Question

Here are the IQ scores of $10$randomly chosen fifth-grade students:

Which of the following statements about this data set is not true?

a. The student with an IQ of $96$is considered an outlier by the $1.5xIQR$rule.

b. The five-number summary of the $10$IQ scores is $96,118,123.5,130,145$.

c. If the value $96$were removed from the data set, the mean of the remaining $9$IQ scores would be greater than the mean of all $10$IQ scores.

d. If the value $96$were removed from the data set, the standard deviation of the remaining $9$IQ scores would be less than the standard deviation of all $10$IQ scores.

e. If the value $96$were removed from the data set, the IQR of the remaining $9$IQ scores would be less than the IQR of all $10$IQ scores.

Short answer

(a) The statement about the data set is true.

(b) The statement about the data set is true.

(c) The statement about the data set is true.

(d) The statement about the data set is true.

(e) The statement about the data set is not true.

Step-by-step solution

Part (a) Step 1: Given information

We need to find whether the given statement "The student with an IQ of $96$is considered an outlier by the $1.5\times IQR$rule is true or not.

Part (a) Step 2: Explanation

From least to largest, order the data values:

$96,110,118,118,122,125,126,130,139,145$

The median is the sorted data set's middle value. Because the number of data values is even, the median is the average of the sorted data set's two middle values:

$M=Q_2=\frac{122+125}{2}=123.5$

The median of the data values below the median is the first quartile (or at $25$ percent of the data). The first quartile is the third data value since there are five data values below the mean.

The median of the data values above the median is the third quartile (or at seventy-five percent of the data). The third quartile is the third data value above the mean, and hence the $3+5=8th$data value, because there are five data values above the mean.

Because $96$ is less than hundred, the $1.5xIQR$ rule considers it an outlier, and hence assertion (a) is TRUE.

Part (b) Step 1: Given information

We need to find whether the given statement "The five-number summary of the $10$IQ scores is $96,118,123.5,130,145$" is true or false.

Part (b) Step 2: Explanation 

The minimum, first quartile, median, third quartile, and maximum are included in the five-number summary.

The minimum is $96$and the highest is $145$, with a median of $123.5$, a first quartile of $118$, and the third quartile of $130$(part (a) of the result).

As a result, the five-number summary is $96,118,123.5,130,145$, making statement (b) TRUE.

Part (c) Step 1: Given information

We need to find whether the given statement "If the value $96$were removed from the data set, the mean of the remaining $9$IQ scores would be greater than the mean of all $10$IQ scores" is true or not.

Part (c) Step 2: Explanation

The mean is calculated by dividing the total of all data values by the number of data items.

The data set's smallest value is $96$.

If $96$is removed from the data set, we may expect the mean to rise since the data's center will grow higher when the minimum is eliminated, proving that assertion (c) is correct.

Part (d) Step 1: Given information

We need to find whether the given statement "If the value $96$were removed from the data set, the standard deviation of the remaining $9$IQ scores would be less than the standard deviation of all $10$IQ scores" Is true or not.

Part (d) Step 2: Explanation

The standard deviation is a measurement of how much variation or spread there is in a data collection.

The data set's smallest value is $96$.

If $96$is removed from the data set, we expect the standard deviation to drop since the spread will reduce when the minimum is removed, and thus statement (d) is TRUE.

Part (e) Step 1: Given information

We need to find whether the given statement "If the value $96$were removed from the data set, the IQR of the remaining $9$IQ scores would be less than the IQR of all $10$IQ scores" is true or not.

Part (e) Step 2: Explanation

The interquartile range (IQR) is a robust measure of spread, which means it is unaffected by outliers.

Part (a) tells us that $96$is an outlier because it is the smallest value in the data set.

If $96$is removed from the data set, we anticipate the IQR to be unaffected, as the IQR should not be influenced by the outlier, and so statement (e) is FALSE.