T TEST EXAMPLES FOR PRACTICE

Dr. Rajeshwar Hendre - Statistics 


     

t-test examples
1.      A random sample of size 16 has 53 as mean. The sum of squares of the deviations takes from mean is 135. Can this sample be regarded as taken from the population having 56 as mean? Obtain fiducial limits of 95 % and 99%.
2.      Raja Restaurant near the railway station at Falna has been having average sales of 500 tea-cups per day. Because of development of bus stand nearby it expects to increase its sales. During the first 12 days after the start of new bus stand, the daily sales were: 550, 570, 490, 615, 505, 580, 570, 460, 600, 580, 530 and 526. Can one conclude that sales have increased at 5% level of significance?
Daily Sales
( X - X bar)
( X - X bar)^2

550
2
4

570
22
484

490
-58
3364

615
67
4489

505
-43
1849

580
32
1024

570
22
484

460
-88
7744

600
52
2704

580
32
1024

530
-18
324

526
-22
484
Average
548
Total =
23978
SD =
46.68852


3.      For random samples of 10 persons, fed on diet A, the increased weight in pounds in a certain period were 10, 6, 16, 17, 13, 12, 8, 14, 15 and 9. For another random sample of 12 persons, fed on diet B, the increase in the same period were, 7, 13, 22, 15, 12, 14, 18, 8, 21, 23, 10 and 17. Test whether the diets A and B differs significance as regards their effect on increase in weight? 
Diet A (10 Person)
( X1 - X1 bar)
( X1 - X1 bar)^2
Diet B (12 Person)
( X2 - X2 bar)
( X2 - X2 bar)^2
10
-2
4
7
-8
64
6
-6
36
13
-2
4
16
4
16
22
7
49
17
5
25
15
0
0
13
1
1
12
-3
9
12
0
0
14
-1
1
8
-4
16
18
3
9
14
2
4
8
-7
49
15
3
9
21
6
36
9
-3
9
23
8
64



10
-5
25



17
2
4
12
0
120
15
0
314
Average
Sum
Sum
Average
Sum
Sum

4.      Let the random variables X and Y denote the growth of pea stem segments for two independent samples. The value of X for n = 11 observations are 0.80, 1.80, 1.0, 0.10, 0.9, 1.7, 1.0, 1.4, 0.9, 1.2, 0.5 and those of Y for m =13 are 1.0, 0.80, 1.6, 2.6, 1.3, 1.1, 2.4, 1.8, 2.5, 1.4, 1.9, 2.0 and 1.20. Compute `X, SX2, `Y, and SY2. Use t-test at µ= 0.01 test hypothesis mx - mY = 0 against mx - mY < 0.
5.      To verify whether a course in accounting improved performance, a similar test given to 12 participants both before and after the course. The original marks recorded in alphabetical order of the participants were- 44, 40, 61, 52, 32, 44, 70, 41, 67, 72, 53 and 72. After the course, the marks were in the same order, 53, 38, 69, 57, 46, 39, 73 48, 73, 74, 60 and 78 was the course useful?
Before Course
After Couse
d
d^2
44
53
9
81
40
38
-2
4
61
69
8
64
52
57
5
25
32
46
14
196
44
39
-5
25
70
73
3
9
41
48
7
49
67
73
6
36
72
74
2
4
53
60
7
49
72
78
6
36


60
578


dbar = 60/12 = 5

6.      A drug is given to 10 patients, and the increments in their blood pressure were recorded to be – 3, 6, -2, 4, -3, 4, 6, 0, 0 and 2. It is responsible to belief that the drug has no effect on charge of blood pressure?

Blood Pressure Recorded
d^2

3
9

6
36

-2
4

4
16

-3
9

4
16

6
36

0
0

0
0

2
4
Total
20
130

7.      A sales data of an item in six shops before and after promotional campaign are as under
Shops
A
B
C
D
E
F
Before
53
28
31
48
50
42
After
58
29
30
55
56
45
Can the campaign be judged as a successful?
8.      Two types of batteries X and Y are tested for their length of life and the following results are obtained.
Battery
Sample Size
Mean Hrs
Variance
X
10
1000
100
Y
12
1020
121
Can you conclude both type of batteries have same mean?
Sample X
100
1080
110
Sample Y
150
1150
144

9.      The life time of electric bulbs for a random sample of 10 from a large consignment gave the following data
Item
1
2
3
4
5
6
7
8
9
11
Life in 000’hrs
4.20
4.6
3.9
4.1
5.2
3.8
3.9
4.3
4.4
5.6
Can we accept the hypothesis that the average life time of bulbs is 4000 hrs?
Item
Life in 000'Hrs
d^2
1
4.2
17.64
2
4.6
21.16
3
3.9
15.21
4
4.1
16.81
5
5.2
27.04
6
3.8
14.44
7
3.9
15.21
8
4.3
18.49
9
4.4
19.36
10
5.6
31.36
Total
44
196.72

10.  The weight of certain kind of hen is 2.5Kg after 3 months. The weights after changing feed type after 3 months found 1.75, 1.90, 2.80, 2.20, 2.00, 2.40, 2.30, 2.2, 2.6 and 2.1. Find out the effectiveness of the new feed.
11.  The heat producing capacity (in millions of calories per ton) of specimen coal from two mines :
Mine 1
8260
8130
8350
8070
8340

Mine 2
7950
7890
7900
8140
7920
7840
Use 0.01 level of significance test whether the difference between the means of these two samples is significant
12.  According to the norms a person who has attained 18year should weigh on an average 52 Kg with S.D. of 5 Kg. In a survey of 18 samples randomly selected weighs were as below:
42, 48, 51, 56, 58, 59, 58, 47, 49, 60, 62 52, 58, 57, 47, 50, 51 and 56
Can we say the samples are drawn from population?
Weights
d= x-50
d^2
42
-8
64
48
-2
4
51
1
1
56
6
36
58
8
64
59
9
81
58
8
64
47
-3
9
49
-1
1
60
10
100
62
12
144
52
2
4
58
8
64
57
7
49
47
-3
9
50
0
0
51
1
1
56
6
36
53.3889
61
731
Average
Total
Total

13.  The mean breaking strength of a wire is 195 pounds. Five pieces of wire gave strength mean 169 and S.D. 15pounds. Test the hypothesis that m = 195 against m < 195 pounds at 0.01 level of significance.
14.  The measurement of tar content in a cigarette is 14.5, 14.3, 14.2, 14.4 and 14.6 mg per cigarette. Show the difference I mean of tar content as claimed by manufacturer,  m =14, is significant?

15.   The defects per day in a company were on an average 30. After starting quality management programs it found 25, 26, 20, 22, 21, 24, 23, 25, 22 and 23. Was the QMP effective?
Defects Per Day
d= x-23
d^2
25
2
4
26
3
9
20
-3
9
22
-1
1
21
-2
4
24
1
1
23
0
0
25
2
4
22
-1
1
23
0
0
23.1
1
33
Average
Sum
Sum


16.  The means of two random samples of size 9 and 7 are 196.42 and 198.82 respectively. The sum of squares of the deviations from the mean are 26.94 and 18.73 respectively. Can the samples be considered to have been drawn from the same normal distribution?

17.  The average number of defectives in a certain factory is claimed to be less than the average of all factories. The average of all factories is 30.5. A random sample of 25 defectives showed following distribution.

Class Limit
No
16-20
2
20-24
7
24-28
11
28-32
4
32-36
1
Total
25

Class Limit
No
m
d
f.d
f.d^2
16-20
2
18
-2
-4
8
20-24
7
22
-1
-7
7
24-28
11
26
0
0
0
28-32
4
30
1
4
4
32-36
1
34
2
2
4
Total
25


-5
23

18.  The breaking strength of copper wire is expected to be 500/Nm2. Test at 5% level of significance that the wires have strength significantly more than expected. Samples tested 10 gave results: 520, 516, 504, 490, 532, 480, 498, 510, 529 and 518. Also explain can it be said equal to 510/Nm2?

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