Chapter 2.1, Problem 9E

(a)

To determine

To estimate: the interquartile range of the distribution

Answer to Problem 9E

The Inter-Quartile range is $27.

Explanation of Solution

Given:

  

Calculation:

The figure below is a cumulative relative frequency graph of the amount spent by 50consecutive grocery shoppers at a store.

  

We are supposed to find the Inter-Quartile range of the amount spent by 50 consecutive grocery shoppers at a store basing on the above cumulative relative frequency graph.

Inter-Quartile range is defined as follows:

  $IQR=Q_3-Q_1$

Where $Q_1$ corresponds to the ${25}^{\text{th }}$ percentile and $Q_3$ corresponds to the ${75}^{\text{th }}$ percentile.

Procedure to find the quartiles is given as follows:

Step1: Draw a horizontal line from the value 25 on the vertical axis and determine the point onthe ogive where the line intersects the ogive.

Step2: Draw a vertical line from this point to the horizontal axis. The line intersects the axis atapproximately $19

Thus, $19 is the estimate of the $Q_1$

Step3: Draw a horizontal line from the value 75 on the vertical axis and determine the point onthe ogive where the line intersects the ogive.

Step4: Draw a vertical line from this point to the horizontal axis. The line intersects the axis at approximately $46

Thus, $46is the estimate of the $Q_3$

The following graph shows the above procedure:

  

Hence, the Inter-Quartile range is calculated as follows:

  $IQR=Q_3-Q_1=46-19=27$

Therefore, the Inter-Quartile range of the amount spent by 50 consecutive grocery shoppers at a store is $27.

Conclusion:

Therefore, the Inter-Quartile range is $27.

(b)

To determine

To find: the percentile for the shopper who spent $19.50

Answer to Problem 9E

Here, $19.50 is at about the ${26}^{\text{th }}$ percentile.

Explanation of Solution

Calculation:

From the above figure we can clearly see that shopper who spent $19.50 is just above the ${25}^{\text{th }}$ percentile. Hence, we can say that $19.50 is at about the ${26}^{\text{th }}$ percentile.

Conclusion:

Hence, $19.50 is at about the ${26}^{\text{th }}$ percentile.

(c)

To determine

To draw: the histogram that correspondsto the given graph

Answer to Problem 9E

  

Explanation of Solution

Calculation:

From the above cumulative graph, we find the cumulative frequencies corresponding todifferent mid values.

From the cumulative graph, the obtained cumulative frequencies are 5,25,55,62,80,87,88 90,96,98 corresponding to mid values 10,20,30,40,50,60,70,80,90,100 .The corresponding frequencies are obtained as 5,20,30,7,18,7,1,2,6,2

Using frequencies, we construct histogram for the given data. The obtained graph is as follows:

  

From the above histogram, we can see that the data of the amount spent by 50 consecutivegrocery shoppers at a store is spread towards right side. Hence, we can say that the abovedata is skewed towards right.

Conclusion:

Therefore, the histogram is drawn.