Page 252 - Statistics for Dummies

236
Part IV: Guesstimating and Hypothesizing with Confidence
the population standard deviation is 0.9 ounces. For Sponge-o-matic (x ),
2
the average absorbency is 3.5 ounces according to your sample; assume the
population standard deviation is 1.2 ounces. Carry out this hypothesis test
by following the 6 steps listed above:
, σ = 1.2,

1. Given the above information, you know
, σ = 0.9,
1
2
n = 50, and n = 50.
1
2
2. The difference between the sample means for (Stats-absorbent – Sponge-

. (A negative difference simply
o-matic) is
means that the second sample mean was larger than the first.)

3. The standard error is
4. Divide the difference, –0.5, by the standard error, 0.2121, which gives

you –2.36. This is your test statistic.
5. To find the p-value, look up –2.36 on the standard normal (Z-) distribu-

tion — see the Z-table in the appendix. The chance of being beyond, .
in this case to the left of, –2.36 is equal to 0.0091. Because H is a not-
a
equal-to alternative, you double this percentage to get 2 ∗ 0.0091 =
0.0182, your p-value. (See Chapter 14 for more on the not-equal-to
alternative.)
6. This p-value is quite a bit less than 0.05. That means you have fairly
strong evidence to reject H .
o
Your conclusion is that a statistically significant difference exists between
the absorbency levels of these two brands of paper towels, based on your
samples. And Sponge-o-matic comes out on top, because it has a higher
average. (Stats-absorbent minus Sponge-o-matic being negative means
Sponge-o-matic had the higher value.)
If one or both of your samples happen to be under 30 in size, you use the
t-distribution (with degrees of freedom equal to n – 1 or n – 1, whichever is
1 2
smaller) to look up the p-value. If the population standard deviations, σ and
1
σ , are unknown, you use the sample standard deviations s and s instead, and
2 1 2
you use the t-distribution with the abovementioned degrees of freedom. (See
Chapter 10 for more on the t-distribution.)
Testing for an Average Difference
(The Paired t-Test)
You can test for an average difference using the test in this section when
the variable is numerical (for example, income, cholesterol level, or miles
per gallon) and the individuals in the sample are either paired up in some
3/25/11 8:14 PM
22_9780470911082-ch15.indd 236 3/25/11 8:14 PM
22_9780470911082-ch15.indd 236