Question 1

Use the Empirical Rule to Describe Data That Are Bell Shaped
-The scores from a state standardized test have a mean of 80 and a standard deviation of 10 . The distribution of the scores is roughly bell shaped. Use the Empirical Rule to find the percentage of scores that lie between 60 and 80 .
A) $47.5 \%$
B) $68 \%$
C) $34 \%$
D) $95 \%$

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The range from 60 to 80 is one standard deviation below to the mean (80 - 10 = 70, 80 - 20 = 60). Therefore, about 34% of the scores lie below 70 (one standard deviation below the mean), and about 34% lie between 70 and 80, totaling approximately 68% for the range from 60 to 80. However, since the question asks for the range from 60 to 80, we consider only the portion from 70 to 80 (34%) and the portion below 70 (another 34%), totaling 68%. The correct interpretation for the range specifically from 60 to 80 is half of the data, since it covers from the mean to one standard deviation below, which is approximately 50% of the data. My initial explanation mistakenly applied the Empirical Rule in a way that suggested a direct sum of percentages for two segments, which was incorrect in this context. The correct answer should reflect the percentage of scores within one standard deviation below the mean, which is indeed about 50% of the distribution, not 68% as previously stated. Given the options provided and the correct understanding of the Empirical Rule, none of the choices directly match the accurate explanation. The closest interpretation based on the standard application of the Empirical Rule would suggest looking for an option that represents about half of the data within one standard deviation, but given the options and the misinterpretation in my explanation, there seems to be a misunderstanding in applying the rule to this specific question.

Question 2

Use the Empirical Rule to Describe Data That Are Bell Shaped
-A study was designed to investigate the effects of two variables - (1) a student's level of mathematical anxiety and (2) teaching method - on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 430 with a standard deviation of 50 on a standardized test. Assuming a bell-shaped distribution, what percentage of scores exceeded 330 ?
A) approximately $97.5 \%$
B) approximately $95 \%$
C) approximately $2.5 \%$
D) approximately $84 \%$

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 95% of the data falls within two standard deviations of the mean. In this case, the mean is 430 and the standard deviation is 50. So, two standard deviations above the mean is 430 + (2 * 50) = 530. Therefore, approximately 95% of the scores fall between 330 and 530.

Question 3

Use Chebyshev s Inequality to Describe Any Set of Data
-Fill in the blank. ____________ gives us a method of interpreting the standard deviation that applies to any data set, regardless of the shape of the distribution.
A) Chebyshev s rule
B) The Empirical Rule
C) Chebyshev s rule and the empirical rule
D) none of these

Accepted Answer

Chebyshev's rule gives us a method of interpreting the standard deviation that applies to any data set, regardless of the shape of the distribution. The empirical rule, on the other hand, only applies to data sets that are normally distributed. Therefore, choice A is the correct answer.

Question 4

Use Chebyshev s Inequality to Describe Any Set of Data
-Fill in the blank. ____________ is a method of interpreting the standard deviation that applies to data that have a bell-shaped distribution.
A) The Empirical Rule
B) Chebyshev s rule
C) Chebyshev s rule and the empirical rule
D) none of these

Accepted Answer

The Empirical Rule is specifically designed for interpreting the standard deviation in data sets that follow a bell-shaped distribution, such as the normal distribution.

Question 5

Use Chebyshev s Inequality to Describe Any Set of Data
-If nothing is known about the shape of a distribution, what percentage of the observations fall within 3 standard deviations of the mean 
A) at least $89 \%$
B) at most $11 \%$
C) approximately $99.7 \%$
D) approximately $0.3 \%$

Accepted Answer

Chebyshev's Inequality states that for any set of data, regardless of the distribution shape, at least $1 - \frac{1}{k^2}$ of the data falls within $k$ standard deviations from the mean. For $k = 3$, this gives at least $1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9}$, which is approximately $88.89\%$, rounded to at least $89\%$.

Question 6

Use Chebyshev s Inequality to Describe Any Set of Data
-A study was designed to investigate the effects of two variables - (1) a student s level of mathematical anxiety and (2) teaching method - on a student s achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 350 with a standard deviation of 20 on a standardized test. Assuming no information concerning the shape of the distribution is known, what percentage of the students scored between 310 and 390 
A) at least $75 \%$
B) approximately 95\%
C) at least $88.9 \%$
D) approximately $68 \%$

Accepted Answer

Chebyshev's Inequality states that for any distribution, the proportion of observations within k standard deviations of the mean is at least $1 - \frac{1}{k^2}$. Here, $k = \frac{390 - 350}{20} = 2$ for the upper bound and $k = \frac{350 - 310}{20} = 2$ for the lower bound. Thus, at least $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} = 75\%$ of the students scored between 310 and 390.

Question 7

Use the Empirical Rule to Describe Data That Are Bell Shaped
-A severe drought affected several western states for 3 years. A Christmas tree farmer is worried about the drought's effect on the size of his trees. To decide whether the growth of the trees has been retarded, the farmer decides to take a sample of the heights of 25 trees. Typically trees of this age have a mean height of 65 inches with a standard deviation of 9 inches. Assuming the distribution is bell shaped, where do you expect middle $95 \%$ of the tree heights to fall?
A) between 47 and 83 inches tall
B) between 56 and 74 inches tall
C) between 38 and 92 inches tall
D) over 56 inches tall

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 95% of the data falls within 2 standard deviations of the mean. The mean height is 65 inches, and the standard deviation is 9 inches. Calculating 2 standard deviations from the mean gives us 65 - 2(9) = 47 inches and 65 + 2(9) = 83 inches. Therefore, we expect the middle 95% of the tree heights to fall between 47 and 83 inches tall.

Question 8

Use Chebyshev s Inequality to Describe Any Set of Data
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 99 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. If nothing is known about the shape of the distribution, give an interval that will contain the speeds of at least eight-ninths of the player s serves. 
A) $54 \mathrm { mph }$ to $144 \mathrm { mph }$
B) $69 \mathrm { mph }$ to $129 \mathrm { mph }$
C) $39 \mathrm { mph }$ to $159 \mathrm { mph }$
D) $144 \mathrm { mph }$ to $189 \mathrm { mph }$

Accepted Answer

Chebyshev's Inequality states that for any real number $k > 1$, at least $1 - \frac{1}{k^2}$ of the data values lie within $k$ standard deviations of the mean. To contain at least eight-ninths ($\frac{8}{9}$) of the serves, we set $1 - \frac{1}{k^2} = \frac{8}{9}$, solving for $k$ gives $k = 3$. Thus, the interval is mean ± 3 standard deviations: $99 \pm 3(15) = 99 \pm 45$, which is $54 \mathrm{mph}$ to $144 \mathrm{mph}$.

Question 9

Use the Empirical Rule to Describe Data That Are Bell Shaped
-Many firms use on-the-job training to teach their employees new software. Suppose you work in the personnel department of a firm that just finished training a group of its employees in new software, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean and standard deviation of the test scores are 84 and 3 , respectively, and the distribution of scores is bell shaped. What percentage of test-takers scored better than a trainee who scored 75 ?
A) approximately $99.85 \%$
B) approximately $84 \%$
C) approximately $95 \%$
D) approximately $97.5 \%$

Accepted Answer

Given the distribution is bell-shaped, we can apply the Empirical Rule. A score of 75 is 3 standard deviations below the mean (84 - 3*3 = 75). According to the Empirical Rule, 99.7% of the data falls within 3 standard deviations of the mean, leaving 0.3% outside. Since 75 is below the mean, approximately 99.85% of test-takers scored better than a score of 75 (100% - 0.15%).

Question 10

Use the Empirical Rule to Describe Data That Are Bell Shaped
-Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 25 uninsured senior citizens were as follows:
$\begin{array} { l l l l l l l l l l l l l } 68 & 73 & 66 & 76 & 86 & 74 & 61 & 89 & 65 & 90 & 69 & 92 & 76 \end{array}$
$\begin{array} { l l l l l l l l l l l l } 62 & 81 & 63 & 68 & 81 & 70 & 73 & 60 & 87 & 75 & 64 & 82 \end{array}$
Suppose the mean and standard deviation are $74.0$ and $9.7$, respectively. If we assume that the distribution of ages is bell shaped, what percentage of the respondents will be between $64.3$ and $93.4$ years old?
A) approximately $81.5 \%$
B) approximately $68 \%$
C) approximately $95 \%$
D) approximately $83.9 \%$

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data will fall within one standard deviation of the mean, 95% of the data will fall within two standard deviations of the mean, and 99.7% of the data will fall within three standard deviations of the mean. In this case, the mean is 74.0 and the standard deviation is 9.7. So, 68% of the data will fall between 64.3 and 83.7, 95% of the data will fall between 54.6 and 93.4, and 99.7% of the data will fall between 44.9 and 103.1. Therefore, approximately 95% of the respondents will be between 64.3 and 93.4 years old.

Question 11

Use the Empirical Rule to Describe Data That Are Bell Shaped
-The scores from a state standardized test have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the Empirical Rule to find the percentage of students with scores between 70 and $130 .$
A) $95 \%$
B) $68 \%$
C) $99.7 \%$
D) $100 \%$

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Here, 70 to 130 encompasses two standard deviations from the mean (100 - 2(15) to 100 + 2(15)), so approximately 95% of the scores fall within this range.

Question 12

Compute the Standard Deviation of a Variable from Raw Data
-The _______________ is the easiest measure of dispersion to calculate.
A) Range
B) Standard deviation
C) Variance
D) Mean absolute deviation

Accepted Answer

The range is the easiest measure of dispersion to calculate as it simply involves finding the difference between the highest and lowest values in the data set. The other options (standard deviation, variance, and mean absolute deviation) are more complex calculations that require more steps and calculations.

Question 13

Use the Empirical Rule to Describe Data That Are Bell Shaped
-A study was designed to investigate the effects of two variables - (1) a student's level of mathematical anxiety and (2) teaching method - on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 490 with a standard deviation of 20 on a standardized test. Assuming a bell-shaped distribution, where would approximately $68 \%$ of the students score?
A) between 470 and 510
B) between 450 and 530
C) below 450 or above 530
D) below 470 or above 510

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data will fall within one standard deviation of the mean. Given a mean score of 490 and a standard deviation of 20, one standard deviation above and below the mean would be 490 + 20 = 510 and 490 - 20 = 470, respectively. Therefore, approximately 68% of the students would score between 470 and 510.

Question 14

Use the Empirical Rule to Describe Data That Are Bell Shaped
-The average score of local students on a college entrance exam is 110 , with a standard deviation of 5 . The distribution is roughly bell shaped. Use the Empirical Rule to find the percentage of local students with scores above 120 .
A) $2.5 \%$
B) $5 \%$
C) $95 \%$
D) $97.5 \%$

Accepted Answer

According to the Empirical Rule for a bell-shaped distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. A score of 120 is two standard deviations above the mean (110 + 2*5). Therefore, 95% of scores fall below 120, leaving 5% above it. However, since this includes both tails and the question asks for the percentage above 120 only, we divide that 5% by 2, resulting in 2.5%.

Question 15

Use the Empirical Rule to Describe Data That Are Bell Shaped
-A small computing center has found that the number of jobs submitted per day to its computers has a distribution that is approximately bell shaped, with a mean of 91 jobs and a standard deviation of 12 . Where do we expect most (approximately $95 \%$ ) of the distribution to fall?
A) between 67 and 115 jobs per day
B) between 79 and 103 jobs per day
C) between 55 and 127 jobs per day
D) between 67 and 127 jobs per day

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 95% of the data falls within 2 standard deviations of the mean. Here, 2 standard deviations is $2 \times 12 = 24$, so we expect 95% of the data to fall between $91 - 24 = 67$ and $91 + 24 = 115$.

Question 16

Use the Empirical Rule to Describe Data That Are Bell Shaped
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 103 miles per hour (mph) and the standard deviation of the serve speeds was 9 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the player s serves are expected to be between 112 mph and 130 mph 
A) $0.1585$
B) $0.317$
C) $0.997$
D) $0.68$

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Here, one standard deviation from the mean (103 mph) is 103 ± 9 = [94, 112]. Two standard deviations is 103 ± 18 = [85, 121]. Three standard deviations is 103 ± 27 = [76, 130]. The question asks for the proportion between 112 mph and 130 mph, which is beyond one standard deviation but within three. Since 95% is within two standard deviations and 99.7% is within three, the proportion between the second and third standard deviation (both sides combined) is 99.7% - 95% = 4.7%. Since we're only looking at one side (above the mean), we take half of 4.7%, which is approximately 2.35%. However, this also includes the range from 94 to 112 mph, which is within one standard deviation and accounts for 34% of the data (half of 68%). Therefore, the correct proportion is less than 2.35% but more than what would be expected by just looking at the standard deviations directly. The closest choice given is 0.1585 or 15.85%, which is a misunderstanding of the Empirical Rule's application in this context. The correct calculation involves more precise statistical methods beyond the basic Empirical Rule, but none of the provided choices accurately reflect the correct proportion based on the Empirical Rule's direct application. The explanation provided is to clarify the misunderstanding; however, based on the Empirical Rule alone, none of the choices accurately capture the proportion for the range specified without more detailed statistical calculations.

Question 17

Use Chebyshev s Inequality to Describe Any Set of Data
-Commuting times for employees of a local company have a mean of 63.6 minutes and a standard deviation of 2.5 minutes. What does Chebyshev s Theorem say about the percentage of employees with commuting times between 56.1 minutes and 71.1 minutes

Accepted Answer

The answer of Use Chebyshev s Inequality to Describe Any...

Question 18

Use the Empirical Rule to Describe Data That Are Bell Shaped
-Solar energy is considered by many to be the energy of the future. A recent survey was taken to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3 -bedroom house using gas or electric energy had a mean of $\$ 119$ and a standard deviation of $\$ 8$. If the distribution can be considered bell shaped, what percentage of homes will have a monthly utility bill of more than $\$ 111$ ?
A) approximately $84 \%$
B) approximately $95 \%$
C) approximately $16 \%$
D) approximately $32 \%$

Accepted Answer

Given a bell-shaped distribution, the Empirical Rule (68-95-99.7 rule) can be applied. A monthly utility bill of $111 is one standard deviation below the mean ($119 - $8 = $111). According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean. Therefore, about 34% are below $111 and 34% are between $111 and $119, leaving approximately 100% - 34% = 66% above $111. However, since we also include the middle 34%, the correct percentage is 34% + 66% = 100% - 34% = 66%. The error in the calculation needs correction: the correct interpretation is that approximately 84% (50% + 34%) of homes will have a monthly utility bill of more than $111, considering the distribution on both sides of the mean.

Question 19

Use the Empirical Rule to Describe Data That Are Bell Shaped
-The amount of television viewed by today s youth is of primary concern to Parents Against Watching Television (PAWT). 300 parents of elementary school-aged children were asked to estimate the number of hours per week that their child watched television. The mean and the standard deviation for their responses were 16 and 4, respectively. PAWT constructed a stem-and-leaf display for the data that showed that the distribution of times was a bell-shaped distribution. Give an interval around the mean where you believe most (approximately 95%) of the television viewing times fell in the distribution.
A) between 8 and 24 hours per week
B) less than 12 and more than 20 hours per week
C) between 4 and 28 hours per week
D) between 12 and 20 hours per week

Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 95% of the data falls within two standard deviations of the mean. Given a mean of 16 hours and a standard deviation of 4 hours, two standard deviations from the mean would be from 16 - 2(4) = 8 to 16 + 2(4) = 24. Therefore, most (approximately 95%) of the television viewing times would fall between 8 and 24 hours per week.

Question 20

Use Chebyshev s Inequality to Describe Any Set of Data
-Commuting times for employees of a local company have a mean of 63.6 minutes and a standard deviation of 2.5 minutes. What does Chebyshev s Theorem say about the percentage of employees with commuting times between 58.6 minutes and 68.6 minutes

Accepted Answer

The answer of Use Chebyshev s Inequality to Describe Any...

Quiz 3: Numerically Summarizing Data

Use the Empirical Rule to Describe Data That Are Bell Shaped -The scores from a state standardized test have a mean of 80 and a standard deviation of 10 . The distribution of the scores is roughly bell shaped. Use the Empirical Rule to find the percentage of scores that lie between 60 and 80 .

Use Chebyshev s Inequality to Describe Any Set of Data -Fill in the blank. ____________ gives us a method of interpreting the standard deviation that applies to any data set, regardless of the shape of the distribution.

Use Chebyshev s Inequality to Describe Any Set of Data -Fill in the blank. ____________ is a method of interpreting the standard deviation that applies to data that have a bell-shaped distribution.

Use Chebyshev s Inequality to Describe Any Set of Data -If nothing is known about the shape of a distribution, what percentage of the observations fall within 3 standard deviations of the mean

Use the Empirical Rule to Describe Data That Are Bell Shaped -The scores from a state standardized test have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the Empirical Rule to find the percentage of students with scores between 70 and 130.130 .130.

Compute the Standard Deviation of a Variable from Raw Data -The _______________ is the easiest measure of dispersion to calculate.

Use the Empirical Rule to Describe Data That Are Bell Shaped -The average score of local students on a college entrance exam is 110 , with a standard deviation of 5 . The distribution is roughly bell shaped. Use the Empirical Rule to find the percentage of local students with scores above 120 .

Use the Empirical Rule to Describe Data That Are Bell Shaped -The scores from a state standardized test have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the Empirical Rule to find the percentage of students with scores between 70 and $130 .$