Question

Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in Table 2.69. Also, include left endpoint, but not the right endpoint.

a. What percentage of the survey answered "not sure"?

b. What percentage think that middle-class is from \(25,000 to \)50,000? c. Construct a histogram of the data.

i. Should all bars have the same width, based on the data? Why or why not?

ii. How should the <20,000 and the 100,000+ intervals be handled? Why? d. Find the 40th and 80th percentiles

e. Construct a bar graph of the data

Short answer

a) $6\%$of the survey answered "not sure"

b) 63 percent of those polled believe that the middle class ranges from $\$25,000to\$50,000.$

c)

i. No, because the width of the class interval varies, all of the bars cannot be the same width. As a result, the width of the bars should be proportional to the class interval.

ii. Because the relative frequency of these incomes is quite low, we can assume the class intervals of the same width as the other class intervals.

d) The${40}^{\text{th}}$percentile lies between $\$30,000-40,000\$.$

The${80}^{\text{th}}$percentile lies between $\$50,000-\$75,000$.

e) The bar graph is :

Step-by-step solution

Part (a) - Step 1: To determine

What percentage of the survey answered "not sure".

Part (a) - Step 2: Explanation

The relative frequency in statistics is the ratio of the frequency of a given class to the sum of all frequencies.

The formula for calculating the relative frequencies is given below:

Relative Frequency $=\frac{\text{Frequency}}{\text{Sum of frequencies}}$

we know that the sum of relative frequencies is 1.

If we find the sum of given frequencies we have:

$0.02+0.09+0.19+0.26+0.18+0.17+0.02+0.01=0.94$

Therefore, the percentage of the survey answered "not sure" is:

$1-0.94=0.06=6\%$

Hence $6\%$of the survey answered "not sure".

Part (b) - Step 3: To determine

What percentage think that middle class is from$\$25,000to\$50,000$?

Part (b) - Step 4: Explanation

The relative frequency in statistics is the ratio of the frequency of a given class to the sum of all frequencies.

The formula for calculating the relative frequencies is given below:

Relative Frequency $=\frac{\text{Frequency}}{\text{Sum of frequencies}}$

The percentage of the survey who thinks that middle class is from$\$25,000to\$50,000is:$

$0.19+0.26+0.18=0.63=63\%$

Therefore, $63\%$of the survey think that middle class is from $\$25,000\$to\$50,000.$

Part (c) - Step 5: To determine

The histogram of the data.

i. Should all bars have the same width, based on the data?

ii. How should the $<20,000andthe\$100,000+$intervals be handled?

Part(c) - Step 6: Explanation

The histogram is a graphical display in statistics that uses rectangles to depict the frequency of data values. It has two axis, one horizontal and one vertical. The data is represented on the horizontal axis, which is labelled. The frequency or relative frequency axis is labelled on the vertical axis. The following is the histogram for the provided data:

i. No, all the bars cannot have the same width because width of class interval is not same. Therefore, we should have bars with width according to the class interval.

ii. We can consider the$<20,000and100,000+$ of the same width as other class interval because the relative frequency of these salaries is very less.

Part (d) - Step 7: To determine

The${40}^{th}and{80}^{th}percentile.$

Part (d) - Step 8: Explanation

In statistics, percentiles are used to divide data into 100 equal parts. Data is compared and interpreted using percentiles. A 50th percentile observation, for example, would be greater than 50% of the other observations in the set.

To find the${40}^{\text{th}}$and${80}^{\text{th}}$percentile for given data, we need to find the cumulative relative frequency for the given data.

Salary ($)	Relative frequency	Cumulative Relative frequency
$<20000$	$0.02$	$0.02$
$20,000-25,000$	role="math" localid="1647976861486" $0.09$	$0.02+0.09=0.11$
$25,000-30,000$	$0.19$	$0.11+0.19=0.3$
$30,000-40,000$	$0.26$	$0.3+0.26=0.56$
$40,000-50,000$	$0.18$	$0.56+0.18=0.74$
$50,000-75,000$	$0.17$	$0.74+0.17=0.91$
$75,000-99,999$	$0.02$	$0.91+0.02=0.93$
$100,000+$	$0.01$	$0.93+0.01=0.94$

From the above table, we clearly see the ${40}^{\text{th}}$percentile lies between $\$30,000-40,000\$$and ${80}^{\text{th}}$percentile lies between$\$50,000-75,000\$.$

Part (e) - Step 9 : To determine

The bar graph.

Part (e) - Step 10: Explanation

The bar graph is a graphical tool used in statistics to show data using bars of various heights or lengths. Bar graphs have vertical or horizontal bars that are separated from one another. Each bar's height is proportionate to that category's values. Bar graphs are quite useful for comparing differences between different types of data. The categories are on the x-axis, and the relative frequency points are on the y-axis in a bar graph.

The bar graph of the given data is: