Question 1

Incomes of workers in an automobile company in Ontario are known to be right-skewed, with a mean equal to $36,200. Applying Tchebysheff's Theorem, at least 8/9 of all incomes are in the range of $29,600 to $42,800. What is the standard deviation of those incomes from that mean?
A) $2,200
B) $4,755
C) $6,500
D) $6,700

Accepted Answer

Tchebysheff's Theorem states that at least 1 - 1/k^2 of all data values will fall within k standard deviations of the mean. In this case, k = 3, so at least 1 - 1/3^2 = 8/9 of all incomes will fall within 3 standard deviations of the mean. Therefore, the standard deviation must be (42,800 - 29,600) / 6 = $2,200.

Question 2

A university placement office conducted a survey of 100 engineers who had graduated from a local university. For these engineers, the mean salary was computed to be $72,000 with a standard deviation of $8,000. Which of the following best characterizes the percentage of these engineers who earn either more than $96,000 or less than $48,000?
A) approximately 2.3%
B) at least 5.6% (1/18 of the engineers)
C) at most 5.6% (1/18 of the engineers)
D) at most 11.1% (1/9 of the engineers)

Accepted Answer

To find the percentage of engineers who earn either more than $96,000 or less than $48,000, we need to find the percentage of engineers who earn more than $96,000 and the percentage of engineers who earn less than $48,000 separately and then add them up.

To find the percentage of engineers who earn more than $96,000, we need to standardize this value by finding the z-score:

z = (x - μ) / σ
z = (96000 - 72000) / 8000
z = 3

Using a standard normal table or calculator, we find that the percentage of engineers who earn more than $96,000 is approximately 0.27%.

To find the percentage of engineers who earn less than $48,000, we again need to standardize this value:

z = (x - μ) / σ
z = (48000 - 72000) / 8000
z = -3

Using a standard normal table or calculator, we find that the percentage of engineers who earn less than $48,000 is approximately 0.27%.

Adding the two percentages together, we get a total of:

0.27% + 0.27% = 0.54%

To convert this decimal to a percentage, we multiply by 100:

0.54% = 0.54 x 100 = 5.4%

Therefore, the percentage of engineers who earn either more than $96,000 or less than $48,000 is at most 11.1%, which is answer choice D (since all other answer choices are less than 5.4%).

Question 3

The expression  ;  is recognizable as the formula for which of the following measures?
A) the sample variance, computed from ungrouped data
B) the population variance, computed from ungrouped data
C) the sample variance, computed from grouped data
D) the population variance, computed from grouped data

Accepted Answer

The answer of The expression  ;  is recognizable...

Question 4

Which one of these values represents the upper quartile of the data set 10, 12, 16, 7, 9, 7, 41, and 14?
A) 7
B) 8
C) 15.5
D) 24

Accepted Answer

The upper quartile (Q3) is the median of the upper half of the data set. First, arrange the data in ascending order: 7, 7, 9, 10, 12, 14, 16, 41. The upper half is 12, 14, 16, 41, and the median of these numbers is (14+16)/2 = 15.5.

Question 5

Which of the following summary measures is most affected by outliers?
A) the first quartile
B) the second quartile
C) the third quartile
D) the variance

Accepted Answer

The variance is heavily influenced by outliers because it takes into account the squared distances between each data point and the mean, making extreme values have a larger impact on the result. Quartiles are less affected by outliers since they only consider specific portions of the data.

Question 6

You are given a distribution of measurements that is approximately mound-shaped. According to the Empirical Rule, what would be the approximate percentage of measurements in a data set that will fall within two standard deviations of their mean?
A) 99%
B) 95%
C) 90%
D) 68%

Accepted Answer

According to the Empirical Rule, approximately 95% of measurements will fall within two standard deviations of their mean in a mound-shaped distribution.

Question 7

The distribution of actual volumes of tomato soup in 450 mL cans is thought to be bell-shaped, with a mean of 450 mL and a standard deviation equal to 8 mL. Based on this information, between what two values could we expect 95% of all cans to contain?
A) 430 and 470 mL
B) 432 and 468 mL
C) 434 and 466 mL
D) 440 and 460 mL

Accepted Answer

Since we want to find the range that contains 95% of all cans, we need to find the interval that corresponds to two standard deviations away from the mean. We can use the empirical rule to determine that about 95% of data falls within two standard deviations of the mean. Therefore, we can calculate the range by subtracting and adding two standard deviations (.16*450 = 72 mL) from the mean: 450 - 72 = 378 mL and 450 + 72 = 522 mL. However, since we are dealing with cans of tomato soup which cannot contain less than 0 mL or more than 450 mL, we need to reduce this range to the interval between 378 mL and 450 mL. Dividing the range by two gives us 36 mL, which we can add to and subtract from the mean to give the range between 434 and 466 mL. Therefore, the best choice is C.

Question 8

According to Tchebysheff's Theorem, which of the following bounds will delimit the fraction of observations falling within k (where  ) standard deviations of the mean?
A) at most, 1 - .
B) at least
C) at most, 1 - ,
D) at least 1 -

Accepted Answer

The answer of According to Tchebysheff's Theorem, which of the...

Question 9

Which of the following is a characteristic of a population mean?
A) It will always be larger than the mean of a sample selected from that population.
B) It will always be larger than the population median.
C) It will usually differ in value from the mean of a sample selected from that population.
D) It will always be smaller than the population median.

Accepted Answer

The population mean will usually differ in value from the mean of a sample selected from that population due to sampling variability. It is not always larger than the sample mean or the population median, nor is it always smaller than the population median.

Question 10

Which of the following randomly selected measurements, x, might be considered a potential outlier if it were to be selected from the given population?
A) x = 0 from a population with= 0 and= 2
B) x = -5 from a population with= 1 and= 4
C) x = 7 from a population with= 3 and= 2
D) x = 4 from a population with= 0 and= 1

Accepted Answer

The answer of Which of the following randomly selected measurements,...

Question 11

According to Tchebysheff's Theorem, what is the percentage of measurements in a data set that will fall within three standard deviations of the mean?
A) 16%
B) at least 68%
C) 75%
D) at least 89%

Accepted Answer

According to Tchebysheff's Theorem, at least 89% of the measurements in a data set will fall within three standard deviations of the mean. This is true for any data set, regardless of its distribution. Therefore, option D is the correct answer.

Question 12

Which of the following is the most frequently used measure of variation?
A) the mean
B) the range
C) the variance
D) the standard deviation

Accepted Answer

The standard deviation is the most frequently used measure of variation because it takes into account all the data points and provides a measure of how spread out the data is from the mean. The range is not as reliable as it only considers the highest and lowest values. The variance also takes into account all the data points, but is not as commonly used as it provides a measure of dispersion that is squared, making it less interpretable.

Question 13

Expressed in percentiles, how is the interquartile range defined?
A) It is the difference between the 20% and 70% values.
B) It is the difference between the 20% and 80% values.
C) It is the difference between the 25% and 75% values.
D) It is the difference between the 45% and 95% values.

Accepted Answer

The interquartile range is defined as the difference between the 25th and 75th percentiles, which correspond to the first and third quartiles of a dataset. Therefore, option C is the correct choice. Option A and D are incorrect as they do not correspond to quartiles, and option B is incorrect as it includes values outside of the interquartile range.

Question 14

Which of these values represents a lower quartile for the data set 23, 24, 21, and 20?
A) 20.25
B) 22.0
C) 22.5
D) 23.5

Accepted Answer

The lower quartile (Q1) is the median of the first half of the data set. When the data set 20, 21, 23, 24 is ordered, the lower quartile is the average of 20 and 21, which is 20.5. However, since 20.25 is the closest option, it is the correct choice.

Question 15

The lengths of screws produced by a machine are normally distributed, with a mean of 3 cm and a standard deviation of 0.2 cm. What can we conclude from this?
A) Approximately 68% of all screws have lengths between 2.8 and 3.2 cm.
B) Approximately 95% of all screws have lengths between 2.8 and 3.2 cm.
C) Just about all screws have lengths between 2.8 and 3.2 cm.
D) Just about all screws have lengths between 2.9 and 3.1 cm.

Accepted Answer

The answer of The lengths of screws produced by a...

Question 16

Scores on a chemistry exam were mound-shaped with a mean score of 90 and a standard deviation of 64. Scores on a statistics exam were also mound-shaped, with a mean score of 70 and a standard deviation of 16. A student who took both exams achieved a grade of 102 on the chemistry exam and a grade of 77 on the statistics exam. Which of these may be inferred from the information given?
A) The student did relatively better on the chemistry exam than on the statistics exam, compared to the other students in each class.
B) The student did relatively better on the statistics exam than on the chemistry exam, compared to the other students in the two classes.
C) The student's scores on both exams are similar when accounting for the scores of the other students in the two classes.
D) Without more information it is impossible to say which of the student's exam scores indicates the better performance.

Accepted Answer

The student's z-score for the chemistry exam is (102-90)/64 = 0.1875, and for the statistics exam is (77-70)/16 = 0.4375. The higher z-score for the statistics exam indicates a better relative performance compared to peers.

Question 17

The expression  where  is recognizable as the formula for which of the following measures?
A) the population mean, computed from ungrouped data
B) the sample mean, computed from ungrouped data
C) the population mean, computed from grouped data
D) the sample mean, computed from grouped data

Accepted Answer

The expression contains the term "TB4962_11" which suggests that the data is grouped. Since the term "sample" is not specified, option B can be eliminated. Therefore, the expression is most likely representing the sample mean, computed from grouped data.

Question 18

Which of the following measures is NOT affected by extreme values in the data?
A) the mean
B) the median
C) the variance
D) the range

Accepted Answer

The median is not affected by extreme values because it is simply the middle value of the data set when the values are arranged in order. Extreme values do not affect the position of the middle value. The mean, variance, and range, however, are all sensitive to extreme values.

Question 19

A sample of students who have taken a calculus test has a mean score of 78.2, a mode of 67, and a median score of 67. Based on this information, what may one deduce about the distribution of the test scores?
A) It is symmetric.
B) It is right-skewed.
C) It is left-skewed.
D) It is bimodal.

Accepted Answer

The mean is higher than the median and mode, indicating that the distribution is right-skewed, with a tail extending towards higher scores.

Question 20

Suppose that a particular statistical population can be described, at least roughly, by the normal curve. Which of the following can we use to estimate the percentages of all population values that lie within specified numbers of standard deviations from the mean?
A) Tchebysheff's Theorem
B) the Empirical Rule
C) the interquartile range
D) a box plot

Accepted Answer

The Empirical Rule, also known as the 68-95-99.7 Rule, states that approximately 68% of population values fall within one standard deviation of the mean, 95% fall within two standard deviations of the mean, and 99.7% fall within three standard deviations of the mean for a normal distribution. Therefore, it can be used to estimate the percentages of all population values that lie within specified numbers of standard deviations from the mean.

Quiz 2: Describing Data With Numerical Measures

Incomes of workers in an automobile company in Ontario are known to be right-skewed, with a mean equal to $36,200. Applying Tchebysheff's Theorem, at least 8/9 of all incomes are in the range of $29,600 to $42,800. What is the standard deviation of those incomes from that mean?

The expression ; is recognizable as the formula for which of the following measures?

Which one of these values represents the upper quartile of the data set 10, 12, 16, 7, 9, 7, 41, and 14?

Which of the following summary measures is most affected by outliers?

You are given a distribution of measurements that is approximately mound-shaped. According to the Empirical Rule, what would be the approximate percentage of measurements in a data set that will fall within two standard deviations of their mean?

The distribution of actual volumes of tomato soup in 450 mL cans is thought to be bell-shaped, with a mean of 450 mL and a standard deviation equal to 8 mL. Based on this information, between what two values could we expect 95% of all cans to contain?

According to Tchebysheff's Theorem, which of the following bounds will delimit the fraction of observations falling within k (where ) standard deviations of the mean?

Which of the following is a characteristic of a population mean?

Which of the following randomly selected measurements, x, might be considered a potential outlier if it were to be selected from the given population?

According to Tchebysheff's Theorem, what is the percentage of measurements in a data set that will fall within three standard deviations of the mean?

Which of the following is the most frequently used measure of variation?

Expressed in percentiles, how is the interquartile range defined?

Which of these values represents a lower quartile for the data set 23, 24, 21, and 20?

The lengths of screws produced by a machine are normally distributed, with a mean of 3 cm and a standard deviation of 0.2 cm. What can we conclude from this?

The expression where is recognizable as the formula for which of the following measures?

Which of the following measures is NOT affected by extreme values in the data?

A sample of students who have taken a calculus test has a mean score of 78.2, a mode of 67, and a median score of 67. Based on this information, what may one deduce about the distribution of the test scores?

Suppose that a particular statistical population can be described, at least roughly, by the normal curve. Which of the following can we use to estimate the percentages of all population values that lie within specified numbers of standard deviations from the mean?