Deck 2: Describing Data With Numerical Measures

Question 1

In a histogram, what may one conclude about the proportion of the total area that must be to the left of the median?

A) It is exactly 0.50.
B) It is less than 0.50 if the distribution is skewed to the left.
C) It is more than 0.50 if the distribution is skewed to the right.
D) It is between 0.25 and 0.75 if the distribution is symmetric.

Answer 1

A

Question 2

Which of the following statements about the mean is NOT always correct?

A) The sum of the deviations from the mean is 0.
B) Half the observations are on either side of the mean.
C) The mean is a measure of the middle of a distribution.
D) The value of the mean times the number of observations equals the sum of all of the observations.

Answer 2

B

Question 3

The following data represent a sample of 10 scores on a 20-point statistics quiz: 16, 16, 16, 16, 16, 18, 18, 20, 20, and 20. After the mean, median, range, and variance were calculated for the scores, it was discovered that one of the scores of 20 should have been an 18. Which of the following pairs of measures will change when the calculations are redone using the correct scores?

A) mean and range
B) median and range
C) mean and variance
D) median and variance

Answer 3

C

Question 4

A random sample from an unknown population had a sample standard deviation of zero. From this piece of information, which one of the following is a reasonable conclusion?

A) The sample range must be zero.
B) An error was made in computing the sample standard deviation. It must always be greater than zero.
C) The population standard deviation must be zero.
D) The population standard deviation must be less than zero

Question 5

Which of the following best describes the relationship between the population mean and the sample mean?

A) The population mean is always larger than the sample mean.
B) The population mean is always smaller than the sample mean.
C) The population mean is always larger than or equal to the sample mean.
D) The population mean can be smaller than, larger than, or equal to the sample mean.

Question 6

Which of the following is a meaningful measure of centre when the data are qualitative?

A) the mean
B) the median
C) the mode
D) the quartile

Question 7

Which of the following statements describes the variance of a data set?

A) The variance is a mean of absolute deviations.
B) The variance is a mean of positive and negative deviations.
C) The variance is a mean of squared deviations.
D) The variance is a mean of only the positive deviations.

Question 8

The following 10 scores were obtained on a 20-point quiz: 4, 5, 8, 9, 11, 13, 15, 18, 18, and 20. The teacher computed the usual descriptive measures of centre and variability for these data, and then discovered an error was made. One of the 18s should have been a 16. Which pair of the following measures, calculated on the corrected data, would change from the original computation?

A) mean and standard deviation
B) mean and median
C) range and median
D) mean and range

Question 9

Which of the following is NOT a measure of variability?

A) the variance
B) the standard deviation
C) the mean
D) the range

Question 10

Which of the following is a property of a symmetric distribution?

A) The mean is greater than the median.
B) The mean and median are equal.
C) The mean is less than the median.
D) The mean is less than the mode.

Question 11

If two data sets have the same range, which of the following characteristics do these data sets also share?

A) The distances from the smallest to the largest observations in both sets will be the same.
B) The smallest and largest observations will be the same in both sets.
C) They will have the same variance.
D) They will have the same interquartile range.

Question 12

Which of the following statements applies to this set of data values: 17, 15, 16, 14, 17, 18, and 22?

A) The mean, median, and mode are all equal.
B) Only the mean and median are equal.
C) Only the mean and mode are equal.
D) Only the median and mode are equal.

Question 13

If a store manager selected a sample of customers and computed the mean income for this sample, what has he computed?

A) a parameter
B) a statistic
C) a qualitative value
D) a categorical value

Question 14

A sample of 26 observations has a standard deviation of 4. What is the sum of the squared deviations from the sample mean?

A) 21
B) 25
C) 100
D) 400

Question 15

Consider this data set: 5, 6, 7, 11, and 15. Which of the following values equals its mean?

A) 7.0
B) 7.1
C) 8.1
D) 8.8

Question 16

In a histogram, what may be said of the proportion of the total area that must be to the right of the mean?

A) It is less than 0.50 if the distribution is skewed to the left.
B) It is always exactly 0.50.
C) It is more than 0.50 if the distribution is skewed to the right.
D) It is exactly 0.50 only if the distribution is symmetric.

Question 17

Which of the following can be used to summarize data about qualitative variables?

A) measures of centre
B) measures of variability
C) proportions
D) measures of relative standing

Question 18

Which of the following refers to numbers that indicate the spread or scatter of observations in a data set?

A) measures of centre
B) measures of location
C) measures of variability
D) measures of shape

Question 19

The average score for a class of 35 students was 70. The 20 male students in the class averaged 73. What was the average score for the 15 female students in the class?

A) 60
B) 66
C) 70
D) 73

Question 20

Which of the following represents a disadvantage of using the sample range to measure dispersion?

A) It produces spreads that are not meaningful for data analysis.
B) The largest or smallest observation (or both) may be an outlier.
C) The sample range is not measured in the same units as the data.
D) The sample range is measured in the same units as the data.

Question 21

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

Question 22

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)

Question 23

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

Question 24

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

Question 25

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

Question 26

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%

Question 27

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

Question 28

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 -

Question 29

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.

Question 30

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

Question 31

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%

Question 32

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

Question 33

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.

Question 34

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

Question 35

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.

Question 36

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.

Question 37

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

Question 38

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

Question 39

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.

Question 40

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

Question 41

A sample of 600 values produced the following summary statistics: and Given this information, which of the following values constitutes the lower fence on a box plot?

A) -4.60
B) 26.80
C) 75.80
D) 102.60

Question 42

A sample of 50 values produced the following summary statistics: and Based on this information, what are the left and right ends, respectively, of the box plot using whiskers?

A) 5.3 and 32.0
B) 10 and 14.6
C) 10 and 16.7
D) 14.6 and 16.7

Question 43

Jessica has been keeping track of what she spends to eat out. Last week's expenditures for meals eaten out were $15.69, $15.95, $16.19, $20.91, $17.49, $24.53, and $17.66. The mean amount Jessica spends on meals is $18.35.

Question 44

Numerical descriptive measures computed from sample measurements are called statistics.

Question 45

If a data set has 15 values that have been sorted in ascending order, which value in the data set will be at the 25th percentile?

A) the fourth value
B) the third value
C) the second value
D) the first value

Question 46

A student scores 89, 75, 94, and 88 on four exams during the semester and 97 on the final exam. If the final is weighted double and the four others weighted equally, the student's final average would be 90.

Question 47

Lily's score on her biochemistry text placed her at the 97th percentile. What does this mean?

A) Lily's score has a z-score of 0.97.
B) Lily was in the bottom 3% of the students who took the test.
C) Lily scored as high as or higher than 97% of the students who took the test.
D) Lily's score has a z-score of -0.97.

Question 48

The mode of a data set or a distribution of measurements, if it exists, is unique.

Question 49

Numerical descriptive measures computed from population measurements are called parameters.

Question 50

Two classes, one with 15 students and the other with 25 students, took the same test and averaged 85 points and 75 points, respectively. If the two classes were combined, the overall average score of the 40 students would be 80 points.

Question 51

The following summary statistics were computed from a sample of size 250: and Given this information, which of the following statements is correct?

A) The distribution of the data is slightly right-skewed.
B) The distribution of the data is symmetric.
C) A data value of 1 is an outlier.
D) A data value of 25 is an outlier.

Question 52

When data have been grouped (as in a frequency table, a relative frequency histogram, etc.), the class with the highest frequency is called the modal class, and the midpoint of that class is taken to be the mode.

Question 53

What percentage of all observations in a data set lie between the 30th percentile and the third quartile?

A) 30%
B) 45%
C) 75%
D) 79%

Question 54

If, from a set of data, the sample mean was found to be 15 but the sample median was only 9, then the data set would be said to be skewed to the right.

Question 55

The mode is generally used to describe large data sets.

Question 56

If the distribution of sales is thought to be symmetric with very little variation, then what may one conclude about the box plot that represents the data set?

A) The whiskers on a box plot the box should be about half as long as the box is wide.
B) The width of the box will be very wide but the whiskers will be very short.
C) The left and right edges will be approximately at equal distance from the second quartile.
D) The width of the box will be very short, but the whiskers will be very long.

Question 57

Which of the following describes a graphical device that displays the highest and lowest values in a data set, as well as the upper quartile, the middle value, and the lower quartile?

A) a box plot
B) a five-number summary
C) a dotplot
D) a stem-and-leaf plot

Question 58

In a mound-shaped distribution, there is no difference in the values of the mean and the median.

Question 59

A sample of 600 values produced the following summary statistics: and Given this information, which of the following values is the upper fence on a box plot of this data set?

A) -4.60
B) 26.80
C) 75.80
D) 102.60

Question 60

A data sample has a mean of 87 and a median of 117. The distribution of the data is positively skewed.

Question 61

In all cases, the sum of the deviations of the measurements from their mean is 0.

Question 62

The sample variance is approximately the average of the squared deviations of the measurements from their mean.

Question 63

The value of the standard deviation will always exceed that of the variance.

Question 64

The median is a measure of centre that divides an ordered array of data into two halves. If the data are arranged in ascending order from smallest to largest, all the observations below the median are smaller than or equal to it, while all the observations above the median are larger than or equal to it.

Question 65

The value of the mean times the number of observations equals the sum of all of the observations.

Question 66

The standard deviation is the positive square root of the variance.

Question 67

The standard deviation is expressed in terms of the original units of measurement, but the variance is not.

Question 68

In a histogram, the proportion of the total area that must be to the left of the median is less than 0.50 if the distribution is skewed to the left.

Question 69

In order to measure the variability in the same units as the original observations, we compute the sample variance.

Question 70

The mode is the sum of a data set's minimum and maximum values, divided by 2.

Question 71

The sample variance calculated with a divisor of n gives a better estimate of the population variance, $\sigma$ ², than does the sample variance, s², with a divisor of n - 1.

Question 72

Measures of centre are values around which observations tend to cluster and which describe the location of what, in some sense, might be called the "centre" of a data set.

Question 73

A sample of 20 observations has a standard deviation of 4. The sum of the squared deviations from the sample mean is 320.

Question 74

The mean is one of the most frequently used measures of variability.

Question 75

Measures of variability describe typical values in the data.

Question 76

The value of the standard deviation may be either positive or negative, while the value of the variance will always be positive or zero.

Question 77

If the variability of a set of data is very small, then the sample variance may be negative.

Question 78

The range is considered the weakest measure of variability.

Question 79

The larger the values of the sample variance, , and the sample standard deviation, s, the greater the variability in the data.

Question 80

When all the numbers in the data set are the same, the standard deviation, s, must be zero.