Exercise 22 Page 750

We are asked to come up with a data set containing 10 numbers which satisfies the following conditions.

• The median is 22.
• The interquartile range is 10.
• The minimum is less than 4.

Let's consider 10 empty bubbles representing our data set. We will add new data values to the data set in order to order to fulfill the conditions one at a time.

For simplicity, we will put the values in increasing order.

Median of 22

Since the data set contains an even number of data values, the median is the mean of the two middle data values, m_1 and m_2. Therefore, the mean of m_1 and m_2 must be equal to 22. m_1 + m_2/2 = 22 Let's first consider some arbitrary number close to 22, for example 20. It will be our first middle data value! To find the second middle value, let's substitute m_1 = 20 into the equation.

The middle values of our data set are 20 and 24. Therefore, the median is indeed equal to 22.

Alternative Solution

Using a Number Line to Find Middle Values

We can also find the middle values by placing the median on a number line.

For the data set to have the appropriate median, it suffices to choose the middle values that are equidistant from the median and placed on either side of the median.

Now, let's calculate the mean of the middle values.

Mean = ( 22- d) + ( 22+ d)/2

▼

Simplify right-hand side

RemovePar

Remove parentheses

Mean = 22-d+22+d/2

AddTerms

Add terms

Mean = 44/2

CalcQuot

Calculate quotient

Mean = 22

Therefore, we can choose any pair of numbers that are equidistant from 22. For example, we could choose 21 and 23, or even 22 and 22.

Interquartile Range of 10

To obtain an interquartile range of 10, we first need to locate the first and third quartiles in the data set. Let's divide the data set into lower and upper halves.

Next, we need to locate the median of each half. Because each half includes odd number of data values, the first quartile (Q_1) is middle value of the lower half and the third quartile (Q_3) is the middle value of the upper half.

Recall that the interquartile range is the difference between Q_3 and Q_1. Q_3 - Q_1 = 10 Furthermore, Q_1 needs to be less than or equal to 20 and Q_3 needs to be greater than or equal to 24. Having all of this information, let's assign 18 to Q_1. Then the value of Q_3 can be found using the equation.

We can now put both Q_1 and Q_3 into our data set. Let's also choose the other values between Q_1 and Q_3, while keeping the increasing order of the data set.

Keep in mind that this is only one possibility for any of the data values.

Minimum Less Than 4

We need to choose the remaining extreme values so that the minimum data value is less than 4. Let's choose 1 as our minimum value.

Finally, we will put some arbitrary numbers for the remaining data values. Remember to keep the increasing order of the data values!