# Assume you have a data set from - Expert Answers

1. Assume you have a data set from a normally distributed random variable. Answer the following questions about it.

a. Will the random variable be discrete, continuous, or neither? How do you know?

b. Will the data be qualitative or quantitative? How do you know?

2. Assume the following data were gathered by a manufacturer of a robotics component, in units of days of continuous use until the component fails. There are 60 measurements in this data set. Show a histogram of this data set with 10 bins of equal size, spanning the range from the data minimum to the data maximum.

3. A university has been tracking the percentage of alumni giving to its annual fund each year for the past 10 years. The data is given here.

14% 13% 15% 21% 19% 24% 25% 28% 25% 31%

Answer the following questions about this data.

a. What are its mean and median?

b. What is the procedure for using mean and median to determine whether the data is skewed, and if so, in what direction?

c. Apply the procedure you described to the mean and median computed in part a.

4. A large department store records the number of returns per day in women's dresses for reasons such as wrong size, didn't like it, color, and so on. The manager of the women's dresses department recalls from her statistics class in college that the Poisson distribution might describe such events. A random sample of 150 days is taken. The number of returns per day in the sample as well as the observed frequencies are show below.

5. In the diagram below, events A, B and C are shown with numbers in various regions of the graph indicating how many sample points lie in each. For example, the number 3 in the top left of the diagram indicates that there are 3 sample points in B that are not also in either A or C.

a. Are the events A and B independent?

b. Are the events and mutually exclusive?

6. Under which of the following conditions would it be appropriate to use a Binomial random variable? In each case, explain why your answer is correct.

a. A department will interview 10 candidates for a position, and call back for second interviews those who answer the interview questions to the satisfaction of all the interviewers. They hope to call back at least 3, but past experience suggests an average of about 1 call back per 4 interviews.

b. A factory posts on the wall the number of days since its last safety infraction or injury. In the past year, the factory has had a safety infraction or injury on 6 different days. The factory is interested in the number of days that can be expected to elapse without an injury.

c. Fifteen of a doctor’s patients have the same ailment. Studies have shown that about 86.5% of patients with this ailment respond to a certain drug. The doctor prescribes the drug to all 15, but the number who will respond in this case is, of course, not known in advance.

7. The mean time for a racecar drivers’ crew to perform a pit stop is 13.2 seconds, with a standard deviation of 0.9 seconds. To maintain his current lead, the driver needs a pit stop in 12.5 seconds or less. Assuming this random variable is normally distributed, what is the probability of the driver getting the pit stop in a short enough time to maintain his lead?

8. From a sample of size 175, the sample mean is x-bar = 54.37 and sample standard deviation is s = 7.07.

a. Construct a 95% confidence interval for the population mean and show your work.

b. Explain how your work in part a would have been different if the sample size had been only 12 instead.

9. A random sample from the population of registered voters in California is to be taken and then surveyed about an upcoming election. What sample size should be used to guarantee a sampling error of 3% or less when estimating p at the 95% confidence level?

10. An elementary school teacher learned that 40% of school children have at least three cavities. The teacher has 30 students in his class. How many students would he expect in his class to have at least three cavities? What is the standard deviation? using the appropriate approximation, determine P(X 20); that is, the probability that more than 20 students in his class will have 3 cavities.

1. The path traveled by a golf ball hit with a 9-iron can be modeled with the quadratic function y = -0.042x2 + 5x, where x is the distance in yards from the point it was hit and y is the height of the golf ball in feet. Assume that the ground is level.

a. Find the maximum height reached by the ball.

b. How far from where it was hit does the ball hit the ground?

2. Translate the following argument into symbolic form and determine whether it’s logically correct by constructing a truth table.

3. To determine the distance to an oil platform in the Pacific Ocean from both ends of a beach, a surveyor measures the angle to the platform from each end of the beach. The angle made with the shoreline from one end of the beach is 83 degrees, from the other end 78.6 degrees. If the beach is 950 yards long, what are the distances to the oil platform from both ends of the beach?

Part B

1. Find the vertex and graph of y = -x2 + 6x – 5.5

2. Suppose a bullet is fired on a distant planet so that its height (in feet) after t seconds is given by h = -4t2 + 16t + 5.

a. When is the bullet at its highest point?

b. When will the bullet strike the ground?

c. What is the maximum height the bullet will attain?

3. On the day of the child’s birth, a proud parent deposits an amount of money into a compound interest savings account, which will earn 5% each year. On the child’s eighteenth birthday, the account will have accumulated $10,000. How much did the parent deposit on the day of the child’s birth?

4. Consider the sequence that begins with 4, 3, … and continues by the rule: Every subsequent element of the sequence is the sum of the two preceding elements.

a. Determine the next five (third through seventh) elements of this sequence.

b. Compute the ratio of the sixth and seventh elements of this sequence. Is it close to the Golden Ratio.

5. Suppose that in a class of 50 students, 10 are married and 18 live at their parents’ homes It’s known that 3 of the married students live at their parents’ homes.

a. Draw a Venn diagram of this information.

b. How many students are single and do not live at their parents’ homes?

6. Make a simple flowchart in which you draw cards from a standard deck until you draw an ace.

7. Determine whether the following argument is correct. If it’s not correct, explain what is wrong with the argument and change the minor premise to make a correct argument.

8. A city block 500 feet by 500 feet has a large building 300 feet by 400 feet in the NW corner. The rest of the block is an unobstructed paved lot. What’s the shortest distance from the SW corner to the NE corner of the city block, going through the paved lot.

9. A pilot, attempting to fly from Middletown to Westburg, a distance of 200 miles, notices after flying 180 miles that Westburg is still 35 miles away because the heading was off by a few degrees. How many degrees was the heading off the desired course (to one decimal place)?

10. Find the dimensions of a rectangular region with the maximum area that can be enclosed by 60 feet of fencing.

a. Will the random variable be discrete, continuous, or neither? How do you know?

b. Will the data be qualitative or quantitative? How do you know?

2. Assume the following data were gathered by a manufacturer of a robotics component, in units of days of continuous use until the component fails. There are 60 measurements in this data set. Show a histogram of this data set with 10 bins of equal size, spanning the range from the data minimum to the data maximum.

3. A university has been tracking the percentage of alumni giving to its annual fund each year for the past 10 years. The data is given here.

14% 13% 15% 21% 19% 24% 25% 28% 25% 31%

Answer the following questions about this data.

a. What are its mean and median?

b. What is the procedure for using mean and median to determine whether the data is skewed, and if so, in what direction?

c. Apply the procedure you described to the mean and median computed in part a.

4. A large department store records the number of returns per day in women's dresses for reasons such as wrong size, didn't like it, color, and so on. The manager of the women's dresses department recalls from her statistics class in college that the Poisson distribution might describe such events. A random sample of 150 days is taken. The number of returns per day in the sample as well as the observed frequencies are show below.

5. In the diagram below, events A, B and C are shown with numbers in various regions of the graph indicating how many sample points lie in each. For example, the number 3 in the top left of the diagram indicates that there are 3 sample points in B that are not also in either A or C.

a. Are the events A and B independent?

b. Are the events and mutually exclusive?

6. Under which of the following conditions would it be appropriate to use a Binomial random variable? In each case, explain why your answer is correct.

a. A department will interview 10 candidates for a position, and call back for second interviews those who answer the interview questions to the satisfaction of all the interviewers. They hope to call back at least 3, but past experience suggests an average of about 1 call back per 4 interviews.

b. A factory posts on the wall the number of days since its last safety infraction or injury. In the past year, the factory has had a safety infraction or injury on 6 different days. The factory is interested in the number of days that can be expected to elapse without an injury.

c. Fifteen of a doctor’s patients have the same ailment. Studies have shown that about 86.5% of patients with this ailment respond to a certain drug. The doctor prescribes the drug to all 15, but the number who will respond in this case is, of course, not known in advance.

7. The mean time for a racecar drivers’ crew to perform a pit stop is 13.2 seconds, with a standard deviation of 0.9 seconds. To maintain his current lead, the driver needs a pit stop in 12.5 seconds or less. Assuming this random variable is normally distributed, what is the probability of the driver getting the pit stop in a short enough time to maintain his lead?

8. From a sample of size 175, the sample mean is x-bar = 54.37 and sample standard deviation is s = 7.07.

a. Construct a 95% confidence interval for the population mean and show your work.

b. Explain how your work in part a would have been different if the sample size had been only 12 instead.

9. A random sample from the population of registered voters in California is to be taken and then surveyed about an upcoming election. What sample size should be used to guarantee a sampling error of 3% or less when estimating p at the 95% confidence level?

10. An elementary school teacher learned that 40% of school children have at least three cavities. The teacher has 30 students in his class. How many students would he expect in his class to have at least three cavities? What is the standard deviation? using the appropriate approximation, determine P(X 20); that is, the probability that more than 20 students in his class will have 3 cavities.

1. The path traveled by a golf ball hit with a 9-iron can be modeled with the quadratic function y = -0.042x2 + 5x, where x is the distance in yards from the point it was hit and y is the height of the golf ball in feet. Assume that the ground is level.

a. Find the maximum height reached by the ball.

b. How far from where it was hit does the ball hit the ground?

2. Translate the following argument into symbolic form and determine whether it’s logically correct by constructing a truth table.

3. To determine the distance to an oil platform in the Pacific Ocean from both ends of a beach, a surveyor measures the angle to the platform from each end of the beach. The angle made with the shoreline from one end of the beach is 83 degrees, from the other end 78.6 degrees. If the beach is 950 yards long, what are the distances to the oil platform from both ends of the beach?

Part B

1. Find the vertex and graph of y = -x2 + 6x – 5.5

2. Suppose a bullet is fired on a distant planet so that its height (in feet) after t seconds is given by h = -4t2 + 16t + 5.

a. When is the bullet at its highest point?

b. When will the bullet strike the ground?

c. What is the maximum height the bullet will attain?

3. On the day of the child’s birth, a proud parent deposits an amount of money into a compound interest savings account, which will earn 5% each year. On the child’s eighteenth birthday, the account will have accumulated $10,000. How much did the parent deposit on the day of the child’s birth?

4. Consider the sequence that begins with 4, 3, … and continues by the rule: Every subsequent element of the sequence is the sum of the two preceding elements.

a. Determine the next five (third through seventh) elements of this sequence.

b. Compute the ratio of the sixth and seventh elements of this sequence. Is it close to the Golden Ratio.

5. Suppose that in a class of 50 students, 10 are married and 18 live at their parents’ homes It’s known that 3 of the married students live at their parents’ homes.

a. Draw a Venn diagram of this information.

b. How many students are single and do not live at their parents’ homes?

6. Make a simple flowchart in which you draw cards from a standard deck until you draw an ace.

7. Determine whether the following argument is correct. If it’s not correct, explain what is wrong with the argument and change the minor premise to make a correct argument.

8. A city block 500 feet by 500 feet has a large building 300 feet by 400 feet in the NW corner. The rest of the block is an unobstructed paved lot. What’s the shortest distance from the SW corner to the NE corner of the city block, going through the paved lot.

9. A pilot, attempting to fly from Middletown to Westburg, a distance of 200 miles, notices after flying 180 miles that Westburg is still 35 miles away because the heading was off by a few degrees. How many degrees was the heading off the desired course (to one decimal place)?

10. Find the dimensions of a rectangular region with the maximum area that can be enclosed by 60 feet of fencing.

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