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Psychological Statistics

1. If an original set of measurements is made in inches and has a mean of 36 and a variance of 144, what will be the mean, variance, and standard deviation if:

a) the unit is changed to feet?

b) 6 inches must be added to each measurement to correct an error?

c) The measurements are converted to centimeters (2.5 cm = 1 inch) and then 15 cm is subtracted from each measurement to correct an error?

2. Calculate the standard scores for each of the members of the following distribution of shoe sizes:

a) 5, 7, 7, 8, 9, 12

3. Determine the proportion of scores falling under the normal curve:

a) between z = -1.5 and z = +1.5

b) between z = 1 and z = 2

c) to the right of z = -.75

d) to the left of z = -2.58 and to the right of z = +2.58

4. Between what two z-scores lie the central 50% of scores in the standard normal distribution?

5. What proportion of a normal distribution with mean 42 and standard deviation of 8 is:
A: between 26 and 58?
B: greater than 66?
C. less than 34?
D. between 10 and 42?
E. Less than 36 or greater than 48?
F. Between 44 and 58?

6. Suppose someone scored an 8 on a statistics homework. The mean for the class was 5.64 and the standard deviation for the class was 1.10. In coming up with final grades for the class, the instructor realized she needed to convert the homework scores so that the mean is 100 and the standard deviation is 10. Your job is to convert the 8 so that it is on this new scale. What is the new score for this student?

Solution Preview

The answers are in the attached Excel File.
1. If an original set of measurements is made in inches and has a mean of 36 and a variance of 144, what will be the mean, variance, and standard deviation if:

If a constant is added to or subtracted from each measurement, there is no effect on resulting variance or standard deviation; the resulting mean is obtained by increasing or decreased the original mean by that constant
If each measurement is multiplied or divided by a constant, the resulting mean is obtained multiplying or dividing the original mean by the constant; the resulting variance is obtained by multiplying or dividing the original variance by the square of the constant; the resulting standard deviation is obtained by multiplying or dividing the original standard deviation by the absolute value (magnitude or number without sign) of the constant.

a) the unit is changed to feet?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

If we change the units to feet, we will have to divide the measurements by 12
Resulting mean and standard deviation are obtained by dividing the original values by 12
and resulting variance is obtained by dividing the original value by square of 12 i.e. 144 =12^2 (^2 means square of)

Resulting values

Mean= 3 feet
Variance = 1 square feet
Standard deviation = square root of variance= 1 feet =√1

b) 6 inches must be added to each measurement to correct an error?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

If we add a constant value, mean increases by the constant value, standard deviation and variance remain constant 12
Resulting mean is obtained by increasing the original value by 6

Resulting values

Mean= 42 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

c) The measurements are converted to centimeters (2.5 cm = 1 inch) and then 15 cm is subtracted from each measurement to correct an error?

Mean= 36 inches
Variance = 144 square inches
Standard deviation = square root of variance= 12 inches =√144

Mean is obtained by multiplying the original mean by 2.5 and subtracting 15
Standard deviation is obtained by multiplying the original standard deviation by 2.5
Variance is obtained by multiplying the original variance by 6.25 =2.5^2

Resulting values

Mean= 75 cm
Variance = 900 square cm
Standard deviation = square root of variance= 30 cm =√900

2. Calculate the standard scores for each of the members of the following distribution of shoe sizes:

a) 5, 7, 7, 8, 9, 12

To convert a measurement to a standard score, first subtract the mean score from it and then divide this remainder by the standard deviation.
z=(x-Mean)/Standard deviation

Therefore, first we calculate the mean and standard deviation of the shoe sizes

Mean and standard deviation

X= X 2 =
5 25
7 49
7 49
8 64
9 81
12 144
Total= 48 412
n=no of ...

Solution Summary

Questions in descriptive statistics- mean, standard deviation, variance, standard score, normal distribution.

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