Questions on Statistics

1. A density curve consists of a straight line segment that begins at the origin (0, 0) and has slope of 1.
a. Sketch this density curve. What are the coordinates of the right endpoint of the segment?
(Note that the right endpoint should be fixed so that the total area under the curve is 1.
This is required for a valid density curve.) b. Determine the median, the first quartile (Q1), and the third quartile (Q3). c. Relative to the median, where would you expect the mean of the distribution to lie?
Explain briefly.
The distribution is skewed left, so the mean will be left of the median.
d. What percent of the observations lie below 0.5? Above 1.5? 2. The following are the salaries of employees in a small business:
15,000 15,000 17,500 23,500 23,750 25,000 26,000 27,500 29,000 45,000 a. Before using your calculator, do you think that the mean or the median for these values will be higher? Why do you think so? b. What are the mean and median of the salaries? Was your answer to (a) correct? c. Find the standard deviation and the IQR of the data d. Describe the shape of the distribution terms of its overall shape, including outliers and skewness. Be sure to give the locations of the main features e. Speculate on the reasons for why the distribution is shaped as it is. 3. The following represents the outcomes of rolling a die 600 times (actually, it was simulated on the TI­83 as follows: RandInt (1,6,600)_L1).
Face

1

2

3

4

5

6

Number

107

120

84

110

92

87

a. What are the mean and median numbers of appearances of each face? b. What are the mean and median face values of the 600 rolls of the die?

c. Suppose the die was perfectly fair, i.e., there were exactly 100 of each outcome. Now, what would be the mean and median face values of the 600 rolls of the die? d. Draw a histogram of the outcomes for the six faces. Describe the shape of the distribution you get. 4. Following are a simple random sample of heights (in inches) of nandina bushes grown in a nursery. All bushes are of the same species:
24.4

32.8

39.9

31.0

33.0

48.7

51.5

54.0

53.0

54.0

31.0

30.1

30.9

33.7

34.0

53.0

47.5

46.7

49.4

51.1

33.7

35.8

32.9

30.7

23.6

49.4

50.9

50.7

52.5

52.2

30.7

22.3

34.9

20.9

31.1

52.5

49.5

49.0

47.4

54.7

20.9

31.9

22.9

32.8

24.4

47.4

54.3

50.9

51.5

48.7

a. Characterize this distribution in terms of center and spread (variation) b. Draw a histogram and describe its shape. c. Offer a theory as to why the shape looks as it does. Be sure to give the locations of the main features. d. What would be a good strategy for sorting and characterizing these data?

e. How do the mean and the median compare? What do the IQR and the standard deviation say about this distribution?

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