Section 9.3
One-sample z-test for a Population Mean
Hypothesis
Tests
Fixed Levels of Significance
This is Greek to me!


The Greek letter alpha, α, is called the
significance level. If the p-value is at or
below the significance level, we consider the
data to be statistically significant.
This is often 0.05, but other times you will see
other levels given to you.
Steps for a Hypothesis Test




Step 1: Write the null and alternative hypotheses. ALWAYS use
parameter symbols in the hypotheses.
Step 2: Choose the appropriate inference procedure. Check the
conditions of the test. (They will match the conditions for
confidence interval).
Step 3: Identify the test by name OR by formula. Calculate the
test statistic and find the p-value. Be sure you graph and shade
here.
Step 4: Link the p-value to the conclusion and write the
conclusion in the context of the problem.
At this time, we’ll only
be studying μ.
Since we are given sigma, our procedure is
the one-sample z-test. The conditions are
the same as for the z-interval.
One sample z-statistic (THE
FORMULA)

This only works when the parameter of interest
is μ.
z
x  0

n
The one-sample t-statistic
x 
t
s
n
The t-statistic does not
follow a normal distribution.
It follows a t-distribution
with n – 1 degrees of
freedom.
Let’s try it!

A score of 275 or higher on the National
Assessment of Educational Progress
corresponds with the ability to balance a
checkbook. In a SRS of 840 young
Americans, the mean score was 272. Is this
sample good evidence that the average score
for young Americans is below 275? Assume
σ = 60.
Back to the cola sweetness


Cola makers test new recipes for loss of
sweetness during storage. Here are the
sweetness losses (sweetness before storage
minus sweetness after storage) found by 10
tasters for one new cola recipe: {2.0, 0.4,
0.7, 2.0, -0.4, 2.2, -1.3, 1.2, 1.1, 2.3}
Are these data good evidence that the cola
lost sweetness?
Two-sided Alternatives



When the alternative hypothesis is two-sided, finding
the p-value is a little different. You don’t know which
way you are looking.
Because you shade on BOTH sides of the normal
curve, you need to DOUBLE the p-value.
Example: Suppose the mean systolic blood
pressure for males is 128 with a standard deviation
of 15. We feel that the mean systolic BP for male
executives differs from that of the entire population.
We take a sample of 72 executives and find that xbar is 126.07. Does this give us good evidence that
the executives have a mean BP that differs from that
of the population?
Try this one on your own!

A pharmaceutical manufacturer forms tablets by
compressing a granular material that contains the
active ingredient and various fillers. The hardness
of a example from each lot of tablets produced is
measured in order to control the compression
process. The target values for the hardness are μ =
11.5 and σ = 0.2. The hardness data for a sample
of 20 tablets are {11.627, 11.613, 11.493, 11.602,
11.360, 11.374, 11.592, 11.458, 11.552, 11.463,
11.383, 11.715, 11.485, 11.509, 11.429, 11.477,
11.570, 11.623, 11.472, 11.531}. Is there significant
evidence at the 5% level that the mean hardness for
the tablets is different from the target value?
Instructions when using your
calculator




Go to Stats. Choose Test. Then pick the
appropriate one.
μ0 is your null hypothesis. You must know σ (for the
population if a z test). Enter x-bar and n. Choose
what alternative hypothesis you want.
If you choose Calculate, you will be given your
information. If you choose Draw you will see a
picture with the z/t-value and p-value.
If you don’t have all the information above, you must
enter your data in L1 and choose Data.
Example

In a discussion of SAT scores, someone comments,
“Because only a minority of high school students
take the test, the scores overestimate the ability of
typical high school seniors. The mean SAT
mathematics score is about 475 with a standard
deviation of 100, but I think that if all seniors took
the test, the mean score would be less than 475.”
You gave the test to an SRS of 500 seniors from
California. These students had a mean score of xbar = 461. Is this good evidence against the claim
that the mean for all California seniors is less than
475?
Example

Here are estimates of the daily intakes of calcium (in
milligrams) for 38 women between the ages of 18
and 24 years who participated in a study of women’s
bone health: {808, 882, 1062, 970, 909, 802, 374,
416, 784, 997, 651, 716, 438, 1420, 948, 1050, 976,
572, 403, 626, 774, 1253, 549, 1325, 446, 465,
1269, 671, 696, 1156, 684, 1933, 748, 1203, 2433,
1255, 1100, 1425}



Display the data using a histogram and then make a
normal probability plot. Describe the distribution of calcium
intakes for these women.
Calculate the mean and the standard deviation of the
sample.
Is the sample of 18-24 year olds calcium intakes
significantly different from that of the population, if the
populations mean was 904.5?
Homework
Chapter9 # 70, 72, 74, 79, 90, 99