Material from:
- W. J. Conover, Practical Nonparametric Statistics 2nd ed, 1980, John Wiley and Sons, Inc.
- Examples from: Bowerman, Bruce L., Richard T. O'Connell, Business Statistics in Practice, 3rd edition, 2003, McGraw-Hill Companies, Inc.
The Sign Test: A Hypothesis Test about the Median
- Sign test is valid for any sample size and population shape
- May use when Population is Highly Skewed to right of left (Use Median)
- May use when If Sample size is small and population is highly skewed or clearly non-mound-shaped (not normally distributed)
Example 15.1 Compact Disc Player Lifetimes (From our book)
Step 1: Design Experiment and Collect Data
Step 2: Stem and Leaf (in thousands) shows us it is highly skewed data and median is best measure.
Counter
2
0 09
2 1
3 2 1
3 3
4 4 8
5 5 8
(7) 6 0245689
8 7 01245678
Step 3: Convert to (+) or (-) counts
Step 4: Record Givens
(+'s) = T
(-'s) = #
N = Grand Total
P = 1/2 (always for testing large samples)
n = # of (+'s) and (-'s)
Ties Don't Count
alpha = assume .05 unless given
In this problem we are looking for the median lifetime of CD players that exceed 6000.
Step 5: Decide on Hypotheses Test
- Two-tailed test Ho: p(+) = p(-)
- One-Tailed Test Ho: p(+) <= p(-)
- One-Tailed Test Ho: p(+) >= p(-)
The Sign Test for a Population Median
S = the number of sample measurements (less/greater) than M0,
x to be a binomial random variable with n = 20, p = 0.5.
We can reject H0: Md = M0 at the alpha level of significance (probability of Type I error equal to ?) if and only if the appropriate p-value is less than ?.
The Large Sample Sign Test for a Population Median
If the sample size n is large (n >= 10), we can reject H0: Md = M0 at the alpha level of significance (probability of Type I error equal to alpha) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than alpha.
Step 6: Using a software package (MiniTab or Excel) Solve
Excel Spreadsheet for Sign Test
Step 7: Use the Binomial Probability or z Tables