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E 270 Homework 8 FINAL Spring 2013

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Question;NAME: COMPREHENSIVE HOMEWORK PROBLEMS 1 In thinking about doing statistical analysis, the sample mean should be interpreted as: a a constant value that is equal to the population mean. b a constant value that is approximately equal to the population mean. c a random variable that is approximately equal to the population mean when sampling is done without replacement. d a random variable that is approximately equal to the population mean if n > 30 and when sampling is done without replacement. e a random variable that when averaged across many samples is approximately equal to the population mean. 2 Which of the following are random? a x? after a sample is taken b x? before a sample is taken c ? after a sample is taken d ? before a sample is taken e More than one answer is correct. 3 The monthly earnings of teachers is normally distributed with a mean of $3,000 and the standard deviation of $250. We select a sample of 87 teachers. The sampling distribution of the sample mean has an expected value and standard deviation of: a 3,000 and 26.8 b 3,000 and 1.69 c 3,000 and 250 d 3,000 and 2.87 e 3,000 and 321.6 4 The following data was collected by taking a simple random sample of a population 13 15 14 16 12 From this we know that, a The population mean is 14. b The point estimate of the population mean is 14. c The population mean must be 14 since the sample mean is 14. d Both a. and b. are correct. e Both a., b., and c. are correct. 6 A quality control expert wants to test car engines. The production manager claims they have an average life of 92 months with a standard deviation of 8. If the claim is true, what is the probability that the mean engine life would be greater than 90.8 months in a sample of 93 engines? a 0.0596 b 0.0735 c 0.4265 d 0.5596 e 0.9265 7 Increasing the size of a sample from 100 to 200 will a reduce the standard error of the mean to one-half its original value. b have no effect on the standard error of the mean. c reduce the standard error of the mean to approximately 70% of its current value. d double the standard error of the mean. e None of the above answers is correct. NEXT TWO ARE RELATED QUESTIONS ABOUT SAMPLING DISTRIBUTIONS. One hundred samples of size 85 each are drawn from an unknown population distribution of x and a sample mean is calculated for each sample. 8 If the number of samples stays at 100, but the size of each sample is increased from 85 to 125, then one would expect the variation in sample means observed across samples to: a increase. b decrease. c remain approximately the same. d change a lot, but not necessary increase or decrease. e be similar to the variation of x values in the population. 9 If the number of samples stays at 100, but the size of each sample is increased from 85 to 125, then one would expect the distribution of sample means observed across samples to: a remain unknown. b to depend upon the population distribution of x. c approximate the normal distribution, but not more closely than when 100 samples were drawn. d less closely approximate the normal distribution. e more closely approximate the normal distribution. 10 Annual part-time earnings in the U.S. average $15,000 and have a standard deviation $3,000. A sample of 62 part-time earners is selected. The standard error of the sample mean is: a $5 b $7 c $48 d $242 e $381 11 A speedboat engine company makes engines with the following specifications: the engine delivers an average power of 220 horsepower with a standard deviation of 16. Assuming that horsepower is normally distributed, if a randomly selected single engine is tested, what is the probability that the horsepower will exceed 224? a 0.4013 b 0.3783 c 0.3520 d 0.3300 e 0.2643 12 In the previous question, if a sample of 80 engines are tested. What is probability that the sample mean will exceed 222 horsepower. a 0.4483 b 0.3446 c 0.2148 d 0.1314 e 0.1056 13 Consider the horsepower average and standard deviation in question 2: ? = 220 and? = 16. If the sample size is n = 100, in the sampling distribution of x? what interval of x? values would contain 95% of all sample means? a 205.98 to 234.02 b 210.08 to 229.92 c 212.99 to 227.01 d 215.57 to 224.43 e 216.86 to 223.14 14 According to the central limit theorem, as the sample size increases, a the expected value of x? approaches 0. b the expected value of standard error of x? approaches 1. c the standard error of x? approaches the population standard deviation. d the distribution of x? approaches the normal. e the distribution of s, the sample standard deviation, approaches 0. 17 A sample of 64 patients in a walk-in clinic showed that they had to wait an average of 48 minutes before they could see a doctor. The sample standard deviation was 20 minutes. What is the 95% confidence Interval for the population average waiting time? a 45.2 50.8 b 44.6 51.4 c 43.1 52.9 d 42.2 53.8 e 41.8 54.2 20 Imagine 526 statisticians each took a different random sample of the population of patients visiting the walk-in clinic in Question 17 (each took a sample size of 64). About how many would produce confidence intervals that contained the population mean? a 100 b 500 c Almost all of them d All of them. e Cannot tell based on the information provided. 21 To build a confidence interval for the average age of the civilian labor force, a sample of 100 people was selected. The sample mean was 38.5 years and the standard deviation was 13.2 years. The lower and upper boundaries of a 95% confidence interval are: a 37.8 to 39.2 b 37.4 to 39.6 c 36.3 to 40.7 d 35.9 to 41.1 e 34.4 to 42.6 22 We can make a confidence interval more precise (narrower) by, a increasing the sample size. b reducing the confidence level (or confidence coefficient). c increasing the confidence level d Both (a) and (b) are correct. e Both (a) and (c) are correct. 25 To build an interval estimate of commuting time from Fishers to downtown Indianapolis in a midweek rush hour period five trial runs were made, obtaining the following results (in minutes). 55 45 43 34 38 Assuming the population commuting time is normally distributed, build a 95% confidence interval for the population mean commuting time. The interval is: a 28.6 to 57.4 b 29.9 to 56.1 c 31.2 to 54.8 d 33.1 to 52.9 e 36.0 to 50.0 26 For another interval estimate of the commuting time a sample of 100 trial runs were made. The lower and upper bounds of the interval were: L = 43.67 and U = 48.33 minutes. The sample standard deviation was s = 10 minutes. What is the confidence level for this interval estimate? a 98 percent. b 96 percent. c 94 percent. d 92 percent. e 90 percent. 27 A survey of 200 individuals who completed four years of college showed that 36 smoked regularly. Using this survey result what is the 95% confidence interval for the proportion of all individuals with four years of college education who smoke? a 0.097 to 0.263 b 0.107 to 0.253 c 0.127 to 0.233 d 0.137 to 0.223 e 0.147 to 0.213 29 The director of admission at a large state university advises parents of incoming students about the cost of textbooks during a typical semester. A sample of 100 students enrolled in the university indicates a sample mean cost of $315.40 with a sample standard deviation of $69. The sample is used to test the hypothesis that the population mean is at most $300. Which of the following is the correct statement of the null and alternative hypotheses? a H0: ?? 300 H1: ? 300 H1: ?? 300 c H0: ?? 300 H1: ? > 300 d H0: ? 5,000 when a sample of size 100 yields a mean of 5,315.4 and a standard deviation of 1400. Conduct the test with a probability of type I error = 0.10. Also compute the probability value. Which of the following is the correct decision: a The probability value is 0.02. Do not reject the null hypothesis that the mean is less than or equal to 5000. b The probability value is 0.10. Do not reject the null hypothesis that the mean is less than or equal to 5000. c The probability value is 0.02. Reject the null hypothesis that the mean is less than or equal to 5000. d The probability value is 0.01. Conclude that the mean is less than 5000. e The probability value is 0.01. Conclude that the mean is greater than 5000. 32 The automobile manufacturer Toyonda substitutes a different engine in cars of a model that were known to have an average miles per gallon (mpg) rating of 30 on the highway. To test whether the new engine changes the average mpg, a random sample of 100 trial runs gives x? = 28.3 mpg and s = 6.6 mpg. At? = 0.05 level of significance, is the average highway mpg rating for new engines different from the rating for the old engines? a The standardized test statistic exceeds the critical value. The average highway mpg rating for new engines is different from that of the old engines. b The probability value is less than the level of significance. The average highway mpg rating for new engines is different from that of the old engines. c The standardized test statistic exceeds the critical value. The average highway mpg rating for new engines is NOT different from that of the old engines. d The probability value is greater than the level of significance. The average highway mpg rating for new engines is NOT different from that of the old engines. e Both (a) and (b) are correct. 34 Use the sample of commuting times from Fishers to downtown Indianapolis in a midweek rush hour period: 55 45 43 34 38 Perform a test of hypothesis that the average time exceeds 36 minutes, using? = 0.05. Based on the sample data, a The test statistic is 1.964 and the critical value is 1.64. The sample mean is significantly greater than 36. Reject the null hypothesis. b The test statistic is 1.64 and the critical value is 1.964. The sample mean is not significantly greater than 36. Do not reject the null hypothesis. c The test statistic is 1.074 and the critical value is 1.64. The sample mean is not significantly less than 50. Do not reject the null hypothesis. d The test statistic is 1.074 and the critical value is 2.132. The sample mean is not significantly above 35. Do not reject the null hypothesis. e The test statistic is 1.964 and the critical value is 2.132. The sample mean is not significantly above 36. Do not reject the null hypothesis. NEXT THREE QUESTIONS ARE BASED ON THE FOLLOWING SCENARIO: The professors at Budget University make $75,000 on average. The professors want to convince the Budget administrators that professors from comparable universities make higher salaries. The Budget professors collect sample data on salaries from comparable universities to provide a test of their hypothesis. 37 The general form of the test should be a H0:? = $75,000 H1:?.? $75,000 b H0:?? $75,000 H1:? = $75,000 c H0:?? $75,000 H1:?? $75,000 d H0:?? $75,000 H1:? > $75,000 e H0:?? $75,000 H1:?< $75,000 38 If the Budget professors economists to use a 1 percent significance level instead of a 5 percent significance level, the critical value (s) will be _______ in absolute value and it is _______ likely that the null hypothesis will be rejected. a larger, more b larger, less c smaller, more d smaller, less e unaffected, equally 39 The economists decide to use a 1 percent significance level. They collect sample data on salaries from 20 comparable universities. The sample mean is $81,000 and the sample standard deviation is $10,000. The test statistic is ________, which causes them to ________ the null hypothesis. a z = 2.68, reject b t = 2.68, fail to reject c z = 0.81, reject d t = 0.81, fail to reject e t = 2.68, reject NEXT FIVE QUESTIONS ARE BASED ON THE FOLLOWING REGRESSION OUTPUT: The following data for a sample of 10 individuals shows the hourly earnings and years of schooling. Hourly Earnings Years of Schooling 17.24 15 15.00 16 14.91 8 4.50 6 18.00 15 6.29 12 19.23 12 18.69 18 7.21 12 42.06 20 The following regression Summary Output is used to study the relationship between hourly earnings and years of schooling: SUMMARY OUTPUT Regression Statistics Multiple R 0.7311 R Square Adjusted R Square 0.4763 Standard Error Observations 10 ANOVA df SS MS F Significance F Regression 1 538.40905 9.1853643 0.016291153 Residual 8 468.93 Total 9 1007.34 Coefficients Std Error t Stat P-value Lower 95% Upper 95% Intercept -7.791 8.3134 -0.9371 0.3761 -26.962 11.38 X Variable 1 1.799 0.5935 0.0163 Answer the next FIVE questions using the information in the Summary Output. 40 What is the predicted hourly earnings for 12 years of schooling? a 3 b 6.6 c 13.8 d 19.19 e 24.59 41 What percentage of hourly earnings is explained by years of schooling? a 74.60% b 73.10% c 69.30% d 53.40% e 47.60% 42 What is the 95% confidence interval for the population slope parameter?1? a 0.8 2.8 b 0.43 3.17 c 0.23 3.37 d 0.13 3.47 e 0.03 3.57 43 To perform a test of hypothesis that the population slope parameter?1 is zero, the test statistic is: a 3.031 b 2.306 c 2.262 d 2.228 e 0.33 44 Given the P-value of 0.0163, we can conclude, at 5% level of significance, that: a The population slope parameter is zero. There is NO relationship between hourly earnings and years of schooling. b The population slope parameter is different than zero. There is NO relationship between hourly earnings and years of schooling. c The population slope parameter is different than zero. There is a relationship between hourly earnings and years of schooling. d The population slope parameter is zero. There is a relationship between hourly earnings and years of schooling. e There is a small probability of a Type II error, accepting the hypothesis that the slope parameter is not equal to zero, when in fact it is. To study the relationship between manufacturers? market share and the quality of product. The following data on market share (in percentage) and product quality (ratings on the scale of 0 to 100) are available. The question is, are the variations in market share explained by the quality of the product? Market share Product Quality (%) (Scale: 0 to 100) 2 27 3 39 10 73 9 66 4 33 6 43 5 47 8 55 7 60 9 68 Using the following calculations complete the relevant parts (the shaded cells) of the Excel regression output below and answer FOUR questions. y? = 6.3 x? = 51.1 ?xy = 3,592 ?x? = 28,331 ?(x? x?)(y? y?) = 372.7 ?(x? x?)? = 2218.9 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations ANOVA df SS MS F Significance F Regression Residual 5.499 Total 68.10 Coefficients Std Error t Stat P-value Lower 95% Upper 95% Intercept X Variable 1 45 The predicted market share for a product quality rating of 90 is: a 11.2 b 12.8 c 13.4 d 13.9 e 14.4 46 The proportion of the variations in market share explained by product quality rating is: a 0.96 b 0.92 c 0.86 d 0.82 e 0.78 47 The upper boundary of the 95% confidence interval for the population slope parameter is: a 0.241 b 0.235 c 0.228 d 0.209 e 0.117 48 The t Stat for the test of hypothesis that the population slope parameter is zero is: a 2.95 b 7.65 c 9.54 d 10.11 e 10.98 Next THREE questions use the following data describing the median annual family income (in $1000s) and the median sale price of a house (in $1000s) for a sample of 12 housing markets. The data are used to regress the median price in a housing market on the median income in that market. The regression output follows the data. Income Price Market ($1000s) ($1000s) Syracuse, NY 41.8 76 Springfield, IL 47.7 91 Lima, OH 40.0 65 Dayton, OH 44.3 88 Beaumont, TX 37.3 70 Lakeland, FL 35.9 73 Baton Rouge, LA 39.3 85 Nashua, NH 56.9 118 Racine, WI 46.7 81 Des Moines, IA 48.3 89 Minneapolis 54.6 110 Wilmington, DE-MD 55.5 110 Average 45.692 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations ANOVA df SS MS F Significance F Regression 2717.86 Residual Total 3158 Coefficients Std Error t Stat P-value Intercept -11.802 X Variable 1 2.1843 49 The point estimate of the median price in a housing market with a median family income of $50,270 per annum is: a $97,000 b $98,000 c $97,270 d $98,990 e $99,000 50 What percent of the total variation in the median sale price of houses is explained by the estimated regression line? a 93 b 84 c 63 d 86 e 80 51 The sum of squared deviations x?s is:?(x? x?)? = 569.669. Calculate the t statistic for testing the null hypothesis of no linear relation (i.e. the slope parameter is zero) at a 10% level of significance (assume that the errors are normal, so there is no problem using the t distribution). The conclusion would be: a reject the null hypothesis, there is a significant linear relationship. b do not reject the null hypothesis, there is no significant linear relationship. c reject the null hypothesis, there is no significant linear relationship. d do not reject the null hypothesis, there is a significant linear relationship. e reject the null hypothesis at 10%, but you would not reject at 1%.

 

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