For the dataset shown here what portion of the data is withi

For the dataset shown here, what portion of the data is within two standard deviations of the mean? Express as a decimal.

0

0

2

5

10

23

49

0

0

3

6

11

23

49

0

0

3

6

11

24

51

0

1

3

6

12

24

52

0

1

3

7

12

30

59

0

1

3

7

15

35

75

0

1

3

10

15

35

100

0

1

4

10

16

38

150

0

1

5

10

20

38

170

0

2

5

10

20

46

250

Mean: 22.60

Standard Deviation: 41.85

0	0	2	5	10	23	49
0	0	3	6	11	23	49
0	0	3	6	11	24	51
0	1	3	6	12	24	52
0	1	3	7	12	30	59
0	1	3	7	15	35	75
0	1	3	10	15	35	100
0	1	4	10	16	38	150
0	1	5	10	20	38	170
0	2	5	10	20	46	250
Mean: 22.60
Standard Deviation: 41.85

Solution

three standard deviations of the mean:

mean -3*standard deviation =22.6-3*41.85 =-102.95

mean +3*standard deviation=22.6+3*41.85=148.15

So the portion of the data is within three standard deviations of the mean is

P(-102.95<X<148.15) = P((-102.95-22.6)/41.85 <(X-mean)/s <(148.15-22.6)/41.85)

=P(-3<Z<3)

=0.9973 (from standard normal table)