Download 12.1

2003 Free Response Question #2
A simple random sample of adults living in a suburb of a large city was
selected. The age and annual income of each adult in the sample were
recorded. The resulting data are summarized in the table below.
Age
$25,000$35,000
$35,000$50,000
Over
$50,000
Total
21-30
8
15
27
50
31-45
22
32
35
89
46-60
12
14
27
53
Over 60
5
3
7
15
Total
47
64
96
207
What is the probability that a person chosen at random from those in
the sample will be in the 31-45 age category?
What is the probability that a person chosen at random from those in
this sample whose incomes are over $50,000 will be in the 31-45 age
category? Show your work.
Based on your answers to parts (a) and (b), is annual income
independent of age category for those in this sample? Explain.
12.1:Tests for the Mean of a
Population
T-Statistic
There is a different tdistribution for each
sample size
We specify a particular
t distribution by giving
its degrees of freedom
= n-1.
Notation: t(k) = tdistribution with k
degrees of freedom.
Facts about t-distributions
Density curve is
similar in shape to
standard normal curve.
Spread is a bit greater
than that of the
standard normal dist.
As d.o.f. k increase,
the t(k) density curve
approaches N(0,1)
more closely.
T-distribution critical values.
Find the p-value from Table C for a tdistribution with n = 20 and t = 1.81. Ha:
mean > 5.
Find the p-value from Table C for a tdistribution with n = 37 and t = -3.17. Ha:
two-sided.
One-sample T Procedures
Draw an SRS of size n from a population
having unknown mean. A level C
confidence interval for  is:
X t
*
s
n
To test the hypothesis H 0 :   0 based on
an SRS of size n, compute the one-sample t
statistic
X  0
t
s
n
Cola Problem
Cola makers test new recipes for loss of sweetness
during storage. Trained tasters selected from an
SRS rate the sweetness before and after storage.
Here are the sweetness losses 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 good data evidence that the cola lost
sweetness? (Sweetness loss = sweetness beforesweetness after storage)
Matched Pairs t-procedures
Recall: In a matched pairs design, subjects are matched
in pairs and each treatment is given to one subject in
each pair, or we can perform before-and-after
observations on the same subjects.
Taste-test example is a matched pairs study in which the
same 10 tasters rated before-and-after sweetness.
To compare the responses to the two treatments in a
matched pairs design, apply the one-sample t procedures
to the observed differences.
The parameter  in a matched pairs t procedure is the
mean difference in the responses to the 2 treatments
within matched pairs of subjects in the entire population.
Floral Scents and Learning
We hear that listening to Mozart improves
students’ performance on tests. Perhaps pleasant
odors have a similar effect. To test this idea, 21
subjects chosen from an SRS worked a penciland-paper maze while wearing a mask; the mask
was either unscented or had a floral scent. The
response variable is their average time on 3 trials.
Each subjected worked the maze with both masks,
in random order. The following table gives the
subjects’ average times with both masks.
Analysis: Subtract the
scented time from the
unscented time for each
subject.
The 21 differences form a
single sample.
First subject: 7.37 s slower
wearing the scented mask,
so difference was -.
Shorter times = better
performance, therefore +
differences = student did
better.
The developer of a new filter for filter-tipped cigarettes claims
that it leaves less nicotine in the smoke than does the current
filter. Because cigarette brands differ in a number of ways, he
tests each filter on one cigarette of each of nine brands and
records the difference in nicotine content (current filter – new
filter). The mean difference is x-bar = 1.32 milligrams (mg),
and the standard deviation of the differences is s = 2.35 mg.
1. What is the parameter being measured?
2. State Ho and Ha for this study in both symbols and words.
3. Describe the population of interest for which inference is
being performed.
4. What conditions are required to carry out the significance
test? Discuss the validity of each.
5. Determine the test statistic and the P-value. Show your
work.
6. What do you conclude?
7. Construct a 90% confidence interval for the mean amount of
additional nicotine removed by the new filter is. Interpret the
interval.