Download Are women`s feet getting bigger? Retailers in the last 20 years have

Are women’s feet getting bigger? Retailers in the last 20 years have had to increase their
stock of larger sizes. Wal-Mart Stores, Inc., and Payless ShoeSource, Inc., have been
aggressive in stocking larger sizes, and Nordstrom’s reports that its larger sizes typically
sell out first. Assuming equal variances, at a = .025, do these random shoe size samples of
12 randomly chosen women in each age group show that women’s shoe sizes have
increased?
ShoeSize1
Born in 1980: 8 7.5 8.5 8.5 8 7.5 9.5 7.5 8 8 8.5 9
Born in 1960: 8.5 7.5 8 8 7.5 7.5 7.5 8 7 8 7 8
Solution:
Size of sample (n) = 11
Sample Mean for Born in 1980(1): 8.227273
Standard Deviation for Born in 1980(s1): 0.64667
Sample Mean for Born in 1960 (2): 7.681818
Standard Deviation for Born in 1960(s2): 0.462208
Degree of freedom = 11+11-2 = 22-2 = 20
We can define two-tailed statistics for above observations as follows:
Null Hypothesis: H0: (1 - 2) =0 vs. Ha: (1 - 2) > 0
Formula for test statistics,
Using above two-tailed statistics,
Rejection region can be defined as
z < -z/2
or z > z/2
Here significance level is 0.025,
So, z0.025 = 2.4231 (Using Statistical Ratio Calculator
From http://www.graphpad.com/quickcalcs/DistMenu.cfm for calculating z with 0.025
significance with DF = 20)
I am assuming the sample is selected independently and randomly from population. Population
size is sufficiently large in both samples.
Calculating test statistics for both samples,
z=
8.227273  7.681818
0.64667 2 0.462208 2

11
11
 2.276
Calculating p-value:
Degree of freedom = DF = 22-2 = 20
P(|z20| < 2.276) = 0.033994
(Using http://www.danielsoper.com/statcalc/calc08.aspx with t=2.276 and DF = 20)
Since calculated test-statistics is equal to 2.276 which is less than 2.4231and calculated
probability of 3.4% is greater than significance level of 2.5%. Hence, we fail to reject the null
hypothesis as there is not enough evidence at 2.5% level of significance that there is a difference
in the mean number of women’s shoe size between 1980 and 1960.