Question 1

In a negatively skewed distribution,the mean is smaller than the median and the median is smaller than the mode.

Accepted Answer

In a negatively skewed distribution, there are extreme values in the low end of the distribution, pulling the tail to the left. This causes the mean to be smaller than the median, since the outliers are below the average. Additionally, the median is smaller than the mode because the tail pulls most of the data to the left, causing the single peak of the distribution to be at a higher value than the median.

Question 2

Suppose a sample of size 50 has a sample mean of 20.In this case,the sum of all observations in the sample is 1,000.

Accepted Answer

To find the sum of all observations in a sample, we need to multiply the sample mean by the sample size. Therefore, if the sample size is 50 and the sample mean is 20, the sum of all observations would be 50 * 20 = 1,000.

Question 3

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

Accepted Answer

This is the definition of the arithmetic mean. Mathematically, it can be expressed as:

mean x number of observations = sum of observations

Or in symbols:

\bar{x} \cdot n = \sum_{i=1}^n x_i

where \bar{x} is the mean, n is the number of observations, and x_i are the individual observations.

Question 4

The mean is a measure of variability.

Accepted Answer

The mean is a measure of central tendency, not variability. Measures of variability include range, variance, and standard deviation.

Question 5

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

Accepted Answer

In a bell shaped distribution, the mean, median, and mode are all at the center of the distribution, which is the highest point of the curve. Therefore, they all have the same value.

Question 6

A data sample has a mean of 107,a median of 122,and a mode of 134.The distribution of the data is positively skewed.

Accepted Answer

The mean is less than the median, which indicates that there are more values on the higher end of the distribution, resulting in positive skewness. The mode being greater than both the mean and median also supports this conclusion.

Question 7

The mean is affected by extreme values but the median is not.

Accepted Answer

The mean is calculated by summing up all the data points and dividing by the number of data points, therefore, extreme values can significantly impact the mean. However, the median is the middle value of a dataset, so it is not affected by extreme values.

Question 8

Which of the following statistics is a measure of central location?

Accepted Answer

All three statistics - the mean, the median, and the mode - are measures of central location. The mean represents the average value of a data set, the median represents the middle value in a sorted data set, and the mode represents the most frequently occurring value in a data set.

Question 9

If the mean,median,and mode are all equal,the histogram must be symmetric and bell shaped.

Accepted Answer

If the mean, median, and mode are all equal, it indicates that the data is well balanced and centered, which often results in a symmetric and bell-shaped histogram.

Question 10

The median of an ordered data set with 30 items would be the average of the 15^th and the 16^th observations.

Accepted Answer

The median is the middle value of an ordered data set. In this case, there are 30 items so the median would be the (30+1)/2 = 15.5th value. Since we cannot have a fractional observation, we take the average of the 15th and 16th observations, which are the 15^th and the 16^th observations. Therefore, the statement is true.

Question 11

In a histogram,the proportion of the total area which must be to the right of the mean is exactly 0.50 if the distribution is symmetric and unimodal.

Accepted Answer

In a symmetric and unimodal distribution, the mean is the same as the median and divides the data into two equal parts. Therefore, exactly 50% of the total area must be to the right of the mean and 50% to the left, which is reflected in the shape of the histogram.

Question 12

Which measure(s)of central location is/are meaningful when the data are ordinal?

Accepted Answer

When the data is ordinal, the most appropriate measures of central location are the median and the mode. This is because the values in ordinal data can be arranged in a specific order, but it may not be meaningful to perform arithmetic operations on them. Hence, the mean is not a suitable measure in this case. The median, which is the middle value of the ordered data, and the mode, which is the most frequently occurring value, are more meaningful and useful in describing the central tendency of the ordinal data.

Question 13

In a histogram,the proportion of the total area which must be to the left of the median is:

Accepted Answer

The median divides a dataset into two equal parts, meaning that 50% of the data falls below the median and 50% falls above it, regardless of the distribution's shape.

Question 14

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

Accepted Answer

In a histogram, regardless of whether the distribution is negatively skewed or not, the median divides the data into two halves, with 50% of the data falling to the left of the median and 50% to the right, assuming continuous data and a perfectly representative histogram. Skewness affects the shape and spread of the data, not the proportion of data on either side of the median.

Question 15

The median of a set of data is more representative than the mean when the mean is larger than most of the observations.

Accepted Answer

When the mean is larger than most of the observations, it indicates the presence of extreme values or outliers that skew the mean towards the higher end. In such cases, the median provides a more accurate estimate of the central tendency because it is not affected by extreme values. Therefore, option A is true.

Question 16

Lily has been keeping track of what she spends to eat out.The last week's expenditures for meals eaten out were $5.69,$5.95,$6.19,$10.91,$7.49,$14.53,and $7.66.The mean amount Lily spends on meals is $8.35.

Accepted Answer

The mean is calculated by adding all expenditures and dividing by the number of meals: ($5.69 + $5.95 + $6.19 + $10.91 + $7.49 + $14.53 + $7.66) / 7 = $8.34, which rounds to $8.35.

Question 17

Which measure of central location can be used for both interval and nominal variables?

Accepted Answer

The mode is the only measure of central location that can be used for both interval and nominal variables. The mean and median can only be used for interval variables.

Question 18

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

Accepted Answer

Half of the observations are not always on either side of the mean. This is only true for symmetric distributions. In skewed distributions, the mean may be pulled in one direction or the other by extreme values, making it so that more or less than half of the observations are on either side.

Question 19

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

Accepted Answer

In a positively skewed distribution, the median is closer to the left of the distribution, meaning less than 50% of the total area is to the left of the median.

Question 20

Which of the following statements is true for the following observations: 9,8,7,9,6,11,and 13?

Accepted Answer

The mean, median, and mode are all equal to 9.

Quiz 4: Numerical Descriptive Techniques

In a negatively skewed distribution,the mean is smaller than the median and the median is smaller than the mode.

Suppose a sample of size 50 has a sample mean of 20.In this case,the sum of all observations in the sample is 1,000.

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

The mean is a measure of variability.

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

A data sample has a mean of 107,a median of 122,and a mode of 134.The distribution of the data is positively skewed.

The mean is affected by extreme values but the median is not.

Which of the following statistics is a measure of central location?

If the mean,median,and mode are all equal,the histogram must be symmetric and bell shaped.

The median of an ordered data set with 30 items would be the average of the 15^th and the 16^th observations.

In a histogram,the proportion of the total area which must be to the right of the mean is exactly 0.50 if the distribution is symmetric and unimodal.

Which measure(s) of central location is/are meaningful when the data are ordinal?

In a histogram,the proportion of the total area which must be to the left of the median is:

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

The median of a set of data is more representative than the mean when the mean is larger than most of the observations.

Lily has been keeping track of what she spends to eat out.The last week's expenditures for meals eaten out were $5.69,$5.95,$6.19,$10.91,$7.49,$14.53,and $7.66.The mean amount Lily spends on meals is $8.35.

Which measure of central location can be used for both interval and nominal variables?

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

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

Which of the following statements is true for the following observations: 9,8,7,9,6,11,and 13?

Quiz 4: Numerical Descriptive Techniques

In a negatively skewed distribution,the mean is smaller than the median and the median is smaller than the mode.

Suppose a sample of size 50 has a sample mean of 20.In this case,the sum of all observations in the sample is 1,000.

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

The mean is a measure of variability.

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

A data sample has a mean of 107,a median of 122,and a mode of 134.The distribution of the data is positively skewed.

The mean is affected by extreme values but the median is not.

Which of the following statistics is a measure of central location?

If the mean,median,and mode are all equal,the histogram must be symmetric and bell shaped.

The median of an ordered data set with 30 items would be the average of the 15th and the 16th observations.

In a histogram,the proportion of the total area which must be to the right of the mean is exactly 0.50 if the distribution is symmetric and unimodal.

Which measure(s) of central location is/are meaningful when the data are ordinal?

In a histogram,the proportion of the total area which must be to the left of the median is:

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

The median of a set of data is more representative than the mean when the mean is larger than most of the observations.

Lily has been keeping track of what she spends to eat out.The last week's expenditures for meals eaten out were $5.69,$5.95,$6.19,$10.91,$7.49,$14.53,and $7.66.The mean amount Lily spends on meals is $8.35.

Which measure of central location can be used for both interval and nominal variables?

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

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

Which of the following statements is true for the following observations: 9,8,7,9,6,11,and 13?

The median of an ordered data set with 30 items would be the average of the 15^th and the 16^th observations.