Question 1

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  ± 2s.
B) Approximately 68% of all observations are in the interval  ± s.
C) Approximately 95% of all observations are in the interval  ± s.
D) Approximately 68% of all observations are in the interval  ± 2s.

Accepted Answer

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

Question 2

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 provides a lower bound on the proportion (or percentage) of observations lying within a certain number of standard deviations from the mean. Specifically, it states that at least 1 - 1/k^2 of the observations will fall within k standard deviations from the mean, where k is any positive constant greater than 1. This means that the proportion of observations within an interval will always be at least as large as the value given by Chebyshev's theorem.

Question 3

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 is applicable only for approximately symmetric and bell-shaped distributions, also known as normal distributions. Therefore, the best choice is option D. It cannot be used for skewed distributions or any other shape of distribution.

Question 4

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

Accepted Answer

The answer of Amounts spent by a sample of 200...

Question 5

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 6

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution.  (See the Excel Data File.) 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 7

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution.  (See the Excel Data File.) 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 frequency. Starting from the first class interval, we add up all the frequencies until we reach the midpoint of the total sample size (n=200/2=100).

Class Interval           |   Frequency  |   Relative Frequency | Cumulative Frequency
--------------------------------------------------------------------------------------
0 up to 10              |      36      |        0.18          |        36
10 up to 20             |      52      |        0.26          |        88
20 up to 30 (median)    |      55      |        0.275         |      143
30 up to 40             |      33      |        0.165         |      176
40 up to 50             |      11      |        0.055         |      187
50 up to 60             |      13      |        0.065         |      200

The median amount will fall in the class interval of 20 up to 30, which has a cumulative frequency of 143.

Question 8

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 9

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution.  (See the Excel Data File.) 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

The following frequency distribution represents the number of hours studied per week by a sample of 50 students.  (See the Excel Data File.) The mean number of hours studied is ________.
A) 7
B) 8
C) 22.8
D) 480

Accepted Answer

The answer of The following frequency distribution represents the number...

Question 11

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 12

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 can be applied to any distribution, regardless of its shape or skewness.

Question 13

What is the difference between Chebyshev's Theorem 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 is applicable to any data set, regardless of its distribution shape, providing a minimum proportion of observations within a certain number of standard deviations from the mean. The empirical rule, on the other hand, specifically applies to data sets with a distribution that is approximately symmetric and bell-shaped, detailing the percentage of data within one, two, and three standard deviations from the mean.

Question 14

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

Chebyshev's theorem states that at least (1 - 1/k^2) of data falls within k standard deviations from the mean. For scores less than 60 or more than 80, k = 2 (since 60 and 80 are 2 standard deviations away from the mean of 70). Thus, at least 1 - 1/4 = 0.75 of the students scored between 60 and 80, meaning at most 25% (or 15 students) scored less than 60 or more than 80.

Question 15

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) 68%
B) 65%
C) 75%
D) 95%

Accepted Answer

Chebyshev's inequality states that at least 1 - (1/k^2) of the data will fall within k standard deviations of the mean, where k is any number greater than 1. To find the percentage of salaries falling between $68,000 and $92,000, we need to find how many standard deviations away from the mean those values are: $68,000 - $80,000 = -$12,000 -$12,000/$6,000 = -2 $92,000 - $80,000 = $12,000 $12,000/$6,000 = 2 Therefore, the range from $68,000 to $92,000 is 2 standard deviations away from the mean. Using the inequality, we can say: P(|X - μ| < kσ) >= 1 - (1/k^2) where X = salary of professors, μ = mean salary ($80,000), σ = standard deviation ($6,000), and k = 2 (since we want to find the range within 2 standard deviations from the mean). Plugging in the values, we get: P(|X - 80,000| < 2(6,000)) >= 1 - (1/2^2) P(68,000 < X < 92,000) >= 3/4 So we can say with at least 75% confidence that salaries fall between $68,000 and $92,000. Therefore, the answer is C.

Question 16

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 are approximately symmetric and bell-shaped, meaning that the majority of the data falls within a certain range around the mean. This assumption allows for the use of specific percentages (68%, 95%, 99.7%) when making predictions or analyzing data. The data can come from either a sample or a population.

Question 17

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

In a normal distribution, the bottom 2.5% corresponds to approximately 1.96 standard deviations below the mean. Calculating: $80,000 - (1.96 * $6,000) = $68,240$, which is closest to $68,000.

Question 18

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 for a normal distribution, approximately 68% of observations fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations of the mean, and 99.7% fall within 3 standard deviations of the mean. Therefore, the best choice is A, which includes these percentages.

Question 19

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,...

Question 20

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

Analysis of Relative Location provides insights into the percentage of data values that fall within certain standard deviations from the mean, often using concepts like z-scores and the empirical rule.

Quiz 3: Numerical Descriptive Measures

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

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

The empirical rule can be used to estimate some proportions ________.

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution. (See the Excel Data File.) The mean amount spent by customers is the closest to ________.

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 50 customers at a retail store are summarized in the following relative frequency distribution. (See the Excel Data File.) The mean amount spent by customers is the closest to ________.

Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution. (See the Excel Data File.) The median amount will fall in the following class interval ________.

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?

Amounts spent by a sample of 50 customers at a retail store are summarized in the following relative frequency distribution. (See the Excel Data File.) The median amount will fall in the following class interval ________.

The following frequency distribution represents the number of hours studied per week by a sample of 50 students. (See the Excel Data File.) The mean number of hours studied is ________.

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?

Chebyshev's theorem is applicable when the data are ________.

What is the difference between Chebyshev's Theorem and the Empirical Rule?

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.

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 ________.

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

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

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.

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