Download Online 03 - Sections 2.3 and 2.4

Student: _____________________
Date: _____________________
Instructor: Doug Ensley
Assignment: Online 03 ­ Sections 2.3
Course: MAT117 01 Applied Statistics
and 2.4
­ Ensley
1. Consider the following three sets of observations.
Set 1: 3 , 5 , 7 , 9 , 11
Set 2: 3 , 5 , 7 , 9 , 100
Set 3: 3 , 5 , 7 , 9 , 1000
a. Find the median for each data set.
b. Find the mean for each data set.
c. What do these data sets illustrate about the resistance of the median and the mean?
a. The median of set 1 is .
The median of set 2 is .
The median of set 3 is .
b. The mean of set 1 is .
The mean of set 2 is .
The mean of set 3 is .
c. What do these data sets illustrate about the resistance of the median and the mean?
A. The mean is resistant while the median is not.
B. Both the median and the mean are resistant.
C. The median is resistant while the mean is not.
D. Neither the median nor the mean are resistant.
2. According to a study, the median household income was $51,605 for one group, and $30,709 for another. The mean for each group was $69,688 and $44,366 , respectively. Does this suggest that the distribution of houshold income for each group is symmetric, skewed to the right, or skewed to the left? Explain.
Choose the correct answer below.
A. The fact that the mean is larger than the median for each group indicates that there are extremely large incomes that are affecting the mean, suggesting that the shape is skewed to the left.
B. The fact that the median is larger than the mean for each group indicates that there are extremely small incomes that are affecting the mean, suggesting that the shape is skewed to the left.
C. The fact that the mean is larger than the median for each group indicates that there are extremely large incomes that are affecting the mean, suggesting that the shape is skewed to the right.
D. The fact that the mean and median of one group is larger than the mean and median of the other group indicates that the shape is skewed to the right.
E. The fact that both means are larger than their respective medians indicates that the shape is symmetric.
3. The dot plot to the right shows cereal sodium values. What aspect of the distribution causes the mean to be greater than the median?
median = 100
mean = 115.6
12
0
298
100
200
300
Sodium (mg)
Choose the correct answer below.
A. The sodium value of 12 is an outlier. It causes the mean to be pulled to the left.
B. The sodium value of 298 is an outlier. It causes the mean to be pulled to the left.
C. The sodium value of 298 is an outlier. It causes the mean to be pulled to the right.
D. The sodium values of 298 and 12 are outliers. They cause the distribution to be symmetric.
4. For the dot plot on the right, determine whether the mean is greater than, less than, or equal to the median. Justify your answer.
2
4 6
Which of the following is correct?
A. The mean is greater than the median because the dot plot is symmetrical.
B. The mean is equal to the median because the dot plot is skewed right. C. The mean is greater than the median because the dot plot is skewed left.
D. The mean is equal to the median because the dot plot is symmetrical. E. The mean is less than the median because the dot plot is skewed right.
F. The mean is less than the median because the dot plot is skewed left.
8
5. The owner of a company would like to promote the use of public transportation. She decides to investigate how many miles her employees travel on public transportation during a typical day. The values for her ten employees (recorded to the closest mile) are shown below.
0
4
0
0
0
11
0
6
0
0
a. Find the mean, median, and mode.
b. She has just hired an additional employee. He lives in a different city and travels 85 miles a day on public transportation. Recompute the mean and median. Describe the effect of this outlier.
a. The mean is The median is The mode is mile(s). (Round to two decimal places as needed.)
mile(s).
mile(s).
b. Recompute the mean and median with the new employee included. Describe the effect of this outlier.
The mean is mile(s).
(Round to two decimal places as needed.)
The median is mile(s).
What effect does the outlier have on the mean and median?
A. The outlier does not effect the mean or the median.
B. The outlier effects the mean but not the median.
C. The outlier effects the median but not the mean.
D. The outlier effects both the mean and the median.
6. A recent national census found that the median household income in a certain country was $62,100 and the mean was $58,100. Based on only this information, what would you predict about the shape of the distribution? Why?
Choose the correct prediction below.
A. The fact that the mean is less than the median indicates that there are extremely low incomes that are affecting the mean, but not the median, suggesting that the shape is skewed to the left. B. The fact that the mean is less than the median indicates that there are extremely low incomes, suggesting that the shape is skewed to the right.
C. The fact that the mean is less than the median indicates that there are extremely high incomes that are affecting the mean, but not the median, suggesting that the shape is skewed to the right.
D. The fact that the mean is less than the median indicates that there are extremely high incomes that are affecting the mean, suggesting that the shape is symmetric.
7. The following table summarizes responses of 4382 subjects in a recent survey to the question, "Within the past month, how many people have you known personally that were victims of homicide?"
Use this information to answer parts a through c.
Number of Victims Frequency
0
3938
1
287
2
90
3
45
4 or more
22
Total
4382
a. To find the mean, it is necessary to give a score to the "4 or more" category. Find it, using the score 4.5.
The mean is victim(s)
(Round to two decimal places as needed.)
b. Find the median. Note the "4 or more" category is not problematic.
The median is victim(s).
c. If 1738 observations shift from 0 to 4 or more, how do the mean and median change?
A. Both the mean and the median increase.
B. Both the mean and the median remain the same.
C. The median increases while the mean remains the same.
D. The mean increases while the median remains the same.
8. Height has an approximately bell­shaped distribution. For a sample of heights of college students collected, the males had x = 71 and s = 4 and the females had x = 67 and s = 4.
Use this information to answer parts a and b.
a. Use the empirical rule to describe the distribution of heights for females.
Approximately 68% of the observations fall within the interval (
(Type a whole number.)
,
).
Approximately 95% of the observations fall within the interval (
(Type a whole number.)
,
).
All or nearly all of the observations fall within the interval (
(Type a whole number.)
,
).
b. The standard deviation for the overall distribution (combining females and males) was 5. Why would you expect it to be larger than the standard deviations for the separate male and female height distributions?
A. This will always be the case because it increases the sample size.
B. The standard deviation for the overall distribution of a combination will usually be larger than the standard deviation for two distributions with different means because it introduces more spread to the data.
C. This is expected because the distributions are bell­shaped.
D. This is the case because the standard deviations for the separate distributions are equal.
9. The linked figure shows histograms for three different samples, each with sample size n = 100.
a. Which sample has the (i) largest and (ii) smallest standard deviation?
b. To which sample(s) is the empirical rule relevant? Why?
1
Click the icon to view the figure. a. Which sample has the (i) largest and (ii) smallest standard deviation?
A. Sample (a) has the largest standard deviation because it is the most bell shaped and sample (b) has the smallest standard deviation because it is the least bell shaped.
B. Sample (a) has the largest standard deviation because it is the most bell shaped and sample (c) has the smallest standard deviation because it is the least bell shaped.
C. Sample (c) has the largest standard deviation because it has the lowest typical distance from the mean and sample (b) has the smallest standard deviation because it has the highest typical distance from the mean.
D. Sample (c) has the largest standard deviation because it has the highest typical distance from the mean and sample (b) has the smallest standard deviation because it has the lowest typical distance from the mean.
b. To which sample(s) is the empirical rule relevant? Why?
A. Sample (c) because it is not bell shaped.
B. Sample (b) because all of the observations are the same.
C. Sample (a), sample (b), and sample (c) because they all have the required sample size.
D. Sample (a) because it is bell shaped.
1: Figure
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
(a)
(b)
(c)
10. Suppose that data for 57 randomly selected female high school athletes was collected on the maximum number of pounds they were able to bench press. The data are roughly bell shaped, with x = 79.1 and s = 12.4. Use the empirical rule to describe the distribution.
Use the empirical rule to describe the distribution.
Approximately 68% of the observations fall within the interval (
(Type integers or decimals.)
,
).
Approximately 95% of the observations fall within the interval (
(Type integers or decimals.)
,
).
All or nearly all of the observations fall within the interval (
(Type integers or decimals.)
,
).
11. Data for 64 male college athletes was collected. The data on weight (in pounds) are roughly bell shaped with x = 152 and s = 15. Complete parts a and b below.
a. Give an interval within which about 95 % of the weights fall.
(
,
) (Type a whole number.)
b. Identify the weight of an athlete who is three standard deviations above the mean in this sample. Would this be a rather unusual observation? Why?
An athlete who is three standard deviations above the mean would weigh (Type a whole number.)
pounds.
Would the weight above be an unusual observation?
A. N
o, this would not be an unusual observation because typically all or nearly all observations fall within three standard deviations from the mean.
B. Y
es, this would be an unusual observation because typically all or nearly all observations fall within three standard deviations from the mean.
C. N
o, this would not be an unusual observation because approximately 68% of all observations fall within three standard deviations from the mean.
D. Y
es, this would be an unusual observation because approximately 95% of all observations fall within three standard deviations from the mean.
12.
The table to the right gives data on a survey of the number of times married for men of age 20 to No. Count Times
24. For the observations, x = 0.22 and s = 0.43.
0
7241
Use this information to answer parts a and b.
1
1963
2
45
Total 9249
a. Find the actual percentages of observations within one, two, and three standard deviations of the mean. How do these compare to the percentages predicted by the Empirical Rule?
of the observations are within one standard deviation of the mean.
%
(Round to one decimal place as needed.)
of the observations are within two standard deviations of the mean.
%
(Round to one decimal place as needed.)
of the observations are within three standard deviations of the mean.
%
(Round to one decimal place as needed.)
How do the actual percentages compare to the percentages predicted by the Empirical Rule?
A. There are fewer observations within 1 standard deviation of the mean and more within 2 standard deviations than predicted by the Empirical Rule.
B. There are more observations within 1 standard deviation of the mean and more within 2 standard deviations than predicted by the Empirical Rule.
C. There are more observations within 1 standard deviation of the mean and fewer within 2 standard deviations than predicted by the Empirical Rule.
b. How do you explain the results in (a)?
A. The data concerns a sample of the population, so the Empirical Rule is only applicable if the data are bell­shaped.
B. The data concerns a sample of the population, so the Empirical Rule may not be applicable.
C. The Empirical Rule is only valid when used with data from a bell­shaped distribution.
D. The results are such because the data is discrete instead of continuous.
1. 7
7
7
7
24.8
204.8
C. The median is resistant while the mean is not.
2. C.
The fact that the mean is larger than the median for each group indicates that there are extremely large incomes that are affecting the mean, suggesting that the shape is skewed to the right.
3. C. The sodium value of 298 is an outlier. It causes the mean to be pulled to the right.
4. D. The mean is equal to the median because the dot plot is symmetrical. 5. 2.1
0
0
9.64
0
B. The outlier effects the mean but not the median.
6. A.
The fact that the mean is less than the median indicates that there are extremely low incomes that are affecting the mean, but not the median, suggesting that the shape is skewed to the left. 7. 0.16
0
D. The mean increases while the median remains the same.
8. 63
71
59
75
55
79
B.
The standard deviation for the overall distribution of a combination will usually be larger than the standard deviation for two distributions with different means because it introduces more spread to the data.
9. D.
Sample (c) has the largest standard deviation because it has the highest typical distance from the mean and sample (b) has the smallest standard deviation because it has the lowest typical distance from the mean.
D. Sample (a) because it is bell shaped.
10. 66.7
91.5
54.3
103.9
41.9
116.3
11. 122
182
197
B.
Yes, this would be an unusual observation because typically all or nearly all observations fall within three standard deviations from the mean.
12. 78.3
99.5
99.5
B.
There are more observations within 1 standard deviation of the mean and more within 2 standard deviations than predicted by the Empirical Rule.
C. The Empirical Rule is only valid when used with data from a bell­shaped distribution.