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STA301 Solution for fall 2009

Question 1: (Marks: 5+10=15)

a) Compute the mean deviation from the following data.

Groups

30-34

35-39

40-44

45-49

50-54

Frequency

10

15

5

11

9

Sol:

Necessary calculation for “mean deviation”.

Groups

Frequency ()

i.e.

30-34

10

32

320

-9.4

9.4

94

35-39

15

37

555

-4.4

4.4

66

40-44

5

42

210

0.6

0.6

3

45-49

11

47

517

5.6

5.6

61.6

50-54

9

52

468

10.6

10.6

95.4

Total

50

2070

320

Mean (Arithmetic mean) is

So mean deviation is

= Ans.

b) The mean of 5 observations is 15 and variance is 9. If two more observations having values -3 and 10 are combined with these 5 observations, what will be the new mean and variance of 7 observations?

Sol:

Now sum of seven observations is i.e.

So the mean of seven observations is

Now variance of seven observations is so

Now

Ans.

Question 2: (Marks: 5+10=15)

a) Mean and the standard deviation of a set of values is 100 and 5. Compute mean ±2S.D and mean ±3S.D. Also interpret the results in the light of (i) empirical rule (ii) Chebychev’s inequality.

Sol: given

So calculations of intervals are

(i) Empirical rule

For interval (90,110)

Approximately 95% of the measurements will fall within standard deviations of the mean.

For interval (85,115)

Approximately 100% of measurements will fall within standard deviations of the mean.

(ii) Chebychev’s inequality

For interval (90,110)

At least of the observations will fall within standard deviations of the mean,

For interval (85,115)

At least of the observations will fall within standard deviations of the mean.

b) Find the first four moments about X= 174.5 as well as about mean for the following frequency distribution.

Weights (pounds)

Frequency

110-119

1

120-129

4

130-139

17

140-149

28

150-159

25

160-169

18

170-179

13

180-189

6

190-199

5

200-209

2

210-219

1

Total

-End-

Sol:


Weight

110-119

114.5

1

-60

-60

3600

-216000

12960000

120-129

124.5

4

-50

-200

10000

-500000

25000000

130-139

134.5

17

-40

-680

27200

-1088000

43520000

140-149

144.5

28

-30

-840

25200

-756000

22680000

150-159

154.5

25

-20

-500

10000

-200000

4000000

160-169

164.5

18

-10

-180

1800

-18000

180000

170-179

174.5

13

0

0

0

0

0

180-189

184.5

6

10

60

600

6000

60000

190-199

194.5

5

20

100

2000

40000

800000

200-209

204.5

2

30

60

1800

54000

1620000

210-219

214.5

1

40

40

1600

64000

2560000

Total

-2200

83800

-2614000

113380000

total/n

698.33

-21783.33

944833.33


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