Statistics Essentials For Dummies (34 page)
See Chapter 1 for calculations of the mean and standard deviation.
2. Find
minus
. (Remember,
is the claimed value of the population mean.)
3. Calculate the standard error:
4.
Divide your result from Step 2 by the standard error found in Step 3.
For the Dr. Phil example, suppose a random sample of 100 working mothers spend an average of 11.5 minutes per day talking with their children, with a standard deviation of 2.3 minutes. That means
is 11.5, where
n
= 100 and
s
= 2.3. Take 11.5 - 11 = +0.5.Take 2.3 divided by the square root of 100 (which is 10) to get 0.23 for the standard error. Divide +0.5 by 0.23, to get 2.17. That's your test statistic.
This means your sample mean is 2.17 standard errors above the claimed population mean. Would these sample results be unusual if the claim (H
o
:
μ
= 11 minutes) were true? To decide whether your test statistic supports H
o
, calculate the
p
-value. To calculate the
p
-value, look up your test statistic (in this case, 2.17) on the standard normal distribution (
Z
-distribution) — see Table A-1 in the appendix — and take 100% minus the percen-tile shown (since we are looking at the right tail), because your H
a
is a greater-than hypothesis. In this case, the percentage would be 100% - 98.50% = 1.50%. So, the
p
-value is 0.0150 (1.50%).
This
p
-value of 0.0139 (1.39%) is much less than 0.05 (5%). So, reject the claim (
μ
= 11 minutes) by rejecting H
o
, and concluding H
a
(
μ
> 11 minutes). Your conclusion: According to this (hypothetical) sample, Dr. Phil's claim of 11 minutes is rejected; the actual average is greater than 11 minutes per day.
If the sample size,
n
, were less than 30 here, or the population standard deviation, s, were unknown, you would look up your test statistic on the
t
-distribution with n - 1 degrees of freedom (see Chapter 9) rather than the (
Z
-distribution).
Testing One Population Proportion
This test is used when the variable is categorical (for example, gender or political party) and only one population is being studied (for example, all U.S. citizens). The test is looking at the proportion (
p
) of individuals in the population who have a certain characteristic — for example, the proportion of people who carry cell phones. The null hypothesis is H
o
:
p
=
p
o
, where
p
o
is a certain claimed value. For example, if the claim is 20% of people carry cell phones,
p
o
is 0.20. The alternative hypothesis is one of the following:
p
>
p
o
,
p
<
p
o
, or
p
≠
p
o
.
The formula for the test statistic for a single proportion is
. To calculate it, do the following:
1. Calculate the sample proportion,
, by taking the number of people in the sample who have the characteristic of interest (for example, the number of people in the sample carrying cell phones) and dividing that by
n,
the sample size.
2. Take
minus
p
o. (Remember
p
o is the claimed number for the population proportion.)
