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Frequency Distribution Assignment
Question 1: For the Academic Community of Higher Education improving student's academic performance is not an easy task. A case study was conducted using descriptive Statistics which is the discipline of quantitatively describing the main features of a collection of information. A data of 200 Students marks as population was collected and out of it 25 students marks are being analyzed for the case study by applying some measures that can be used to describe a data set.
Let us consider the marks obtained from a sample of 25students listed as follows:
70 80 86 46 56 66 76 86 90 70 50 45
94 65 55 60 90 80 70 71 72 62 64 76 70
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Solution:
(i) Construct a frequency distribution & relative frequency distribution for the above data with 5 classes. (1)
Classes

Frequency

Relative frequency = frequency/total

4555

3

0.12

5565

5

0.2

6575

7

0.28

7585

5

0.2

8595

5

0.2


25

1

(ii) Construct a Histogram with appropriate scale for the data. (1)
The following is the histogram:
(iii) Calculate the mean for the data using the frequency distribution table. (1)
The mean is as follows:
Classes

Frequency

Relative frequency = frequency/total

midpoint = (lower limit+upper limit)/2

midpoint * frequency

4555

3

0.12

50

150

5565

5

0.2

60

300

6575

7

0.28

70

490

7585

5

0.2

80

400

8595

5

0.2

90

450


25

1


1790

Mean = 1790/25 = 71.6
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Question 2: Three males with an Xlinked genetic disorder have one child each. The random variable x is the number of children among the three who inherit the Xlinked genetic disorder.
x

0

1

2

3

P(x)

0.15

0.20

0.30

0.35

Determine whether it is a probability distribution or not. If it is a probability distribution, find its mean and standard deviation.
Solution:
It is a probability distribution as the sum of p(x) = 0.15+0.20+0.30+0.35 = 1 and each probability is between 0 and 1.
Mean=∑xpx=(0*0.15)+(1*0.20)+(2*0.30)+(3*0.35)=0+0.20+0.60+1.05=1.85
Standard deviation=√((xmean)^2*p(x) )=√1.1275=1.06
x p(x) x.p(x) (xmean)^2 ((xmean)^2).p(x)
x

p(x)

x.p(x)

(xmean)^2

((xmean)^2).p(x)

0

0.15

0

3.4225

0.513375

1

0.2

0.2

0.7225

0.1445

2

0.3

0.6

0.0225

0.00675

3

0.35

1.05

1.3225

0.462875



1.85


1.1275

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Question 3: One Market Research company determined that 13% of college students work parttime during the academic year. For a random sample of 5 students, what is the probability that at least 3 students work parttime?
Solution:
n=5
p=0.13
q=1p=0.87
Probability at least 3 works part time=P(3)+P(4)+P(5)
Required probability=C_{3}^{5} 0.13^{3} 0.87^{2} +C_{4}^{5} 0.13^{4} 0.87^{1} + C_{5}^{5} 0.13^{5}0.87^{0}=0.18
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