Question

57 randomly selected students were asked the number of pairs of shoes they have. Let X represent the number of pairs of shoes. The results are as follows:

# of Pairs of Shoes	4	5	6	7	8	9	10	11
Frequency	7	8	14	4	3	4	7	10

Round all your answers to 4 decimal places where possible.

The mean is: __________________

The median is: __________________

The sample standard deviation is: __________________

The first quartile is: __________________

The third quartile is: __________________

What percent of the respondents have at least 6 pairs of Shoes? __________________%

71% of all respondents have fewer than how many pairs of Shoes? __________________

Answer 1

Solution(a)
Data is given in the question as follows:

Number of Pairs of Shoes(N)	Frequency(F)
4	7
5	8
6	14
7	4
8	3
9	4
10	7
11	10

Mean of data can be calculated as
Mean = Number of Pairs of shoes * Frequency / Frequency = ((4*7)+(5*8)+(6*14)+(7*4)+(8*3)+(9*4)+(10*7)+(11*10))/(7+8+14+4+3+4+7+10)

Number of Pairs of Shoes(N)	Frequency(F)	N*F
4	7	28
5	8	40
6	14	84
7	4	28
8	3	24
9	4	36
10	7	70
11	10	110

Mean = (28+40+84+28+24+36+70+110)/57 = 420/57 = 7.3684
Solution(b)
Median of data can be calculated as
Median = (n+1)/2 th value = (57+1)/2 = 58/2 = 29th value i.e. 6
Solution(c)
Sample standard deviation can be calculated as
Sample standard deviation = sqrt((F*(N-mean)^2/(n-1))

Number of Pairs of Shoes(N)	Frequency(F)	N*F	(N-mean)	(N-mean)^2 * F
4	7	28	-3.3684	79.42282992
5	8	40	-2.3684	44.87454848
6	14	84	-1.3684	26.21525984
7	4	28	-0.3684	0.54287424
8	3	24	0.6316	1.19675568
9	4	36	1.6316	10.64847424
10	7	70	2.6316	48.47722992
11	10	110	3.6316	131.8851856

Sample standard deviation = sqrt((79.4228+44.8745+26.2152+0.5428+1.1967+10.6484+48.4772+131.8852)/56) = sqrt(343.2632/56) = 2.4758
Solution(d)
First quartile Q1 can be calculated as
Q1 = (n+1)/4 = (57+1)/4 = 14.5th value, In the data set 14th value is 5 and 15th value is 5 so 14.5th value is 5 so Q1 = 5
Solution(e)
Third quartile can be calculated as
Q3 = 3*(n+1)/4 = 3*(57+1)/4 = 43.5th value. 43rd value is 10 and 44th value is 10 so 43.5th value is also 10. Q3 = 10
Solution(f)
P(Atleast 6 pairs of shoes) = Number of respondents have number of pair of shoes equal and more than 6 pair of shoes/ total number of respondents = (14+4+34+7+10)/57 = 42/57 = 0.7368
So there is 73.68% of the respondents have at least 6 pairs of Shoes.
Solution(g)
We need to calculate "71% of all respondents have fewer than how many pairs of Shoes"
kth value = Percent * Total number of respondents = 0.71*57 = 40.47 or 41 th
So 41th respondent have 10 pair of shoes so we can say 71% of all respondents have fewer than 10 pairs of Shoes