Download Math 251 Review for Exam 1

Math 251 Science Supplement to Chapter 2
1. Suppose that the following numbers represent a sample of 35 consecutive birthweights in ounces at a hospital:
68, 120, 110, 95, 98, 142, 106, 102, 46, 119, 108, 107, 84, 92, 101, 103, 106, 104, 90, 111, 113, 115, 118, 132, 128,
125, 102, 106, 109, 108, 103, 99, 98, 100, 38.
(a) Make a dotplot of the 35 birthweights and describe the shape.
(b) Make a stem-and-leaf plot of the 35 birthweights and describe the shape.
(c) Compute the mean, median, and mode of the 35 birthweights.
(d) Compute the 40th percentile of the 35 birthweights.
(e) Compute the first and third quartiles of the 35 birthweights.
(f) Find the 5-number summary of the 35 birthweights.
(g) Make a boxplot of the 35 birthweights and describe the shape.
(h) Find the range, IQR (interquartile range), and standard deviation of the 35 birthweights.
(i) Group the birthweight data into intervals of equal length.
(j) Make a histogram of the grouped birthweights and describe the shape.
2. Suppose the following data are from a sample of patients discharged from a hospital.
ID no.
Duration of
hospital stay
Received antibiotic
(1=yes, 2=no)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2
2
1
2
2
1
1
3
8
1
4
5
11
1
1
2
3
3
1
1
2
2
2
2
1
2
2
1
1
2
2
2
1
2
2
2
1
2
1
2
(a) Make a dotplot of the duration of hospital stay for the 20 patients and describe the shape.
(b) Make a stem-and-leaf plot of the duration of hospital stay for the 20 patients and describe the shape.
(c) Compute the mean, median, and mode of the duration of hospital stay for the 20 patients.
(d) Compute the 67th percentile of the duration of hospital stay for the 20 patients.
(e) Compute the first and third quartiles of the duration of hospital stay for the 20 patients.
(f) Find the 5-number summary of the duration of hospital stay for the 20 patients.
(g) Make a boxplot of the duration of hospital stay for the 20 patients and describe the shape.
(h) Find the range, IQR, and standard deviation of the duration of hospital stay for the 20 patients.
Math 251 Science Supplement to Chapter 2
page two
(i) Compute the mean and standard deviation of duration of hospital stay for those patients who
received antibiotics.
(j) Compute the mean and standard deviation of duration of hospital stay for those patients who
did not receive antibiotics.
(k) What seems to be the relationship between the receiving of antibiotics and the length of hospital stay?
3. Suppose that the following data represents serum cholesterol levels in mg/dL of a sample of 16 people before and
after they adopted a vegetarian diet.
Subject
Before
After
Before  After
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
148
200
194
188
236
256
210
198
187
175
190
154
165
168
192
200
160
166
154
180
198
188
180
202
172
166
160
150
162
166
178
158
 12
34
40
8
38
68
30
4
15
9
30
4
3
2
14
42
(a) Make a dotplot of the change in cholesterol and describe the shape.
(b) Make a stem-and-leaf plot of the change in cholesterol and describe the shape.
(c) Compute the mean, median, and mode of the change in cholesterol.
(d) Compute the 40th percentile of the change in cholesterol.
(e) Compute the first and third quartiles of the change in cholesterol.
(f) Find the 5-number summary of the change in cholesterol.
(g) Make a boxplot of the change in cholesterol and describe the shape.
(h) Find the range, IQR (interquartile range), and standard deviation of the change in cholesterol.
(i) Group the change in cholesterol data into intervals of equal length.
(j) Make a histogram of the grouped change in cholesterol and describe the shape.
(k) Spit the data into 2 groups according to whether their baseline cholesterol is above or below the median.
Compute the mean and standard deviation of the cholesterol change for each group. Does the diet appear to have
different effects on the two groups?
Math 251 Science Supplement to Chapter 2
page three
4. Suppose the following table gives the distribution of astigmatism in a sample of 1000 men ages 18 to 22.
Degree of
astigmatism(diopters)
Frequency
0 - 0.1
0.1 - 0.3
0.3 - 0.5
0.5 - 1
1-2
2-3
3-4
4-5
5-6
442
264
148
76
41
15
8
4
2
(a) Find the grouped mean.
(b) Find the grouped standard deviation.
(c) In which interval does the median lie?
(d) Is the set of data symmetric, skewed left, or skewed right?
(e) Is the mean a good indicator of the degree of astigmatism of a typical man aged 18 to 22?
Why or why not?
5. Prove that if each frequency is 1 then the formula for the grouped mean (page 61) is equivalent to the other
formula for the mean (page 56). i.e. Prove that if each f i  1 and if k  n then
k
 f i xi
i 1
k
f
i 1
n

x
i 1
i
n
i
6. Prove tha the two formulas for the standard deviation given on page 70 of the text are equivalent.
n
 x
i.e. Prove that
i 1
 x
2
i
n 1
 n 
n   xi     xi 
i 1
 i 1 

nn  1
n
2
2
Math 251 Science Supplement to Chapter 2 Partial Answers
1. (a),(b),(g),(j) Overall the distribution is slightly skewed to the left although if you remove outliers it is slightly
skewed to the right.
(c) The mean is 103 ounces. The median is 106 ounces. The mode is 106 ounces.
(d) 102.5 ounces.
(e) Q1 is 98 ounces, Q3 is 113 ounces.
(f) min 38 ounces, Q1 is 98 ounces, median 106 ounces, Q3 is 113 ounces, max 142 ounces.
(h) The range is 104 ounces, the IQR is 15 ounces, and the standard deviation is 20.5 ounces.
2. (a),(b),(g) The distribution is very skewed to the right.
(c) The mean is 2.75 days. The median is 2 days. The mode is 1 day.
(d) 3 days.
(e) Q1 is 1 day. Q3 is 3 days.
(f) min 1 day, Q1 is 1 day, median 2 days, Q3 is 3 days, max 11 days.
(h) The range is 10 days, the IQR is 2 days, and the standard deviation is 2.61 days.
(i) The mean is 4.67 days. The standard deviation is 3.93 days.
(j) The mean is 1.93 days. The standard deviation is 1.27 days.
(k) It appears that patients receiving antibiotics tend to stay in the hospital longer.
3. (a),(b),(g),(i) The distribution is skewed to the right and has two peaks.
(c) The mean change is 20 mg/dL. The median change is 14.5 mg/dL. The mode change is 30 mg/dL.
(d) 7 mg/dL
(e) Q1 is 3.5 mg/dL. Q3 is 36 mg/dL.
(f) min -12 mg/dL, Q1 is 3.5 mg/dL, median 14.5 mg/dL, Q3 is 36 mg/dL, and max 68 mg/dL..
(h) The range is 80 mg/dL, the IQR is 32.5 mg/dL, and the standard deviation is 21.1 mg/dL.
(k) The median baseline cholesterol is 191 mg/dL. For the group with baseline cholesterol below 191 mg/dL the
mean change was 7.4 mg/dL with a standard deviation of 12.0 mg/dL.. For the group with baseline cholesterol above
191 mg/dL the mean change was 32.8 mg/dL and the standard deviation was 21.1 mg/dL. The diet appears to be
more effective with people who have high baseline cholesterol levels.
4. (a) 0.3471 diopters
(b) 0.619273841 diopters
(c) 0.1 to 0.3 diopters
(d) Skewed right.
(e) Because the data is skewed right the median is a better indication of the degree of astigmatism of a typical young
man than the mean. The large degree of astigmatism of a few young men raise the mean.
k
 f i xi
5.
i 1
k
 fi
i 1
n

1 xi
i 1
n
1
i 1
n

 xi
i 1
1  1  1  1
n times
n

x
i 1
n
i
Math 251 Science Supplement to Chapter 2 Partial Answers (continued)
n
2
 xi  x 
i 1
6.
n 1
n

n

n
n xi  x 

i 1
nn  1
n
n
i 1
i 1
i 1
n xi   2nx  xi  nx 2 1
2
i 1
nn  1
n
n

n xi  x xi  x 
2
n   xi   2 n
2
i 1
x
 n 
n   xi     xi 
i 1
 i 1 

nn  1
n
2
n
 xi  n
n i 1
nn  1
i 1
n
2


i 1

nn  1
n
n xi   2nx  xi  n 2 x 2

2
i 1
n
i
nn  1
n
n xi2  2 x xi  x 2
i 1
nn  1
n
x x
2 i 1
n
i
i 1
n
i
n

n   xi 
i 1
2
2
 n   n 
 2  xi     xi 
 i 1   i 1 
nn  1
2