Question

1)A group of 110 students sat an aptitude test, their resulting scores are presented:

66
61
66
76
70
64
67
66
71
64
63
61
65
67
67
72
62
64
69
65
72
53
76
69
60
76
70
62
70
71
71
71
64
63
69
65
79
63
64
66
61
58
80
74
61
67
70
62
71
69
79
75
73
72
66
68
72
72
67
63
76
61
75
64
84
73
53
76
71
65
64
61
74
74
72
78
70
83
77
79
67
69
79
66
62
70
75
66
61
75
77
69
75
63
68
69
74
76
79
72
72
58
67
65
58
75
53
62
64
76

a)Calculate the mean and standard deviation for the sample. Give your answers to 2 decimal places.

sample mean = ?

sample standard deviation = ?

b)Find the proportion of scores that are within 1 standard deviation of the sample mean and also the proportion that are within 2 standard deviations of the sample mean. Use the unrounded values for the mean and standard deviation when doing this calculation. Give your answers as decimals to 2 decimal places.

Proportion of scores within 1 standard deviation of the mean = ?

Proportion of scores within 2 standard deviations of the mean = ?

c)Select the appropriate description for the data:

a)the data are APPROXIMATELY normal
b)the data are CLEARLY not normal

d)Calculate the standardized value for the sample value 75. Note that, for a value x within a sample that is approximately distributed as N(x,s), a standardized value can be calculated as z = (x - x) / s

standardized value (to 2 decimal places) for the sample value 75 = ?

e)Calculate the probability that a standard normal random variable Z takes a values less than the standardized value calculated in part d). Give your answer as a decimal to 4 decimal places.

Probability Z less than standardized value = ?

f)Find the proportion of values in the sample that are less than 75. Give you answer as a decimal to 2 decimal places.

Proportion of values less than 75 = ?

2)

A group of mutual funds earned varying annual rates of return in the last year. These rates of return are normally distributed with a mean of 6% and a standard deviation of 17%.

One mutual fund in this group managed to earn a rate of return that was double that of this group's average that year. This performance would put the fund in the top X% of those funds in that year.

Calculate X%. Give your answer to 1 decimal place.

X% = %

Answer 1

1.a)The mean of the scores is .Then the sample mean of the above data is 68.6.Also the standard deviation is given by .Then the sample standard deviation of the above data is 6.44

b.)The proportion of scores within is =0.6636

The proportion of scores within is =0.9545

c.)

From the histogram,we can conclude that the data is approximately normal

d.)The standardized value for the sample value of 75 is is 0.99

e.)If Z~N(0,1),

f.)The proportion of sample values<75=

2.Let R be the random variable representing the annual rate of returns.Then R~N(6,=289)

Then P(R>12)=,i.e,36.2%