Q 88. - The Practice of Statistics for AP

Question

We all know that fruit is good for us. Below is a histogram of the number of servings of fruit per day claimed by $74$ seventeen-year-old girls in a study in Pennsylvania?

(a) With a little care, you can find the median and the quartiles from the histogram. What are these numbers? How did you find them?

(b) Estimate the mean of the distribution. Explain your method clearly.

Step-by-Step Solution

Answer

Part(a) Median $74$

$$\begin{gathered} Q_1=1 \\ {Q}_3=4 \end{gathered}$$

Part (b) $\overline{X}\approx 2.59$

Part (a) Step 1: Given information

Given graph : 

Part (a) Step 2: Concept

The mean of a data set is determined by adding all of the numbers in the set and dividing by the number of values in the set. The median is the value in the middle of a data set when it is arranged from smallest to largest. The mode is the number that appears the most frequently in a data set.

Part (a) Step 3: Calculation

According to a study in Pennsylvania, the median amount of servings of fruit per day claimed by $74$ seventeen-year-old females is$2$, the first quartile is $1$ and the third quartile is $4$To find the median and quartiles, use the formulas below:

Domain main length	Frequency(f)	Commulative frequency (Cf)
0	15	15
1	11	26
2	15	41
3	11	52
4	8	60
5	5	65
6	3	68
7	3	71
8	3	74

The median can be calculated as follows: 

$$\begin{gathered} M = \frac{N+1}{2} \\ N = Cumilative frequency \\ M = \frac{74+1}{2} \\ M = 37.5 th item \\ =2 \\ {Q }_1=\frac{N+1}{4} th item \\ {Q }_1 =\frac{75}{4} th item \\ {Q }_1= 18.75 th item \\ =1 \\ {Q }_3=\frac{3(N+1)}{4} th item \\ =\frac{3\left(75\right)}{4} th item \\ = 56.25 th item \\ =4 \end{gathered}$$

Therefore, the median is $2$ and the quartiles are $1$ and $4$ respectively.

Part (b) Step 1: Calculation

From the given histogram of number of servings of fruit per day claimed by $74$ seventeen years old girls is a study in Pennsylvania the mean

Domain name length (X)	Frequency (f)	tx
0	15	0
1	11	11
2	15	30
3	11	33
4	8	32
5	5	25
6	3	18
7	3	21
8	3	24

The mean of the distribution is $$\begin{gathered} \overline{X}= \frac{\Sigma f x}{\Sigma f} \\ =\frac{194}{74} \\ \overline{X}=2.59 \end{gathered}$$

Therefore, mean is $2.59$