Answer to Question #282144 in Statistics and Probability for wei

Question

The IQ test results of a group of students in a certain college were taken and are presented in a frequency distribution below.

Classes f

70-75 2

76-81 8

82-87 19

88-93 21

94-99 28

100-105 38

106-111 15

112-117 9

1. Use the frequency distribution above, determine the following:

a. Size of the class interval

b. Lower boundary of the median class

C. Median class

d. Number of students with an IQ greater than 99.5

2. Compute the value of:

A. Mean

B. Median

C. Mode

Answer 1

1.

a. The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class limit of the next class

$$76-70=6$$

b.

Calculate the total frequency

$$n=2+8+19+21+28+38+15+9=140$$

Calculate $n/2$

$$n/2=140/2=70$$

Calculate cumulative frequency. Select Cumulative frequency just greater than or equal to $n/2.$

$$2+8+19+21=50,$$

$$2+8+19+21+28=78,$$

We find that the 70th observation lies in the class $94-99.$

Lower boundary of the median class is $l=93.5.$

c. Median class is $93.5-99.5.$

d.

$$38+15+9=62$$

There are 62 students with an IQ greater than 99.5.

2.

A.

$$\bar{x}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nf_ix_i$$

$$=\dfrac{1}{140}(2(72.5)+8(78.5)+19(84.5)+21(90.5)$$

$$+28(96.5)+38(102.5)+15(108.5)+9(114.5))$$

$$=\dfrac{13534}{140}\approx96.6714$$

B.

$l=$ lower boundary point of median class $=93.5$

$n=$ Total frequency $=140$

$cf=$ Cumulative frequency of the class preceding the median class $=50$

$f=$ Frequency of the median class $=28$

$c=$ class length of median class $=6$

$$Median\ M=l+\dfrac{n/2-cf}{f}\cdot c$$

$$=93.5+\dfrac{70-50}{28}\cdot 6\approx97.7857$$

C.

To find Mode Class

Here, maximum frequency is 38.

The mode class is $99.5-105.5.$

$L=$ lower boundary point of mode class $=99.5$

$f_1=$  frequency of the mode class $=38$

$f_0=$ frequency of the preceding class $=28$

$f_2=$  frequency of the succedding class $=15$

$c=$ class length of mode class $=6$

$$Z=L+(\dfrac{f_1-f_0}{2f_1-f_0-f_2})\cdot c$$

$$=99.5+(\dfrac{38-28}{2(38)-28-15})\cdot 6c$$

$$\approx101.3182$$