Math 227 – Elementary Statistics: A Brief Version, 5/e ...

Math 227 ? Elementary Statistics: A Brief Version, 5/e Bluman Ch. 3.1 # 3, 4, 7, 30, 31, 32

Find (a) the mean, (b) the median, (c) the mode, and (d) the midrange. 3) High Temperatures The reported high temperatures (in degrees Fahrenheit) for selected world cities on an October day are shown below. Which measure of central tendency do you think best describes these data?

62 72 66 79 83 61 62 85 72 64 74 71 42 38 91 66 77 90 74 63 64 68 42 (a) Mean

Mean = = (x) 1= 566 68.1 n 23 (b) Median 38, 42, 42, ..., 68, ..., 85, 90, 91 Median=68 (c) Mode= 42, 62, 64, 66, 72, 74 (d) Midrange Midrange= 38 + 91 = 64.5

2 For the best measure of average, answers will vary. 4) Observers in the Frogwatch Program The number of observers in the Frogwatch USA program (a wildlife conservation program dedicated to helping conserve frogs and toads) for the top 10 states with the most observers is 484, 483, 422, 396, 378, 352, 338, 331, 318, and 302. The top 10 states with the most active watchers list these numbers of visits: 634, 464, 406, 267, 219, 194, 191, 150, 130, and 114. Compare the measures of central tendency for these two groups of data.

7) Earthquake Strengths twelve major earthquakes had Richter magnitudes shown here.

5.4 5.4 6.2 6.2 6.4 6.4 6.5 7.0 7.2 7.2 7.7 8.0

a) Mean =

(x) =

7= 9.6

6.63

n 12

b) Median = 6.4 + 6.5 = 6.45 2

c) Mode = none

d) Midrange = 5.4 + 8.0 = 6.7 2

Since almost all data is around the mean value (6.63), the mean is a better choice for the average of measurement.

30) Final Grade An instructor grades exams, 20%; term paper, 30%; final exam, 50%. A student had grades of 83, 72, and 90, respectively, for exams, term paper, and final exam. Find the student's final average. Use the weighted mean.

31) Final Grade Another instructor gives four 1-hour exams and one final exam, which counts as two 1-hour exams. Find a student's grade if she received 62, 83, 97, and 90 on the 1-hour exams and 82 on the final exam.

32) For these situations, state which measure of central tendency--mean, median, or mode--should be used. a. The most typical case is desired. Mode b. The distribution is open-ended. Median c. There is an extreme value in the data set. Median d. The data are categorical. Mode e. Further statistical computations will be needed. Mean f. The values are to be divided into two approximately equal groups, one group containing the larger values and one containing the smaller values. Median

Section 3-2 # 7, 9, 11, 13,30, 31, 33, 34, 35,37, 41, 42 For exercises 7-13, find the range, variance, and standard deviation. Assume the data represent samples, and use the

shortcut formula for the unbiased estimator to compute the variance and standard deviation.

mean= X= ( X ) ;

n

= variance = s2 = ( X - X ) or s2

X 2 -[( X )2 / n]

n -1

n -1

standard deviation = s = variance

7) Police Calls in Schools the number of incidents where police were needed for a sample of 10 schools in Allegheny County is 7, 37, 3, 8, 48, 11, 6, 0, 10, 3. Are the data consistent or do they vary? Explain your answer.

x

x 2

0

0

3

9

3

9

6

36

7

49

8

64

10

100

11

121

37 1369

48 2304

Sum 133 4061

Mean =

(

X) =

1= 33

13.3

n

10

Range = 48 ? 0 = 48 n=10

Variance = s2 = X 2 -[( X )2 / n] n -1

= 4061- (133)2 / 10 9

= 4061-1= 768.9 2= 292.1 254.7

9

9

Standard deviation = 254.7 16 The data vary widely.

9) Precipitation and High Temperatures the normal daily high temperatures (in degrees Fahrenheit) in January for 10 selected cities are as follows. 50, 37, 29, 54, 30, 61, 47, 38, 34, 61 The normal monthly precipitation (in inches) for theses same 10 cities is listed here. 4.8, 2.6, 1.5, 1.8, 1.8, 3.3, 5.1, 1.1, 1.8, 2.5

x

x 2

29 841

30 900

34 1156

37 1369

38 1444

47 2209

50 2500

54 2916

Which set is more variable?

61 61 Sum 441

3721 3721 20,777

For the Temperature:

Mean =

(

X) =

4= 41

44.1

Range = 61 ? 29 = 32

n=10

n

10

Variance = s2 = X 2 -[( X )2 / n] = 20, 777 -[(441)2 / 10]

n -1

9

= 20, 777 -19,= 448.1 1= 328.9 147.6

9

9

Standard deviation = 147.6 = 12.15

C= Var=

1= 2.15 44.1

27.6%

For the Precipitation:

x

x 2

4.8 23.04

2.6 6.76

1.5 2.25

1.8 3.24

1.8 3.24

3.3 10.89

5.1 26.01

1.1 1.21

1.8 3.24

2.5 6.25

Sum 26.3 86.13

= 86.13 - 6= 9.17 1= 6.96 1.88

9

9

Standard deviation = 1.88 = 1.37

C= Var=

1= .373 2.63

52.2%

Mean =

(

X) =

2= 6.3

2.63

n

10

Range = 5.1 ? 1.1 = 4.0

n=10

Variance =

s2 = X 2 -[( X )2 / n] = 86.13 -[(26.3)2 /10]

n -1

9

Precipitation is more variable since CVar is higher.

11) Stories in the Tallest Buildings The number of stories in the 13 tallest buildings for two different cities is listed below. Which set of data is more variable?

Houston: 75, 71, 64, 56, 53, 55, 47, 55, 52, 50, 50, 50, 47 Pittsburgh: 64, 54, 40, 32, 46, 44, 42, 41, 40, 40, 34, 32, 30

13) Hardcover Bestsellers the number of weeks on The New York Times Best Sellers list for hardcover fiction is

1 4 2 2 3 18 5 5 10 4 3 6 2 2 22

Use the range rule of thumb to estimate the standard deviation. Compare the estimate to the actual standard

deviation.

=s

ra= nge 4

22 - 4

1

5.25

,

x x2

1 1

4 16

2 4

2 4

3 9

s2 =s

X 2 -[( X )2 / n]

=

1061-[(89)2 ] 15 38.1

n -1

15 -1

38.1 6.2

18 324 5 25 5 25 10 100 4 16

3 9

6 36

2 4

2 4

22 484

Sum 89 1061

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