Question 1

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution cannot be regarded as bell-shaped.What can be said about the percentage of salaries that are less than $68,000 or more than or more than $92,000?
A) It is at least 75%.
B) It is at most 75%.
C) It is at least 25%.
D) It is at most 25%.

Accepted Answer

According to Chebyshev's theorem, for any distribution (regardless of its shape), at least $1 - \frac{1}{k^2}$ of the data values fall within k standard deviations from the mean. Here, $68,000 and $92,000 are 2 standard deviations away from the mean ($80,000 ± 2 \times $6,000). Thus, at least $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}$ or 75% of the salaries fall within this range. Therefore, at most 25% of the salaries are outside this range, making option D correct.

Question 2

The empirical rule can be used to estimate some proportions_________________.
A) when data has any shape.
B) when it is skewed to the left.
C) when it is skewed to the right.
D) when it is approximately symmetric and bell-shaped.

Accepted Answer

The empirical rule applies to data that is approximately symmetric and bell-shaped, as it relies on the assumption of normal distribution. It cannot be used for skewed data.

Question 3

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution is approximately bell-shaped.What can be said about the percentage of salaries that are at least $74,000?
A) It is about 68%.
B) It is about 84%.
C) It is about 95%.
D) It is about 97.5%.

Accepted Answer

Given the distribution is approximately bell-shaped (normal distribution), we can use the empirical rule. A salary of $74,000 is one standard deviation below the mean ($80,000 - $74,000 = $6,000). According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, which means about 34% is below and 34% is above one standard deviation from the mean. Since we are looking for salaries at least $74,000, we add the 34% above one standard deviation to the 50% that falls within the first standard deviation to the mean, totaling approximately 84%.

Question 4

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution.  The median amount will fall in the following class interval ____________.
A) 0 up to 10
B) 10 up to 20
C) 20 up to 30
D) 30 up to 40

Accepted Answer

To find the median, we need to first calculate the cumulative relative frequency, which is the sum of each frequency and all the frequencies below it.

Class Interval | Frequency | Relative Frequency | Cumulative Frequency | Cumulative Relative Frequency
--- | --- | --- | --- | ---
0 up to 10 | 36 | 0.18 | 36 | 0.18
10 up to 20 | 72 | 0.36 | 108 | 0.54
20 up to 30 | 56 | 0.28 | 164 | 0.82
30 up to 40 | 28 | 0.14 | 192 | 0.96
40 up to 50 | 8 | 0.04 | 200 | 1.00

The median is the middle value in the data set, which in this case is the 100th value (since there are 200 customers). Looking at the cumulative relative frequencies, we can see that the median amount falls in the third class interval, 20 up to 30, since the cumulative frequency at the end of that interval (164) is closest to (but not greater than) the middle value of 100. Therefore, the answer is choice C.

Question 5

When applicable,the empirical rule provides the approximate percentage of observations that fall within A) 1,2,or 3 standard deviations. B) 2,3,or 4 standard deviations. C) k standard deviations for every k> 1. D) 1 - 1/k² standard deviations for every k> 1.

Accepted Answer

The empirical rule states that approximately 68% of observations fall within 1 standard deviation of the mean, 95% of observations fall within 2 standard deviations of the mean, and 99.7% of observations fall within 3 standard deviations of the mean. Therefore, option A is the best choice as it correctly states the percentage of observations within 1, 2, or 3 standard deviations.

Question 6

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution.  The mean amount spent by customers is the closest to _____.
A) $17.50
B) $20.00
C) $22.50
D) $50.00

Accepted Answer

To find the mean amount spent by customers, we need to multiply each value by its corresponding relative frequency, then add up all the products and divide by the total number of customers (200).

Using the midpoint of each class interval as the representative value, we can create the following table:

| Amount Spent | Relative Frequency | Product |
|--------------|--------------------|---------|
| $10-$15      | 0.10               | $12.50  |
| $15-$20      | 0.25               | $17.50  |
| $20-$25      | 0.30               | $22.50  |
| $25-$30      | 0.20               | $27.50  |
| $30-$50      | 0.15               | $40.00  |

To find the mean, we add up all the products:

$12.50 + $17.50 + $22.50 + $27.50 + $40.00 = $120.00

Then we divide by the total number of customers (200):

$120.00 ÷ 200 = $0.60

Therefore, the mean amount spent by customers is $0.60, which corresponds to the midpoint of the $20-$25 class interval. So the best choice is C) $22.50.

Question 7

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution.  The mean amount spent by customers is the closest to ______.
A) $0.36
B) $18.00
C) $20.00
D) $25.00

Accepted Answer

The answer of Amounts spent by a sample of 50...

Question 8

Chebyshev's theorem is applicable when the data are___________________.
A) any shape
B) skewed to the left
C) skewed to the right
D) approximately symmetric and bell-shaped

Accepted Answer

Chebyshev's theorem is applicable to any shape of data, regardless of whether it is skewed or symmetric. It provides a general rule for the proportion of data that falls within a certain number of standard deviations from the mean, regardless of the shape of the distribution.

Question 9

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution.  The median amount will fall in the following class interval _____________.
A) 0 up to 10
B) 10 up to 20
C) 20 up to 30
D) 30 up to 40

Accepted Answer

The answer of Amounts spent by a sample of 50...

Question 10

When using the empirical rule,which of the following assumptions is made?
A) The data only comes from a sample.
B) The data only comes from a population.
C) The data are exactly symmetric and bell-shaped.
D) The data are approximately symmetric and bell-shaped.

Accepted Answer

The empirical rule assumes that the data is approximately symmetric and bell-shaped, meaning that the majority of the data falls within one, two, or three standard deviations of the mean. This is based on the normal distribution, which is a theoretical model for data that is approximately symmetric and bell-shaped. The empirical rule can be used with data from both a sample and a population.

Question 11

Which of the following is true when using the empirical rule for a set of sample data?
A) Almost all observations are in the interval 
B) Approximately 68% of all observations are in the interval 
C) Approximately 95% of all observations are in the interval 
D) Approximately 68% of all observations are in the interval

Accepted Answer

The answer of Which of the following is true when...

Question 12

Which of the following capabilities does Analysis of Relative Location provide?
A) They make statements regarding the shape of the data.
B) They make statements regarding the central location of the data.
C) They make statements regarding the dispersion of the data around the median.
D) They make statements regarding the percentage of data values that fall within some number of standard deviations from the mean.

Accepted Answer

The answer of Which of the following capabilities does Analysis...

Question 13

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution is approximately bell-shaped.Because of budget limitations,it has been decided that only those whose salaries are approximately in the bottom 2.5% would get a raise.What is the maximum current salary that qualifies for the raise?
A) It is about $58,000.
B) It is about $62,000.
C) It is about $68,000.
D) It is about $74,000.

Accepted Answer

1. We know that the distribution is bell-shaped or approximately normal with a mean of $80,000 and a standard deviation of $6,000.
2. We want to find the maximum current salary that qualifies for the raise, which means we need to find the value that corresponds to the bottom 2.5% of the distribution.
3. To do this, we can use a standard normal distribution table (also known as a z-score table), which gives us the proportion of values below a certain z-score.
4. Since we know the mean and standard deviation, we can standardize any salary value in the distribution to a z-score using the formula:

z-score = (x - mean) / standard deviation

where x is the salary value we are interested in.
5. We want to find the z-score that corresponds to the 2.5th percentile, which is the same as the area to the left of that percentile on the standard normal distribution table. This is approximately equal to -1.96 (you can look this up on the table).
6. We can solve for x (the salary value) by rearranging the z-score formula:

z-score = (x - mean) / standard deviation
-1.96 = (x - 80,000) / 6,000
-11,760 = x - 80,000
x = $68,240

Therefore, the maximum current salary that qualifies for the raise is about $68,000 (rounded to the nearest thousand).

Question 14

What is the difference between Analysis of Relative Location and the empirical rule?
A) The empirical rule applies to all data sets,whether their distributions are symmetric and bell-shaped or not.
B) Chebyshev's theorem applies to all data sets except those that have approximately a symmetric and bell-shaped distribution.
C) The empirical rule applies to all data sets,whereas Chebyshev's theorem is appropriate when the data have approximately a symmetric and bell-shaped distribution.
D) Chebyshev's theorem applies to all data sets,whereas the empirical rule is only appropriate when the data have approximately a symmetric and bell-shaped distribution.

Accepted Answer

Chebyshev's theorem applies to all data sets regardless of their distribution shape, while the empirical rule is specifically for data sets with a symmetric, bell-shaped distribution.

Question 15

In its standard form,Chebyshev's theorem provides a lower bound on __________________________________
A) the number of observations lying within a certain interval
B) the number of observations lying outside a certain interval
C) the proportion (or percentage)of observations lying within a certain interval
D) the proportion (or percentage)of observations lying outside a certain interval

Accepted Answer

Chebyshev's theorem states that for any data set no matter how it is distributed, at least (1 - 1/k^2) proportion (or percentage) of the observations must lie within k standard deviations of the mean. Therefore, it provides a lower bound on the proportion (or percentage) of observations lying within a certain interval (within k standard deviations of the mean), not on the number of observations.

Question 16

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.Using Chebyshev's inequality,the percentage of salaries falling between $68,000 and $92,000 is at least __________.
A) 65%
B) 68%
C) 75%
D) 95%

Accepted Answer

Chebyshev's inequality states that at least (1 - 1/k^2) of the data falls within k standard deviations of the mean. Here, $68,000 and $92,000 are $12,000 away from the mean, which is 2 standard deviations ($12,000/$6,000 = 2). So, at least (1 - 1/2^2) = 75% of the salaries fall within this range.

Question 17

In an accounting class of 200 students,the mean and standard deviation of scores was 70 and 5,respectively.Use the empirical rule to determine the number of students who scored less than 65 or more than 75.
A) It is about 32.
B) It is about 64.
C) It is about 68.
D) It is about 136.

Accepted Answer

The answer of In an accounting class of 200 students,the...

Question 18

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution is approximately bell-shaped.What can be said about the percentage of salaries that are less than $68,000 or more than $92,000?
A) It is about 5%.
B) It is about 32%.
C) It is about 68%.
D) It is about 95%.

Accepted Answer

According to the empirical rule (68-95-99.7 rule) for a bell-shaped distribution, approximately 95% of the data falls within two standard deviations of the mean. Since $68,000 and $92,000 are two standard deviations away from the mean ($80,000 ± 2*$6,000), the percentage of salaries outside this range (less than $68,000 or more than $92,000) is about 5%.

Question 19

For k> 1,Chebyshev's theorem is useful in estimating the proportion of observations that fall within ________________________________. A) k standard deviations from the mean B) k² standard deviations from the mean C) (1 - 1/k)standard deviations from the mean D) (1 - 1/k²)standard deviations from the mean

Accepted Answer

Chebyshev's theorem states that for any data set, regardless of the shape of the distribution, at least 1-1/k^2 of the data values will fall within k standard deviations from the mean.

Question 20

In a marketing class of 60 students,the mean and the standard deviation of scores was 70 and 5,respectively.Use Chebyshev's theorem to determine the number of students who scored less than 60 or more than 80.
A) At least 45
B) At most 15
C) At most 45
D) At least 15

Accepted Answer

The answer of In a marketing class of 60 students,the...

Quiz 3: Numerical Descriptive Measures

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution cannot be regarded as bell-shaped.What can be said about the percentage of salaries that are less than $68,000 or more than or more than $92,000?

The empirical rule can be used to estimate some proportions_________________.

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution is approximately bell-shaped.What can be said about the percentage of salaries that are at least $74,000?

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution. The median amount will fall in the following class interval ____________.

When applicable,the empirical rule provides the approximate percentage of observations that fall within

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution. The mean amount spent by customers is the closest to _____.

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution. The mean amount spent by customers is the closest to ______.

Chebyshev's theorem is applicable when the data are___________________.

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution. The median amount will fall in the following class interval _____________.

When using the empirical rule,which of the following assumptions is made?

Which of the following is true when using the empirical rule for a set of sample data?

Which of the following capabilities does Analysis of Relative Location provide?

What is the difference between Analysis of Relative Location and the empirical rule?

In its standard form,Chebyshev's theorem provides a lower bound on __________________________________

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.Using Chebyshev's inequality,the percentage of salaries falling between $68,000 and $92,000 is at least __________.

In an accounting class of 200 students,the mean and standard deviation of scores was 70 and 5,respectively.Use the empirical rule to determine the number of students who scored less than 65 or more than 75.

Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000.The salary distribution is approximately bell-shaped.What can be said about the percentage of salaries that are less than $68,000 or more than $92,000?

For k> 1,Chebyshev's theorem is useful in estimating the proportion of observations that fall within ________________________________.

In a marketing class of 60 students,the mean and the standard deviation of scores was 70 and 5,respectively.Use Chebyshev's theorem to determine the number of students who scored less than 60 or more than 80.