Question

Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work? In one study, 16 student subjects took a test of sensitivity to contrast after three hours facing into bright sun, both with and without eye grease. (Greater sensitivity to contrast improves vision and glare reduces sensitivity to contrast.) This is a matched pairs design. Here are the differences in sensitivity, with eye grease minus without eye grease.
0.07, 0.64, ?0.13, ?0.06, ?0.19, 0.16, ?0.16, 0.04
0.06, 0.02, 0.42, 0.25, ?0.11, 0.27, 0.05, 0.3
How much more sensitive to contrast are athletes with eye grease than without eye grease? Give a 99% confidence interval to answer this question. (Round your answer to three decimal places.)
______________ to ______________?

Answer 1

(a) test H0:  ?= 0 vs. Ha: ? > 0, where ? is the mean sensitivity difference in the population.

(b) The mean of the 16 differences is 0.101875,

so the test statistic is z = (0.101875-0) / (0.22/sqrt(16))=  1.8523.

The one-sided P-value for this value of z is P = 0.0329.

CONCLUDE: The sample gives significant evidence (at the a = 0.05 level) that eye grease increases sensitivity.

Calculation of Confidence Interval

M = 0.101875
t = 2.58 for 99% confidence
s_M = ?(0.22777²/16) = 0.06

? = M ± Z(s_M)
? = 0.101875 ± 2.58*0.06
? = 0.101875 ± 0.14667416

Result

M = 0.101875,

99% CI [-0.04479916 to 0.24854916].

You can be 99% confident that the population mean (?) falls between -0.04479916 and 0.24854916.