[SOLVED] Business and Economic Statistics
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True / False Questions (2 points each)
1. The reason sample variance has a divisor of n-1 rather than n is that it makes the sample variance an unbiased estimate of the population variance.
2. The sampling distribution of (X-bar) is always a normal distribution according to the Central limit theorem.
3. The standard deviation of the sampling distribution of the sample mean increases as the sample size increases.
4. If the sampled population is a normal distribution, then the sampling distribution of (X-bar is normal only for a large enough sample size.
5. If a population is known to be normally distributed, then it follows that the sample mean must equal the population mean.
6. The mean of the sampling distribution of (X-bar) is always equal to the mean of the sampled population.
7. Assuming the same level of significancea, as the sample size increases, the critical t-value becomes smaller and becomes lower than Z-value.
8. When constructing a confidence interval for a sample proportion, the t distribution is appropriate whenever the population standard deviation is not known, whether the sample size is large or small.
9. When the population is normally distributed and the population standard deviation s is unknown, then for any sample size n, the sampling distribution of (X-bar) is based on the t distribution.
10. When the level of confidence and sample standard deviation remain the same, a confidence interval for a population mean based on a sample of n = 100 will be wider than a confidence interval for a population mean based on a sample of n = 150.
11. When the level of confidence and the sample size remain the same, a confidence interval for a population mean µ will be narrower, when the sample standard deviation s is smaller than when s is larger.
12. The closer is the hypothesized mean is from the actual mean the higher is the power of the test.
13. The manager of the quality department for a tire manufacturing company wants to know the average tensile strength of rubber used in making a certain brand of radial tire. She knows the population standard deviation and uses a Z test to test the null hypothesis that the mean tensile strength is 800 pounds per square inch. The calculated Z test statistic is a positive value that leads to a p-value of .067 for the test. If the significance level is .01, the null hypothesis would be rejected. Assume that the population of pressure values is normally distributed.
14. The smaller the p-value, the more we doubt the null hypothesis.
15. You cannot make a Type II error when the null hypothesis is true.
16. Type I error is failing to reject a False Null Hypothesis.
17. When conducting a hypothesis test about a single mean, other relevant factors held constant, changing the level of significance from .05 to .01 will reduce the probability of a Type II error.
18. When the null hypothesis is false, you cannot make Type II error.
19. Everything else being constant, increasing the sample size decreases the probability of committing a Type I error.
20. The error term is the difference between the actual value of the dependent variable and the corresponding mean value of the dependent variable.
21. The Coefficient of Determination does not show the direction of relationship between the dependent and the independent variables.
22. The intercept of the simple linear regression equation represents the average change in the value of the dependent variable per unit change in the independent variable (X).
23. The correlation coefficient is the ratio of explained variation to total variation.
24. If there is a strong correlation between the independent and dependent variable, we expect that an increase in the value of the independent variable is associated with an increase in the value of the dependent variable.
25. In simple linear regression analysis, if the variance of the error term is correlated with the independent variable, then the assumption of Homoscedasticity is violated.
Multiple Choices (4 Points Each)
1. A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will be less than 94 lbs. is:
2.If the sampled population has a mean 48 and standard deviation 18, then the mean and the standard deviation for the sampling distribution of (X-bar) for n = 9 are:
A. 48 and 18
B. 48 and 9
C. 16 and 6
D. 48 and 6
E. 48 and 2
3. If a population distribution is known to be normal, then it follows that:
A. The sample mean must equal the population mean
B. The sample mean must equal the population mean for large samples
C. The sample standard deviation must equal the population standard deviation
D. All of the above
E. None of the above
4. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 14 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that at least 30 accounts will be classified as delinquent?
5. In a manufacturing process a machine produces bolts that have an average length of 3 inches with a variance of .03. If we randomly select three bolts from this process: What is the probability the mean length of the bolt is more than 3.16 inches?
6. The width of a confidence interval will be:
A. Narrower for 99% confidence than 95% confidence
B. Wider for a sample size of 100 than for a sample size of 50
C. Wider for 95% confidence than 90% confidence
D. Wider when the sample standard deviation (s) is small than when s is large
7. A confidence interval increases in width as
A. The level of confidence decreases
B. N increases
C. S decreases
D. All of the above
E. None of the above
8. As standard deviation increases, the samples size has to _____________ in order to achieve a specified level of confidence.
C. Remain the same
9. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are current (between 0 and 60 days after billing). The historical records show that over the past 8 years 70 percent of the accounts have been current. Determine the sample size needed in order to be 95% confident that the sample proportion of the current customer accounts is within .03 of the true proportion of all current accounts for this company.
10. When a confidence interval for a population proportion is constructed for a sample size n= 60 and the value of p = 0.4, the interval is based on:
A. the Z distribution
B. the t distribution
C. a Skewed distribution
D. None of the above
11. In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a variance of .09. What is the 90% confidence interval for the true mean length of the bolt?
A. 2.8355 to 3.1645
B. 2.5065 to 3.4935
C. 2.8140 to 3.1860
D. 2.4420 to 3.5580
E. 2.9442 to 3.0558
12. The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent (more than 90 days overdue). For this quarter, the auditing staff randomly selected 400 customer accounts and found that 80 of these accounts were delinquent. What is the 95% confidence interval for the proportion of all delinquent customer accounts at this manufacturing company?
A. 0.1608 to 0.2392
B. 0.1992 to 0.2008
C. 0.1671 to 0.2329
D. 0.1485 to 0.2515
E. 0.1714 to 0.2286
13. Which statement is incorrect?
A. The null hypothesis contains the equality sign
B. When a false null hypothesis is not rejected, a Type II error has occurred
C. If the null hypothesis is rejected, it is concluded that the alternative hypothesis is true
D. If we reject the null hypothesis, we cannot commit Type I error
14. If a null hypothesis is rejected at a significance level of 0.01, it will ______ be rejected at a significance level of 0.05
15. A study investigated the relationship of employment status to mental health. A sample of 49 unemployed men took a mental health examination measuring present mental health with lower values indicating better mental health. Their mean score was 10.94 and a standard deviation of 4.90. How much evidence do we have that the mean score exceeds 10?
A. No evidence (not even at 10%)
B. Some evidence (significant at 10% only)
C. Strong evidence (significant at 5% but not 1%)
D. Very strong evidence (significant even at 1%)
16. If a one-sided null hypothesis is not rejected for a single mean at a given significance level, the corresponding two-sided null hypothesis ( with the same sample size, the same standard deviation and the same mean) will _________ be rejected at the same significance level.
17. A major airline company is concerned that its proportion of late arrivals has substantially increased in the past month. Historical data shows that on the average 18% of the company airplanes have arrived late. In a random sample of 1,250 airplanes, 250 airplanes have arrived late. If we are conducting a hypothesis test of a single proportion to determine if the proportion of late arrivals has increased: What is the correct statement of null and alternative hypothesis?
A. H0: p <.18 and HA: p ³.18
B. H0: p £.18 and HA: p >.18
C. H0: p =.18 and HA: p ¹ .18
D. H0: p >.18 and HA: p £.18
E. H0: p £.20 and HA: p >.20
18. When carrying out a large sample test of H0: m £ 10 vs. Ha: m > 10 by using a critical value approach, we reject H0 at level of significance a when the calculated test statistic is:
A. Less than za
B. Less than – za
C. Greater than za
D. Greater than za/2
E. Less than the p value
19. What value(s) of alpha would we reject H0 for m greater than 10 if = 11, s =2 and n=36?
A. .05 and .01
B. .01 and .001
C. .05 and .10 but not .01
D. All of the above
E. None of the above
20. Which of the following is a violation of one of the major assumptions of the simple regression model?
A. The error terms are independent of each other
B. Histogram of the residuals form a bell-shaped, symmetrical curve
C. The error terms show no pattern
D. As the value of x increases, the value of the error term also increases
21. The least squares regression line minimizes the sum of the
A. Differences between actual and predicted Y values
B. Squared differences between actual and predicted Y values
C. Absolute deviations between actual and predicted X values
D. Absolute deviations between actual and predicted Y values
E. Squared differences between actual and predicted X values
22. The point estimate of the in a regression model is
23. In simple regression analysis the quantity that gives the amount by which Y (dependent variable) changes for a unit change in X (independent variable) is called the
A. Coefficient of determination
B. Standard error
C. The Y intercept of the regression line
D. Correlation coefficient
E. Slope of the regression line
24.The correlation coefficient may assume any value between
A. 0 and 1
B. -1 and 1
C. -infinity and + infinity
D. 0 and infinity
E. -1 and 0
25. In simple regression analysis, if the correlation coefficient is a positive value, then
A. The Y intercept must also be a positive value
B. The coefficient of determination can be either positive or negative, depending on the value of the slope
C. The least squares regression equation could either have a positive or a negative slope
D. The slope of the regression line must also be positive
E. The standard error of estimate can either have a positive or a negative value
Essay type Questions (10 points each)( You must show all your relevant work in Essay type to receive full points)
1. Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of 0.24 ounces. The weights of the sugar bags are normally distributed. What is the probability that 16 randomly selected packages will have an average weight less than 15.97 ounces?
2. The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.0 inches and a standard deviation of 0.9 inches. A sample of 36 metal sheets is randomly selected from a batch. What is the probability that the average length of a sheet is between 29.82 and 30.27 inches long?
3. The quality control manager of a tire company wishes to estimate the tensile strength of a standard size of rubber used to make a class of radial tires. A random sample of 49 pieces of rubber from different production batches is subjected to a stress test. The test measures the force needed to break the rubber in pounds. According to the sample results, the average pressure is 250 pounds and the standard deviation is 42 pounds. Determine the 95% confidence interval.
4. The production manager for the XYZ manufacturing company is concerned that the customer orders are being shipped late. He asked one of his planners to check the timeliness of shipments for 900 orders. The planner randomly selected 900 orders and found that 180 orders were shipped late. Construct the 95% confidence interval for the proportion of orders shipped late.
5. A cable TV company wants to estimate the percentage of cable boxes in use during an evening hour. An approximation based on previous surveys is 25 percent. The company wants the new estimate to be at the 90 percent confidence level and within 2 percent of the actual proportion. What sample size is needed?
6. Given sample data: 61.2, 61.9, 62.8, 63.1, 64.0, 64.3, 64.9, 65.5, 66.3 and 67.9,
test H0: m £ 62.89 versus H1: m > 62.89 at a = 0.05.
7. Test H0: ? = 0.25 versus HA: ? ¹ 0.25 with p = 0.33 and n = 100 at alpha = 0.05 and 0.10.
8. Test at ? =.05 and 0.10 the hypothesis that a majority (more than 50%) of students favor the plus/minus grading system at a university if in a random sample of 500 students, 265 favor the system?
9. An experiment was performed on a certain metal to determine if the strength is a function of heating time (hours). Results based on 25 metal sheets are given below. Use the simple linear regression model.
?X = 50
?X2 = 200
?Y = 75
?Y2 = 1600
?XY = 400
Find the estimated y intercept and slope. Write the equation of the least squares regression line and explain the coefficients. Estimate Y when X is equal to 4 hours. Also determine the standard error, the Mean Square Error, the coefficient of determination and the coefficient of correlation. Check the relation between correlation coefficient and Coefficient of Determination.Test the significance of the slope.
10. Consumer Reports provided extensive testing and ratings for more than 100 HDTVs. An overall score, based primarily on picture quality, was developed for each model. In general, a higher overall score indicates better performance. The following (hypothetical) data show the price and overall score for the ten 42-inch plasma televisions (Consumer Report data slightly changed here):
Use the above data to develop and estimated regression equation. Compute Coefficient of Determination and correlation coefficient and show their relation. Interpret the explanatory power of the model. Estimate the overall score for a 42-inch plasma television with a price of $3600 and perform significance test for the slope.