Question

In: Statistics and Probability

Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these...

Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers.

(b) List all possible samples and calculate the value of the sample mean ¯(X) and variance (s 2 ) for each sample?

(c) Obtain the sampling distribution of X¯ from this list by creating a frequency distribution table. You can create a frequency distribution table using Excel and share it on your file. Then calculate relative frequencies, i.e., f(x), which give the probabilities of the sampling distribution of X¯, and calculate the mean of the sampling distribution, i.e., xf(x). Check if that equals to (µ).

(d) Make a Histogram for the sampling distribution of X¯ you have obtained in (c). Use the Data Analysis Toolpak in Excel to make the Histogram. For Bins, use the Row Labels of the frequency tables you have created in (c).

Solutions

Expert Solution

ANSWER::

b.)

X1 X2
1 1 1 0
1 2 1.5 0.5
1 3 2 2
1 4 2.5 4.5
1 5 3 8
1 6 3.5 12.5
1 7 4 18
1 8 4.5 24.5
1 9 5 32
1 10 5.5 40.5
1 11 6 50
1 12 6.5 60.5
1 13 7 72
1 14 7.5 84.5
1 15 8 98
1 16 8.5 112.5
1 17 9 128
1 18 9.5 144.5
1 19 10 162
1 20 10.5 180.5
1 21 11 200
1 22 11.5 220.5
1 23 12 242
1 24 12.5 264.5
1 25 13 288
2 1 1.5 0.5
2 2 2 0
2 3 2.5 0.5
2 4 3 2
2 5 3.5 4.5
2 6 4 8
2 7 4.5 12.5
2 8 5 18
2 9 5.5 24.5
2 10 6 32
2 11 6.5 40.5
2 12 7 50
2 13 7.5 60.5
2 14 8 72
2 15 8.5 84.5
2 16 9 98
2 17 9.5 112.5
2 18 10 128
2 19 10.5 144.5
2 20 11 162
2 21 11.5 180.5
2 22 12 200
2 23 12.5 220.5
2 24 13 242
2 25 13.5 264.5
3 1 2 2
3 2 2.5 0.5
3 3 3 0
3 4 3.5 0.5
3 5 4 2
3 6 4.5 4.5
3 7 5 8
3 8 5.5 12.5
3 9 6 18
3 10 6.5 24.5
3 11 7 32
3 12 7.5 40.5
3 13 8 50
3 14 8.5 60.5
3 15 9 72
3 16 9.5 84.5
3 17 10 98
3 18 10.5 112.5
3 19 11 128
3 20 11.5 144.5
3 21 12 162
3 22 12.5 180.5
3 23 13 200
3 24 13.5 220.5
3 25 14 242
4 1 2.5 4.5
4 2 3 2
4 3 3.5 0.5
4 4 4 0
4 5 4.5 0.5
4 6 5 2
4 7 5.5 4.5
4 8 6 8
4 9 6.5 12.5
4 10 7 18
4 11 7.5 24.5
4 12 8 32
4 13 8.5 40.5
4 14 9 50
4 15 9.5 60.5
4 16 10 72
4 17 10.5 84.5
4 18 11 98
4 19 11.5 112.5
4 20 12 128
4 21 12.5 144.5
4 22 13 162
4 23 13.5 180.5
4 24 14 200
4 25 14.5 220.5
5 1 3 8
5 2 3.5 4.5
5 3 4 2
5 4 4.5 0.5
5 5 5 0
5 6 5.5 0.5
5 7 6 2
5 8 6.5 4.5
5 9 7 8
5 10 7.5 12.5
5 11 8 18
5 12 8.5 24.5
5 13 9 32
5 14 9.5 40.5
5 15 10 50
5 16 10.5 60.5
5 17 11 72
5 18 11.5 84.5
5 19 12 98
5 20 12.5 112.5
5 21 13 128
5 22 13.5 144.5
5 23 14 162
5 24 14.5 180.5
5 25 15 200
6 1 3.5 12.5
6 2 4 8
6 3 4.5 4.5
6 4 5 2
6 5 5.5 0.5
6 6 6 0
6 7 6.5 0.5
6 8 7 2
6 9 7.5 4.5
6 10 8 8
6 11 8.5 12.5
6 12 9 18
6 13 9.5 24.5
6 14 10 32
6 15 10.5 40.5
6 16 11 50
6 17 11.5 60.5
6 18 12 72
6 19 12.5 84.5
6 20 13 98
6 21 13.5 112.5
6 22 14 128
6 23 14.5 144.5
6 24 15 162
6 25 15.5 180.5
7 1 4 18
7 2 4.5 12.5
7 3 5 8
7 4 5.5 4.5
7 5 6 2
7 6 6.5 0.5
7 7 7 0
7 8 7.5 0.5
7 9 8 2
7 10 8.5 4.5
7 11 9 8
7 12 9.5 12.5
7 13 10 18
7 14 10.5 24.5
7 15 11 32
7 16 11.5 40.5
7 17 12 50
7 18 12.5 60.5
7 19 13 72
7 20 13.5 84.5
7 21 14 98
7 22 14.5 112.5
7 23 15 128
7 24 15.5 144.5
7 25 16 162
8 1 4.5 24.5
8 2 5 18
8 3 5.5 12.5
8 4 6 8
8 5 6.5 4.5
8 6 7 2
8 7 7.5 0.5
8 8 8 0
8 9 8.5 0.5
8 10 9 2
8 11 9.5 4.5
8 12 10 8
8 13 10.5 12.5
8 14 11 18
8 15 11.5 24.5
8 16 12 32
8 17 12.5 40.5
8 18 13 50
8 19 13.5 60.5
8 20 14 72
8 21 14.5 84.5
8 22 15 98
8 23 15.5 112.5
8 24 16 128
8 25 16.5 144.5
9 1 5 32
9 2 5.5 24.5
9 3 6 18
9 4 6.5 12.5
9 5 7 8
9 6 7.5 4.5
9 7 8 2
9 8 8.5 0.5
9 9 9 0
9 10 9.5 0.5
9 11 10 2
9 12 10.5 4.5
9 13 11 8
9 14 11.5 12.5
9 15 12 18
9 16 12.5 24.5
9 17 13 32
9 18 13.5 40.5
9 19 14 50
9 20 14.5 60.5
9 21 15 72
9 22 15.5 84.5
9 23 16 98
9 24 16.5 112.5
9 25 17 128
10 1 5.5 40.5
10 2 6 32
10 3 6.5 24.5
10 4 7 18
10 5 7.5 12.5
10 6 8 8
10 7 8.5 4.5
10 8 9 2
10 9 9.5 0.5
10 10 10 0
10 11 10.5 0.5
10 12 11 2
10 13 11.5 4.5
10 14 12 8
10 15 12.5 12.5
10 16 13 18
10 17 13.5 24.5
10 18 14 32
10 19 14.5 40.5
10 20 15 50
10 21 15.5 60.5
10 22 16 72
10 23 16.5 84.5
10 24 17 98
10 25 17.5 112.5
11 1 6 50
11 2 6.5 40.5
11 3 7 32
11 4 7.5 24.5
11 5 8 18
11 6 8.5 12.5
11 7 9 8
11 8 9.5 4.5
11 9 10 2
11 10 10.5 0.5
11 11 11 0
11 12 11.5 0.5
11 13 12 2
11 14 12.5 4.5
11 15 13 8
11 16 13.5 12.5
11 17 14 18
11 18 14.5 24.5
11 19 15 32
11 20 15.5 40.5
11 21 16 50
11 22 16.5 60.5
11 23 17 72
11 24 17.5 84.5
11 25 18 98
12 1 6.5 60.5
12 2 7 50
12 3 7.5 40.5
12 4 8 32
12 5 8.5 24.5
12 6 9 18
12 7 9.5 12.5
12 8 10 8
12 9 10.5 4.5
12 10 11 2
12 11 11.5 0.5
12 12 12 0
12 13 12.5 0.5
12 14 13 2
12 15 13.5 4.5
12 16 14 8
12 17 14.5 12.5
12 18 15 18
12 19 15.5 24.5
12 20 16 32
12 21 16.5 40.5
12 22 17 50
12 23 17.5 60.5
12 24 18 72
12 25 18.5 84.5
13 1 7 72
13 2 7.5 60.5
13 3 8 50
13 4 8.5 40.5
13 5 9 32
13 6 9.5 24.5
13 7 10 18
13 8 10.5 12.5
13 9 11 8
13 10 11.5 4.5
13 11 12 2
13 12 12.5 0.5
13 13 13 0
13 14 13.5 0.5
13 15 14 2
13 16 14.5 4.5
13 17 15 8
13 18 15.5 12.5
13 19 16 18
13 20 16.5 24.5
13 21 17 32
13 22 17.5 40.5
13 23 18 50
13 24 18.5 60.5
13 25 19 72
14 1 7.5 84.5
14 2 8 72
14 3 8.5 60.5
14 4 9 50
14 5 9.5 40.5
14 6 10 32
14 7 10.5 24.5
14 8 11 18
14 9 11.5 12.5
14 10 12 8
14 11 12.5 4.5
14 12 13 2
14 13 13.5 0.5
14 14 14 0
14 15 14.5 0.5
14 16 15 2
14 17 15.5 4.5
14 18 16 8
14 19 16.5 12.5
14 20 17 18
14 21 17.5 24.5
14 22 18 32
14 23 18.5 40.5
14 24 19 50
14 25 19.5 60.5
15 1 8 98
15 2 8.5 84.5
15 3 9 72
15 4 9.5 60.5
15 5 10 50
15 6 10.5 40.5
15 7 11 32
15 8 11.5 24.5
15 9 12 18
15 10 12.5 12.5
15 11 13 8
15 12 13.5 4.5
15 13 14 2
15 14 14.5 0.5
15 15 15 0
15 16 15.5 0.5
15 17 16 2
15 18 16.5 4.5
15 19 17 8
15 20 17.5 12.5
15 21 18 18
15 22 18.5 24.5
15 23 19 32
15 24 19.5 40.5
15 25 20 50
16 1 8.5 112.5
16 2 9 98
16 3 9.5 84.5
16 4 10 72
16 5 10.5 60.5
16 6 11 50
16 7 11.5 40.5
16 8 12 32
16 9 12.5 24.5
16 10 13 18
16 11 13.5 12.5
16 12 14 8
16 13 14.5 4.5
16 14 15 2
16 15 15.5 0.5
16 16 16 0
16 17 16.5 0.5
16 18 17 2
16 19 17.5 4.5
16 20 18 8
16 21 18.5 12.5
16 22 19 18
16 23 19.5 24.5
16 24 20 32
16 25 20.5 40.5
17 1 9 128
17 2 9.5 112.5
17 3 10 98
17 4 10.5 84.5
17 5 11 72
17 6 11.5 60.5
17 7 12 50
17 8 12.5 40.5
17 9 13 32
17 10 13.5 24.5
17 11 14 18
17 12 14.5 12.5
17 13 15 8
17 14 15.5 4.5
17 15 16 2
17 16 16.5 0.5
17 17 17 0
17 18 17.5 0.5
17 19 18 2
17 20 18.5 4.5
17 21 19 8
17 22 19.5 12.5
17 23 20 18
17 24 20.5 24.5
17 25 21 32
18 1 9.5 144.5
18 2 10 128
18 3 10.5 112.5
18 4 11 98
18 5 11.5 84.5
18 6 12 72
18 7 12.5 60.5
18 8 13 50
18 9 13.5 40.5
18 10 14 32
18 11 14.5 24.5
18 12 15 18
18 13 15.5 12.5
18 14 16 8
18 15 16.5 4.5
18 16 17 2
18 17 17.5 0.5
18 18 18 0
18 19 18.5 0.5
18 20 19 2
18 21 19.5 4.5
18 22 20 8
18 23 20.5 12.5
18 24 21 18
18 25 21.5 24.5
19 1 10 162
19 2 10.5 144.5
19 3 11 128
19 4 11.5 112.5
19 5 12 98
19 6 12.5 84.5
19 7 13 72
19 8 13.5 60.5
19 9 14 50
19 10 14.5 40.5
19 11 15 32
19 12 15.5 24.5
19 13 16 18
19 14 16.5 12.5
19 15 17 8
19 16 17.5 4.5
19 17 18 2
19 18 18.5 0.5
19 19 19 0
19 20 19.5 0.5
19 21 20 2
19 22 20.5 4.5
19 23 21 8
19 24 21.5 12.5
19 25 22 18
20 1 10.5 180.5
20 2 11 162
20 3 11.5 144.5
20 4 12 128
20 5 12.5 112.5
20 6 13 98
20 7 13.5 84.5
20 8 14 72
20 9 14.5 60.5
20 10 15 50
20 11 15.5 40.5
20 12 16 32
20 13 16.5 24.5
20 14 17 18
20 15 17.5 12.5
20 16 18 8
20 17 18.5 4.5
20 18 19 2
20 19 19.5 0.5
20 20 20 0
20 21 20.5 0.5
20 22 21 2
20 23 21.5 4.5
20 24 22 8
20 25 22.5 12.5
21 1 11 200
21 2 11.5 180.5
21 3 12 162
21 4 12.5 144.5
21 5 13 128
21 6 13.5 112.5
21 7 14 98
21 8 14.5 84.5
21 9 15 72
21 10 15.5 60.5
21 11 16 50
21 12 16.5 40.5
21 13 17 32
21 14 17.5 24.5
21 15 18 18
21 16 18.5 12.5
21 17 19 8
21 18 19.5 4.5
21 19 20 2
21 20 20.5 0.5
21 21 21 0
21 22 21.5 0.5
21 23 22 2
21 24 22.5 4.5
21 25 23 8
22 1 11.5 220.5
22 2 12 200
22 3 12.5 180.5
22 4 13 162
22 5 13.5 144.5
22 6 14 128
22 7 14.5 112.5
22 8 15 98
22 9 15.5 84.5
22 10 16 72
22 11 16.5 60.5
22 12 17 50
22 13 17.5 40.5
22 14 18 32
22 15 18.5 24.5
22 16 19 18
22 17 19.5 12.5
22 18 20 8
22 19 20.5 4.5
22 20 21 2
22 21 21.5 0.5
22 22 22 0
22 23 22.5 0.5
22 24 23 2
22 25 23.5 4.5
23 1 12 242
23 2 12.5 220.5
23 3 13 200
23 4 13.5 180.5
23 5 14 162
23 6 14.5 144.5
23 7 15 128
23 8 15.5 112.5
23 9 16 98
23 10 16.5 84.5
23 11 17 72
23 12 17.5 60.5
23 13 18 50
23 14 18.5 40.5
23 15 19 32
23 16 19.5 24.5
23 17 20 18
23 18 20.5 12.5
23 19 21 8
23 20 21.5 4.5
23 21 22 2
23 22 22.5 0.5
23 23 23 0
23 24 23.5 0.5
23 25 24 2
24 1 12.5 264.5
24 2 13 242
24 3 13.5 220.5
24 4 14 200
24 5 14.5 180.5
24 6 15 162
24 7 15.5 144.5
24 8 16 128
24 9 16.5 112.5
24 10 17 98
24 11 17.5 84.5
24 12 18 72
24 13 18.5 60.5
24 14 19 50
24 15 19.5 40.5
24 16 20 32
24 17 20.5 24.5
24 18 21 18
24 19 21.5 12.5
24 20 22 8
24 21 22.5 4.5
24 22 23 2
24 23 23.5 0.5
24 24 24 0
24 25 24.5 0.5
25 1 13 288
25 2 13.5 264.5
25 3 14 242
25 4 14.5 220.5
25 5 15 200
25 6 15.5 180.5
25 7 16 162
25 8 16.5 144.5
25 9 17 128
25 10 17.5 112.5
25 11 18 98
25 12 18.5 84.5
25 13 19 72
25 14 19.5 60.5
25 15 20 50
25 16 20.5 40.5
25 17 21 32
25 18 21.5 24.5
25 19 22 18
25 20 22.5 12.5
25 21 23 8
25 22 23.5 4.5
25 23 24 2
25 24 24.5 0.5
25 25 25 0

c.)

x Frequency Relative frequency (=f(x)=frequency/625) xf(x)
1 1 0.0016 0.0016
1.5 2 0.0032 0.0048
2 3 0.0048 0.0096
2.5 4 0.0064 0.016
3 5 0.008 0.024
3.5 6 0.0096 0.0336
4 7 0.0112 0.0448
4.5 8 0.0128 0.0576
5 9 0.0144 0.072
5.5 10 0.016 0.088
6 11 0.0176 0.1056
6.5 12 0.0192 0.1248
7 13 0.0208 0.1456
7.5 14 0.0224 0.168
8 15 0.024 0.192
8.5 16 0.0256 0.2176
9 17 0.0272 0.2448
9.5 18 0.0288 0.2736
10 19 0.0304 0.304
10.5 20 0.032 0.336
11 21 0.0336 0.3696
11.5 22 0.0352 0.4048
12 23 0.0368 0.4416
12.5 24 0.0384 0.48
13 25 0.04 0.52
13.5 24 0.0384 0.5184
14 23 0.0368 0.5152
14.5 22 0.0352 0.5104
15 21 0.0336 0.504
15.5 20 0.032 0.496
16 19 0.0304 0.4864
16.5 18 0.0288 0.4752
17 17 0.0272 0.4624
17.5 16 0.0256 0.448
18 15 0.024 0.432
18.5 14 0.0224 0.4144
19 13 0.0208 0.3952
19.5 12 0.0192 0.3744
20 11 0.0176 0.352
20.5 10 0.016 0.328
21 9 0.0144 0.3024
21.5 8 0.0128 0.2752
22 7 0.0112 0.2464
22.5 6 0.0096 0.216
23 5 0.008 0.184
23.5 4 0.0064 0.1504
24 3 0.0048 0.1152
24.5 2 0.0032 0.0784
25 1 0.0016 0.04
Sum: 625 1 13

d.)

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orchestra answered 1 year ago
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