Chapter 10.1, Problem 22PFA

a.

To determine

To calculate the mean

Answer to Problem 22PFA

The mean population is 16.67 (in millions)

Explanation of Solution

Given information:

Population of states.

State	Population (in hundred thousand)
North Carolina	9.5
Georgia	9.7
Michigan	9.9
Ohio	11.5
Pennsylvania	12.7
Illinois	12.8
Florida	18.8
New York	19.4
Texas	25.1
California	37.3

Formula Used:

Mean formula is,

  

Where, f is frequency

N is total number of items

X is item.

Calculation:

Substituting the values,

  $\overline{X}=\frac{37.3+18.8+9.7+12.8+9.9+19.4+9.5+11.5+12.7+25.1}{10}$

So,

  $\overline{X}=\frac{166.7}{10}$

On solving,

  $\overline{X}=16.67$

Hence, mean is 16.67

Conclusion:

The mean population is 16.67 (in millions)

b.

To determine

To calculate the median of the population of the states.

Answer to Problem 22PFA

Median of the population of states is 12.8

Explanation of Solution

Given information:

Population of the states

Formula Used:

Median formula is,

  $M=\frac{{(\frac{n}{2})}^{th}+{(\frac{n}{2}+1)}^{th}}{2}$

Where, n cumulative frequency

Calculation:

Rearranging the states in an ascending order of their population

State	Population (in hundred thousand)	Cumulative frequency
North Carolina	9.5	9.5
Georgia	9.7	19.2
Michigan	9.9	29.1
Ohio	11.5	40.6
Pennsylvania	12.7	53.3
Illinois	12.8	66.1
Florida	18.8	84.9
New York	19.4	104.3
Texas	25.1	129.4
California	37.3	166.7

Substituting values in median formula,

  $median=\frac{(\frac{{10}^{th}}{2})term+(\frac{10}{2}+1)term}{2}$

On solving,

  $median=\frac{5^{th}term+6^{th}term}{2}$

Substituting the values of 5^th and 6^th term,

  $M=\frac{12.7+12.8}{2}$

On solving,

  $M=12.75$

On rounding of the value,

  $\approx 12.8$

Hence, median is 12.8

Conclusion: Median of the population of states 12.8

c.

To determine

To calculate the percentile of North Carolina

Answer to Problem 22PFA

Percentile rank of North Carolina is 0.

Explanation of Solution

Given information:

Population of states,

State	Population (in million)
North Carolina	9.5
Georgia	9.7
Michigan	9.9
Ohio	11.5
Pennsylvania	12.7
Illinois	12.8
Florida	18.8
New York	19.4
Texas	25.1
California	37.3

Formula Used: Percentile formula is,

  $percentile=\frac{NumberOfValuesBeforeX}{n}\times 100$

Calculation:

Substituting the values,

  $percentile=\frac{0}{10}\times 100$

On solving,

  $percentile=0$

Conclusion:

Percentile rank of North Carolina is 0.

d.

To determine

To calculate the percentile of Georgia

Answer to Problem 22PFA

Percentile rank of Georgia is 10.

Explanation of Solution

Given information:

Population of states,

State	Population (in million)
North Carolina	9.5
Georgia	9.7
Michigan	9.9
Ohio	11.5
Pennsylvania	12.7
Illinois	12.8
Florida	18.8
New York	19.4
Texas	25.1
California	37.3

Formula Used: Percentile formula is,

  $percentile=\frac{NumberOfValuesBeforeX}{n}\times 100$

Calculation:

Substituting the values,

  $percentile=\frac{1}{10}\times 100$

On solving,

  $percentile=10$

Hence, percentile rank of Georgia is 10.

Conclusion:

Percentile rank of Georgia is 10.

e.

To determine

To calculate if there is any state at 50^th percentile.

Answer to Problem 22PFA

Illinois is at 50 percentile

Explanation of Solution

Given information:

Population in states

Formula Used:

  $value=\frac{percentile}{100}\times (n+1)$

Calculation:

Substituting the values,

  $value=\frac{50}{100}\times (10+1)$

So,

  $value=\frac{50}{100}\times 11$

On solving,

  $value=5.5$

In this situation, taking the average of 5^th and 6^th value. Hence, substituting the values of 5^th and 6^th observations,

  $=\frac{12.7+12.8}{2}$

On solving,

  $=12.75$

On rounding of the value,

  $\approx 12.8$

Comparing the value to the table.

So, Illinois is at 50 percentile

Conclusion:

Illinois is at 50 percentile

f.

To determine

To calculate which states are under 20 percentile?

Answer to Problem 22PFA

The state below 20^th percentile is North Carolina.

Explanation of Solution

Given information:

Population of states,

State	Population (in million)
North Carolina	9.5
Georgia	9.7
Michigan	9.9
Ohio	11.5
Pennsylvania	12.7
Illinois	12.8
Florida	18.8
New York	19.4
Texas	25.1
California	37.3

Formula Used:

  $value=\frac{percentile}{100}\times (n+1)$

Calculation:

Substituting the values,

  $value=\frac{20}{100}\times (10+1)$

So,

  $value=\frac{20}{100}\times 11$

On solving,

  $value=2.2$

On rounding of the value,

  $value\approx 2$

Comparing with the table.

So, the state below 20^th percentile is North Carolina

Conclusion: The state below 20^th percentile is North Carolina