Chapter 10.3, Problem 21PFA

a.

To determine

To calculate minimum, maximum, median, upper quartile, and lower quartile.

Answer to Problem 21PFA

Median is 1518, lower quartile is 2060, upper quartile is 1342, minimum is 1215 and maximum is 2788.

Explanation of Solution

Given information: the worldwide grosses for top 10 grossing films

Formula used: Median is,

  $\frac{{(\frac{n}{2})}^{th}+{(\frac{n}{2}+1)}^{th}term}{2}$ ,

Lower quartile is,

  $Q1=(n+1)\times \frac{1}{4}$

Upper quartile is,

  $Q3=(n+1)\times \frac{3}{4}$

Where, n is number of tiles/ data points

Calculation:

Rank	Gross( in millions)
1	2788
2	2187
3	2060
4	1670
5	1520
6	1516
7	1405
8	1342
9	1277
10	1215

For median, substituting the values in median formula

  $=\frac{{(\frac{10}{2})}^{th}+{(\frac{11}{2}+1)}^{th}term}{2}$

On solving,

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

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

  $=\frac{1520+1516}{2}$

So,

  $=\frac{3036}{2}$

On solving,

  $=1518$

Hence, median is 1518

For lower quartile, substituting the values,

  $Q1=(10+1)\times \frac{1}{4}$

So,

  $Q1=11\times \frac{1}{4}$

On solving,

  $Q1=2.75$

Rounding off the value,

  $Q1\approx 3$

Substituting the value,

  $Q1=2060$

Hence, lower quartile is 2060

For upper quartile, substituting the values,

  $Q3=(10+1)\times \frac{3}{4}$

So,

  $Q3=11\times \frac{3}{4}$

On solving,

  $Q3=8.25$

Rounding off,

  $Q3\approx 8$

Substituting the value,

  $Q3=1342$

Hence, upper quartile is 1342

Minimum is the smallest number of the dataset.

Hence, minimum= 1215

Maximum is the largest number of the dataset.

Hence, maximum= 2788

Hence, Median is 1518, lower quartile is 2060, upper quartile is 1342, minimum is 1215 and maximum is 2788.

b.

To determine

To evaluate the five number summary.

Answer to Problem 21PFA

Five number summary is 1215, 1342, 1518, 2060, 2788

Explanation of Solution

Given information: The data of worldwide grosses for the top 10 highest grossing films.

Rank	Gross( in millions)
1	2788
2	2187
3	2060
4	1670
5	1520
6	1516
7	1405
8	1342
9	1277
10	1215

Median is 1518, lower quartile is 2060, upper quartile is 1342, minimum is 1215 and maximum is 2788.

Formula used: Five number summary is Minimum, lower quartile, median upper quartile and maximum

Calculation:

Substituting the value of minimum, lower quartile, median, upper quartile and maximum. Hence, the five number summary of the given data set is,

1215, 1342, 1518, 2060, 2788