Final Exam

Fall 2020

Introduction to Statistics – DL

• Exercises can be solved in any order.

• Most exercises are made up of a few independent items, thus, if you are not able to solve one of them, do not skip the whole exercise but read and try the next item.

Part A

Please, solve all the following five exercises!

1. IQ scores are distributed normally with a mean of 100 and a standard deviation of 15.

(a) For many school systems, an IQ of 70 indicates that the child may be eligible for special education. What percentage of the general population has an IQ of 70 or less? [2 points]

(b) In a school district of 4000 students, how many would be expected to have IQs of 70 or less. [2 points]

(c) Mensa is an organization of people who have high IQs. To be eligible for membership a person must have an IQ ”higher than 98 percent of the population.” What IQ is required to qualify? [2 points]

2. A fast food restaurant manager claims that they have an average time of 4.2 minutes with a standard deviation of 1.3 minutes as the duration between a customer entering the restaurant and the food being served to them. Brandon visited the restaurant every day and disagrees with the information provided by the manager. To check this, Brandon conducted a survey by selecting a random sample of 42 customers from the restaurant. He discovered an average time of five minutes from his sample.

(a) Calculate the probability that the average time until a customer is served will be at least five minutes. [7 points]

(b) What conclusion about the managers claim should Brandon make based on his find- ings? [2 point]

(c) If the sample size had been 8 customers instead of 42, what further assumption(s) would have been necessary in order to solve this problem? [1 point]

3. The mean of the sample of 65 customer satisfaction ratings is 2.95. If we assume that the population standard deviation equals 2.64: Calculate and interpret (in context) a 95% confidence interval for the mean of all possible customer satisfaction ratings for the XYZ-Box video game system. [8 points]

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4. An automobile rental agency has the following mileages (in thousands) for a simple ran- dom sample of 19 cars rented last year. Given this information, and assuming the data are from a population that is approximately normally distributed, construct and interpret the 90% confidence interval for the population mean: 55, 35, 65, 69, 37, 88, 80, 39, 61, 54, 50, 74, 92, 59, 50, 38, 59, 29, 60. [8 points]

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Part B

Please, choose two out of the following six exercises (each 10 points)! Solve only these ones (and only these two exercises will be corrected)!

My chosen exercises: � � 1. Here you can find the amount that a sample of nine customers spent for lunch (CHF) at

a fast-food restaurant: 4.20, 5.03, 5.86, 6.45, 7.38, 7.54, 8.46, 8.47, 9.87.

(a) How would you visualize data? Draw a graph!

(b) Find median and mean. Give interpretation!

(c) Find measures of spread (range, IQR, standard deviation, variance) and explain in the given context.

2. Regression is a technique that economists and businesspeople rely on heavily. Think about the relationship between advertising expenditures and sales. Use the data in the table below to answer the following questions:

Advertising, X ($ thousands) 3 4 3 5 6 5 4 Sales, Y ($ thousands) 70 120 110 100 140 120 110

Is there a correlation between the advertising expenditures and sales? Answer to the question by drawing a scatter plot and calculating the linear correlation coefficient. Write down the equation for the regression line. Based on this model predict sales for an advertising expenditure of $ 10,000.

3. A department store reports that 30% of its customer transactions are in cash, 20% are by check, and the rest are by credit card. 20% of the cash transactions, 90% of the check transactions, and 60% of the credit card transactions are for more than $75. A costumer has just made a $125 purchase. What is the probability that she paid cash?

4. A manufacturer of clothing knows that the probability of a button flaw (broken, sewed on incorrectly, or missing) is 0.002. An inspector examines 50 shirts in an hour, each with 6 buttons.

(a) Is it binomial or Poisson distribution?

(b) What is the probability that she finds no button flaws?

(c) What is the probability that she finds at least one?

(d) How many buttons would you expect to flaw?

all cars.

5. Ableson’s anthropology professor announced that the poorest exam grade for each student would be dropped. Ableson scored 79 on the first anthropology exam. The mean was 67 and the standard deviation 4. On the second exam, he made 125. The class mean was 105 and the standard deviation 15. On the third exam, the mean was 45 and the standard deviation 3. Ableson got 51. Which test should be dropped?

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