PSY 555 Homework 4

3.1(a).   Original data:

1 2 2 3 3 3 4 4 4 4 5 5 5 6 6 7

or

3.1(b).   To convert the distribution to a distribution of X-m, subtract m=4 from each score.

-3 -2 -2 -1 -1 -1 0 0 0 0 1 1 1 2 2 3

3.1(c).   To complete the conversion to z, divide each score by s=1.63:

-1.84 -1.23 -1.23 -.61 -.61 -.61 0 0 0 0 .61 .61 .61 1.23 1.23 1.84

3.2.      Converting specific scores from the distribution in Exercise 3.1 into z scores.

Z=

Z=-.92=+1.35=+3.07

Score (X)     Z Score

2.5           -.92      18%of the distribution lies below X=2.5

6.2           +1.35     91% of the distribution lies below X=6.2

9.0           +3.07     99.9% of the distribution lies below X=9.0

3.3(a).   Errors counting shoppers in a major department store:

z=

z=  Between +1 and m lie .3413

z= Between -1 and m lie .3413

.3413+.3413=.6826

Therefore between 960 and 990 are found approximately 68% of the scores.

3.3(b).   975=m; therefore, 50% of the scores lie below 975.

3.3(c).   .5000 lie below 975

.3413 lie between 975 and 990

.8413 (or 84%) lie below 990.

3.4(a).   Using the data in Exercise 3.3, from Appendix Z:

Z Score       Area between Z and Mean

.67           .2486

.6745         .2500              [interpolation from Appendix Z]

.68           .2517

Therefore, z=.6745 encompasses the middle 50%.

z=

.6745=

X=958.12 and 964.88

50% of the scores lie between counts of 965 and 985.

3.4(b).  75% of the counts would be less than 985 because we just calculated the middle 50%, 25% of which lie on either side of the mean.  Since 50% lie below the mean, 50+25=75% lie below 985.

3.4(c).   What scores would 95% of the counts lie between?

z=

.6745=

945.6 and 1004.4

95% of the counts would lie between 946 and 1004.

3.6(a).

3.6(b).   z=

The smaller portion for z=1.00 is .1587.  Therefore 16% of the 4th graders score better than the average 9th grader.

3.6(c).   z=

The smaller portion of the 9th graders score worse than the average 4th grader.

1(a).

1(b).

 X Freq Cum Freq Cum % 6 2 22 100% 5 4 20 91% 4 6 16 73% 3 4 10 45% 2 4 6 27% 1 2 2 9%

The 45th percentile is 4.

1(c).    4.31 is the 70th percentile (convert to a z).

1(d).    5 is the 73% percentile.

1(e).    5 is the minimum score needed to be in the top 25% of the distribution.

1(f).    Yes.  The distribution is fairly normal.

2.       m=80

s=20

 X Calculations and Z-score 85 Z= 100 Z= 120 Z= 75 Z= 65 Z= 60 Z= 130 Z= 82 Z= 68 Z= 80 Z= 105 Z= 30 Z=

3.

 X (X-m)2 Z-score 13 56.25 Z= 17 12.25 Z= 21 .25 Z= 15 30.25 Z= 16 20.25 Z= 31 110.25 Z= 23 6.25 Z= 14 42.25 Z= 27 42.25 Z= 28 56.25 Z=

m=

4.

m=85

X=72

Z=-1.0

Z

sZ=

Standard Deviation Calculation Problem Answers

1.

 X (X-)2 13 139.24 21 14.44 27 4.84 31 38.44 35 104.04 24 .64 28 10.24 32 51.84 17 60.84 20 23.04

2.

 X (X-)2 100 4.41 115 166.41 112 98.01 113 118.81 95 50.41 87 228.01 90 146.41 104 3.61 107 24.01 98 16.81

3.

 X (X-)2 55 0 54 1 59 16 55 0 52 9 51 16 57 4 49 36 61 36 57 4