E270 hw 8 final | Numerical analysis homework help

  NAME:           Your LETTER answers COMPREHENSIVE HOMEWORK PROBLEMS               1 In thinking about doing statistical analysis, the sample mean should be interpreted as:     1 a a constant value that is equal to the population mean.           2 b a constant value that is approximately equal to the population mean.       3 c a random variable that is approximately equal to the population mean when sampling is done without replacement.   4     5 d a random variable that is approximately equal to the population mean if n > 30 and when sampling is done without replacement.   6     7 e a random variable that when averaged across many samples is approximately equal to the population mean. 8                       9 2 Which of the following are random?             10 a x̄ after a sample is taken               11 b x̄ before a sample is taken               12 c µ after a sample is taken               13 d µ before a sample is taken               14 e More than one answer is correct.               15                       16 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: 17   18   19 a 3,000 and 26.8                  20 b 3,000 and 1.69                  21 c 3,000 and 250                 22 d 3,000 and 2.87                  23 e 3,000 and 321.6                 24                       25 4 The following data was collected by taking a simple random sample of a population     26   13 15 14 16 12           27   From this we know that,               28 a The population mean is 14.               29 b The point estimate of the population mean is 14.           30 c The population mean must be 14 since the sample mean is 14.         31 d Both a. and b. are correct.               32 e Both a., b., and c. are correct.               33                       34 5 A direct mail company wishes to estimate the proportion of persons on a large mailing list that will purchase a product. Suppose the true proportion is 0.07. If 486 are sampled, what is the probability that the sample proportion will differ from the population proportion within ±0.03?   35     36     37 a 0.024                   38 b 0.1428                   39 c 0.4952                   40 d 0.9994                   41 e None of the above answers is correct.             42                       43 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?   44     45     46 a 0.0596                   47 b 0.0735                   48 c 0.4265                   49 d 0.5596                   50 e 0.9265                   51                         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:  µ = 220 and σ = 16.  If the sample size is n = 100, in the sampling distribution of x̄ what interval of x̄ 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 x̄  approaches 0.               b the expected value of standard error of x̄ approaches 1.             c the standard error of x̄ approaches the population standard deviation.         d the distribution of x̄ approaches the normal.               e the distribution of s, the sample standard deviation, approaches 0.                                   15 The actual proportion of defective jeans in a large warehouse is 0.20.  What is the probability that in a random sample of 500 jeans the sample proportion p̄ will be within a margin of ±0.04 of the population proportion of defective jeans?                 a 0.8198                     b 0.8415                     c 0.8859                     d 0.9164                     e 0.9749                                             16 In the previous question, what interval of p̄ values contains 90% of all  p̄’s?         a 0.171 to 0.229                   b 0.165 to 0.235                   c 0.154 to 0.246                   d 0.145 to 0.255                   e 0.122 to 0.278                                           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                                           18 What is the minimum sample size to estimate the 95% confidence interval in the previous question to within plus or minus 3 minutes. Use a planning value of 20 minutes for the standard deviation.           a 131                     b 141                     c 151                     d 161                     e 171                                             19 Here is a problem similar to the previous question, but now the sample of patients is only 9. In this sample the patients had to wait an average of 45 minutes and the sample standard deviation was 12 minutes. What is the margin of error for a 95% confidence Interval for the population average waiting time? (Assume that the population of the waiting times in normal).                       a 7.8                     b 9.2                     c 10.6                     d 12.0                     e 13.4                                             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.                                         23 To estimate the population average age of the civilian labor force to within a margin of error of 0.5 years at 95% level of confidence, what is the minimum sample size?  Assume the population standard deviation is known to be 12.1 years.                 a 2,250                     b 2,145                     c 1,972                     d 1,576                     e 1,255                                             24 Using the standard deviation 2.61 as a planning value, what sample size should be used in order to obtain a 95% confidence interval with a margin of error of ±0.5?           a 105                     b 94                     c 85                     d 64                     e 56                                             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                                           28 In the previous question, what is the minimum sample size to estimate the population proportion of all individuals with four years of college education who smoke to within a margin of error of ±0.03.  We expect 19 out of every 20 such interval estimates to contain the population proportion.  Use the sample proportion in the previous question as the planning value.                       a 227                     b 355                     c 631                     d 993                     e 1418                                             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 : µ ≥ 300  H1 : µ < 300               b H0 : µ > 300  H1 : µ ≤ 300               c H0 : µ ≤ 300  H1 : µ > 300               d H0 : µ < 300  H1 : µ ≥ 300               e H0 : µ = 300  H1 : µ ≠ 300                                       30 Regardless how you answered the previous question, which of the following statements is correct?     a If the mean cost of text books is in fact greater than $300 and the hypothesis test leads you to conclude that it is at most $300, the you have committed a Type II error.           b If the mean cost of text books is in fact greater than $300 and the hypothesis test leads you to conclude that it is at most $300, the you have committed a Type I error.           c If the mean cost of text books is at most $300 and the hypothesis test leads you to conclude that it is greater than $300, the you have committed a Type II error.           d If the mean cost of text books is in less than $300 and the hypothesis test leads you to conclude that it is greater than $300, the you have committed a Type II error.           e If the mean cost of text books is in less than $300 and the hypothesis test leads you to conclude that it is at least $300, the you have committed a Type II error.                                   31 Test H0 : µ ≤  5,000 versus H1 : µ > 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 x̄ = 28.3 mpg and s = 6.6 mpg.  At α = 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.                                         33 The engineering team of Honota Motors has designed a new engine for cars of Model X240 which it claims will improve (increase) the gas mileage while maintaining the same horsepower.  The current average highway mileage is 25 mpg.  A sample of 40 trial runs gives x̄ = 26.4 mpg.  Based on the average trial-run mpg the management rejects the engineering team’s claim and does not adopt the new design.  Suppose the new design does in fact improve the gas mileage.  Which of the following correctly describes the management’s decision:                                   a The management has rejected a true null hypothesis.  Therefore, it has committed a Type I Error.     b The management has rejected a true null hypothesis.  Therefore, it has committed a Type II Error.     c The management has not rejected a false alternative hypothesis.  Therefore, it has committed a Type I Error.   d The management has not rejected a false null hypothesis.  Therefore, it has committed a Type II Error.     e Both (a) and (c) 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 α = 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.                                   35 To test the hypothesis that the percentage of individuals with four years of college education who smoke has decreased from 21% a decade ago, a random sample of 1200 such individuals revealed that 222 smoked.  Use α = 0.05.  Based on the sample result,                 a The sample proportion is not significantly less than 21%.  Do not reject the null hypothesis.  Conclude that the proportion of college educated individuals who smoke has not decreased compared to a decade ago.                 b The sample proportion is significantly less than 21%.  Reject the null hypothesis.  Conclude that the proportion of college educated individuals who smoke has decreased compared to a decade ago.           c The sample proportion is significantly less than 21%.  Do not reject the null hypothesis.  Conclude that the proportion of college educated individuals who smoke has decreased compared to a decade ago.           d The sample proportion is not significantly less than 21%.  Reject the null hypothesis.  Conclude that the proportion of college educated individuals who smoke has not decreased compared to a decade ago.           e The test statistic is less than the critical value.  Do not reject the null hypothesis.  Conclude that the proportion of college educated individuals who smoke has not decreased compared to a decade ago.                                   36 In the previous question, the probability value is:             a 0.095                     b 0.032                     c 0.017                     d 0.010                     e 0.004                                             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: μ = $75,000  H1: μ.≠ $75,000               b H0: μ ≠ $75,000  H1: μ = $75,000               c H0: μ ≤ $75,000  H1: μ ≠ $75,000               d H0: μ ≤ $75,000  H1: μ > $75,000               e H0: μ ≥ $75,000  H1: μ< $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 0.0162912             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 β1?         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 β1 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.                             ȳ = 6.3                     x̄ = 51.1                     ∑xy = 3,592                     ∑x² = 28,331                     ∑(x − x̄)(y − ȳ) = 372.7                     ∑(x − x̄)² = 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 − x̄)² = 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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