STATISTICS 1403
PROBABILITY AND STATISTICS FOR BIOSCIENCES
PROBLEM SET: CORRELATION AND Z-SCORES
Abstract. Solve these problems, and prepare a clean, annotated handwritten
copy of them. Scan your write-up, and submit it via Blackboard in .PDF or
.DOC format.
1. An Alternate View of Correlation
An interesting interpretation of sample correlation is the average cross-product
of z-scores for two paired random variables:
n n
1 X xi − x̄
yi − ȳ
1 X
r=
z(xi )z(yi )
=
n − 1 i=1
sx
sy
n − 1 i=1
2. Find the Correlation
Use the z-score formula above to find the correlation between height and weight
for this sample of n = 15 height-weight observations. Start by converting each
height and weight to their appropriate z-scores, then averaging the subject-bysubject products of those scores.
height weight
60
100
62
115
63
128
66
133
67
170
68
155
69
147
70
128
70
182
70
178
70
118
73
175
74
193
74
211
75
227
Hint: the minimum height z-score is -1.94, the maximum weight z-score is +1.86.
1
2
CORRELATION AND Z-SCORES
3. Confidence Interval
Use Fisher’s transformation to estimate a confidence interval for the correlation.
• First, transform the correlation to something having a Gaussian distribution
1
1+r
w(r) = ln
2
1−r
• Next, find an α-level confidence interval for the transformed value (find the
± values)
1
= (wL , wU )
w = w(r) ± zα/2 √
n−3
• Finally, use the inverse transformation to get a (rL , ru ) confidence interval
for the correlation
e2w − 1
r(w) = 2w
e +1
4. Your Analysis
Your analysis must be legibly handwritten. Include a brief problem
statement (in your own words, do NOT copy these instructions), and a brief
narrative explaining your calculations and findings. BE SURE TO PUT YOUR
NAME ON THE WRITE-UP. Submit your write-up via Blackboard as a
scanned or saved .PDF or Word .DOC file.