Chapter 4, Problem 48E

(a)

To determine

To construct: a suitable display for the growth rates.

Explanation of Solution

Given:

  

Graph:

  

By seeing the data, it is observed that the data is quantitative so the histogram is the appropriate to display the data. On the basis of given data plot the histogram.

(b)

To determine

To find: the summary the growth rate among these cities with a median and mean.

Explanation of Solution

Given:

  

Formula used:

  $Mean=\frac{{\displaystyle \sum x}}{n}$

Calculation:

Arranging in the ascending order

1.4, 1.5, 1.7, 1.9, 2.2, 2.3, 2.6, 2.6, 2.6, 2.8, 2.9, 3.1, 3.2, 3.3, 3.3, 3.4, 4.2, 4.4, 4.5, 7.5 (in percent)

These are 20 values so median is the average of 10^th and 11^th values. therefore median would be

  $\begin{array}{c}\text{Median=}\frac{2.8\%+2.9\%}{2}\\ =2.85\%\end{array}$

Sum of all the job growth percentages is 61.4% so mean would be

  $\begin{array}{c}\text{Mean=}\frac{61.4\%}{20}\\ =3.1\%\end{array}$

Hence, median is 2.85% and mean is 3.1%. Distribution is skewed to right so mean is higher than median.

(c)

To determine

To find: the better job of summarizing the growth rates.

Answer to Problem 48E

Median is better

Explanation of Solution

Median would be better than mean because the distribution is skewed to right. The distribution has outliers’ therefore median works better to explain the growth rates of job.

(d)

To determine

To find: the summary of the spread of the growth rate distribution with a standard deviation and with an IQR.

Answer to Problem 48E

Standard deviation of the distribution is 1.4% and IQR is 1.1%.

Explanation of Solution

Given:

  

Formula used:

  $\text{SD}\left(s\right)=\sqrt{\frac{{\left(x-\overline{x}\right)}^2}{n-1}}$

  $\text{IQR}=Q_3-Q_1$

Calculation:

  ${{\displaystyle \sum \left(x-\overline{x}\right)}}^2=35.58$

  $n=18$

So standard deviation will be

  $\begin{array}{c}\text{SD}\left(s\right)=\sqrt{\frac{{\left(x-\overline{x}\right)}^2}{n-1}}\\ =\sqrt{\frac{35.58}{19}}\\ =\sqrt{1.87}\\ =1.4\end{array}$

So standard deviation is 1.4%.

To find IQR let us first find $Q_1$ and $Q_3$ . Lower quartile is middle value of lower half.

Lower half of data has 10 values. Lower half of data is

1.4, 1.5, 1.7, 1.9, 2.2, 2.3, 2.6, 2.6, 2.6, 2.6, 2.8

So the lower quartile is

  $\begin{array}{c}{Q}_1=\frac{5\text{th}\text{value}\text{+}\text{6}\text{th}\text{value}}{2}\\ =\frac{2.2+2.3}{2}\\ =2.25\end{array}$

Upper half of data has 10 values. Upper half of data is

2.9, 3.1, 3.2, 3.3, 3.3, 3.4, 4.2, 4.4, 4.5, 7.5

So the lower quartile is

  $\begin{array}{c}{Q}_3=\frac{5\text{th}\text{value}\text{+}\text{6}\text{th}\text{value}}{2}\\ =\frac{3.2+3.4}{2}\\ =3.35\end{array}$

Therefore IQR is

  $\begin{array}{c}\text{IQR}=Q_3-Q_1\\ =3.35-2.25\\ =1.1\end{array}$

Hence, standard deviation of the distribution is 1.4% and IQR is 1.1%.

(e)

To determine

To find: the measures in (d) the better job of summarizing the growth rates.

Explanation of Solution

Inter quartile range would be better than standard deviation. The distribution is skewed to right and has few outliers therefore inter quartile range would be better to explain the growth rates of job.

(f)

To determine

To find: the change the values of the summary statistics calculated above.

Explanation of Solution

Suppose first median, Subtracting 1.20% from every value of data does not change the order of data vales. Therefore the median would also be decrease by 1.20%. Therefore the new median would be 2.85% - 1.20% = 1.65%.

Now taking the standard deviation, Mean decreases by 1.20% and every value also decreases by 1.20% so deviation would not change. Therefore standard deviation remains unchanged.

Therefore, new standard deviation is 1.4%.

Lower and upper quartile both decrease by 1.20% when every value of data decreases by 1.20% so IQR remains would be unchanged. Therefore new IQR is equal to 1.1%.

(g)

To determine

To Explain: the expected the mean, median, standard deviation and IQR to change.

Explanation of Solution

1.4, 1.5, 1.7, 1.9, 2.2, 2.3, 2.6, 2.6, 2.6, 2.8, 2.9, 3.1, 3.2, 3.3, 3.3, 3.4, 4.2, 4.4, 4.5

Total number of values is odd so median is the 10^th value that is 2.8%.

  $\begin{array}{c}\text{Mean=}\frac{61.4\%-7.5\%}{19}\\ =2.8\%\end{array}$

So new mean is equal to 2.8%

Standard deviation decreases. Las Vegas job growth percentage works as an outlier in the distribution so omitting Las Vegas decreases standard deviation also.

Las Vegas value is last in order data set so omitting it does not change IQR.

(h)

To determine

To Explain: the report about all of these growth rates.

Explanation of Solution

The distribution is skewed to right and has an outlier so median shows the data correctly. Las Vegas job growth is significantly larger than any other city. Spread of the correctly represents by IQR. Almost 20 states have scores between 240 and 244. One of the state has scores less than 225.