.Y. .45 .42 L.A. .50 .50 Chicago .52 .52 Philadelphia .45 .45 Detroit .46 .43 San Francisco .55 .55 Boston .60 .45 Pitt. .49 .34 St. Louis .35 .45 Connecticut .55 .54 Wash., D.C. .52 .42 Cinn. .53 .51 Baltimore .57 .49 Newark .53 .54 Minn/St. Paul .59 .50 Buffalo .64 .58 Houston .50 .49 Patterson .57 .56 Dallas .64 .63 It may seem reasonable to compare LFPR rates in the two years with a pooled t-test since the United States did not change much from 1968 to 1972. However, the data are also naturally paired because the measurements were made in the same cities for each of the two years. We can compare each city in 1972 to its own value in 1968. a) Construct a histogram for both years. Be sure to use the same number of groups or bins for each year. Compare the two histograms. Do they appear the same or different? If you can plot each histogram on the same graph, it might make it easier to see similarities or differences. b) What is the sampling distribution for the mean? Would you use a normal or a t-distribution? Why? Find the 95% confidence intervals for the mean for each year (1968 and 1972). Based on the two confidence intervals, would you conclude the LFPR in 1968 and 1972 are the same or different? c) In order to perform a paired t-test, we need to make sure the data is approximately normal and the variances are the same. Construct and interpret box-plots for both years. d) Now perform a pooled t-test to test the hypothesis that the mean LFPR for 1968 and the mean LFPR for 1972 are the same, using significance level =0.05. State the null and the alternative hypothesis, and find the critical value. Find the p-value. What is your conclusion? e) Now perform a paired t-test on the data, using the same null and alternative hypothesis, and the same significance level as part d. State the null and the alternative hypothesis, and find the critical value. Find the p-value. What is your conclusion? How does this compare to the results from the pooled test? Explain the reasons for the differences, if any. 3. Results of a study of gas chromatography, a technique which is used to detect very small amounts of a substance. Five measurements were taken for each of four specimens containing different amounts of the substance. The amount of the substance in each specimen was determined before the experiment. The response variable is the output reading from the gas chromatograph. The purpose of the study is to calibrate the chromatograph by relating the actual amount of the substance to the chromatograph reading. Variable Names: 1. amount: amount of substance in the specimen (nanograms) 2. response: output reading from the gas chromatograph amount response 0.25 6.55 0.25 7.98 0.25 6.54 0.25 6.37 0.25 7.96 1.00 29.7 1.00 30.0 1.00 30.1 1.00 29.5 1.00 29.1 5.00 211 5.00 204 5.00 212 5.00 213 5.00 205 20.00 929 20.00 905 20.00 922 20.00 928 20.00 919 a) Perform a linear regression with amount as the independent variable and response as the dependent variable. What is the fitted equation? What are 95% confidence intervals for the regression coefficients? b) Perform a goodness of fit test, stating the null and alternative hypotheses. Interpret the results of the test. c) Plot the residuals and assess whether the variance is constant. Create a qq-normal plot and determine whether the errors appear normally distributed. d) Based on the goodness of fit test and the residual diagnostics, do you have a good model? Why or why not? 4. The data below consists of heights of singers in the NY Choral Society in 1979, self-reported, to the nearest inch. Voice parts in order from highest pitch to lowest pitch are Soprano, Alto, Tenor, Bass. The first two are female voices and the last two are male voices. One can examine how height varies across voice range, or make comparisons of sopranos and altos and separate comparisons of tenors and basses. There is some evidence of the shortest singers reporting greater heights, possibly to avoid standing in the front row in a concert. a) Perform an ANOVA, with height as the response and voice range as the factor, and interpret the results. b) Perform a hypothesis test to determine if female singers are shorter on average than male singers. Variable Names: 1. Soprano: Heights of sopranos (in inches) 2. Alto: ***** ***** altos (in inches) 3. Tenor: Heights of tenors (in inches) 4. Bass: Heights of basses (in inches) The Data: Soprano Alto Tenor Bass 64 65 69 72 62 62 72 70 66 68 71 72 65 67 66 69 60 67 76 73 61 63 74 71 65 67 71 72 66 66 66 68 65 63 68 68 63 72 67 71 67 62 70 66 65 61 65 68 62 66 72 71 65 64 70 73 68 60 68 73 65 61 73 70 63 66 66 68 65 66 68 70 62 66 67 75 65 62 64 68 66 70 71 62 65 70 65 64 74 63 63 70 65 65 75 66 69 75 65 61 69 62 66 72 65 65 71 66 61 70 65 63 71 61 64 68 65 67 70 66 66 75 65 68 72 62 66

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