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Hypothesis Tests
One Sample Means
How canagency
I tell ifhas
they really
A government
are
underweight?
received numerous complaints
A hypothesis test
that will
a particular
restaurant
has
allow me to
been
selling
underweight
decide
if the
claim
is true or not!
hamburgers.
restaurant
Take The
a sample
& find x.
advertises that it’s patties are
“a quarter
pound”
(4 ounces).
But how
do I know
if this x is one
that I expect to happen or is it one
that is unlikely to happen?
Steps for doing a
hypothesis test
“Since the p-value < (>) a, I reject
1) Assumptions
(fail to reject) the H0. There is (is
not) sufficient evidence to suggest
thathypotheses
Ha (in context).”
2) Write
& define parameter
H0: m = 12 vs Ha: m (<, >, or ≠) 12
3) Calculate the test statistic & p-value
4) Write a statement in the context of the
problem.
Assumptions for z-inference
• Have an SRS from population (or
randomly assigned treatments)
• s known
• Normal (or approx. normal)
distribution
– Given
– Large sample size
Use only one of
these methods to
check normality
Assumptions for t-inference
• Have an SRS from population (or
randomly assigned treatments)
• s unknown
• Normal (or approx. normal)
distribution
– Given
– Large sample size
– Check graph of data
Use only one of
these methods to
check normality
Formulas:
s known:
statistic - parameter
test statistic 
standard deviation of statistic
z=
x m
s
n
Formulas:
s unknown:
statistic - parameter
test statistic 
standard deviation of statistic
t=
x m
s
n
Calculating p-values
• For z-test statistic –
– Use normalcdf(lb,ub)
– [using standard normal curve]
– Follow the same guidelines given previously
based on the type of test
• For t-test statistic –
– Use tcdf(lb, ub, df)
– Follow the same guidelines given previously
based on the type of test
Draw & shade a curve &
calculate the p-value:
1) right-tail test
t = 1.6; n = 20
P-value = .0630
2) two-tail test
t = 2.3; n = 25
P-value = (.0152)2 = .0304
Example 1: Bottles of a popular cola are
supposed to contain 300 mL of cola.
There is some variation from bottle to
bottle. An inspector, who suspects that
the bottler is under-filling, measures the
contents of six randomly selected bottles.
Is there sufficient evidence that the
bottler is under-filling the bottles?
Use a = .1
299.4 297.7 298.9 300.2 297 301
• I have an SRS of bottles
SRS?
Normal?
•Since the boxplot is approximately symmetrical with
no
outliers, the sampling distribution is approximatelyHow do you
know?
normally distributed
Do you
know s?
What are your
H0: m = 300 where m is the true mean amount
hypothesis
statements? Is
Ha: m < 300 of cola in bottles
there a key word?
299 .03  300
t 
 1.576 p-value =.0880
a = .1
1.503
Plug p-value
values to
Compare your
6
into decision
formula.
a & make
Since p-value < a, I reject the null hypothesis.
Writethat
conclusion
in
There is sufficient evidence to suggest
the true
context
in terms of Ha.
mean cola in the bottles is less than
300 mL.
• s is unknown
Example 2: The Degree of Reading Power
(DRP) is a test of the reading ability of
children. Here are DRP scores for a random
sample of 44 third-grade students in a
suburban district:
(See Data in Power Point Notes.)
At the a = .1, is there sufficient evidence to
suggest that this district’s third graders
reading ability is different than the national
mean of 34?
• I have an SRS of third-graders
SRS?
Normal?
•Since the sample size is large, the sampling distribution
is
How do you
approximately normally distributed
know?
OR
Do you
•Since the histogram is unimodal
withs?no outliers, the
know
What are your
sampling distribution is approximately normally
hypothesis
distributed
• s is unknown
statements? Is
H0: m = 34
a key word?
where m is the true mean there
reading
Ha: m ≠ 34
ability of the district’s third-graders
35.091  34
Plug values
t
 .6467
into formula.
11.189
44
p-value = tcdf(.6467,1E99,43)=.2606(2)=.5212
Use tcdf to
calculate p-value.
a = .1
Compare your p-value to
a & make decision
Since p-value > a, I fail to reject the null
hypothesis.
Conclusion:
There is not sufficient evidence to suggest that the
true mean reading ability of the district’s third-graders
is different than the national mean of 34.
Write conclusion in
context in terms of Ha.
A type II error – We decide that the true mean
reading ability is not different from the national
What type of error could you
average when it really is different.
potentially have made with this
decision? State it in context.
What confidence level should you
use so that the results match this
hypothesis test?
90%
Compute the interval.
(32.255, 37.927)
What do you notice about the
hypothesized mean?
Example 3: The Wall Street Journal
(January 27, 1994) reported that based
on sales in a chain of Midwestern grocery
stores, President’s Choice Chocolate Chip
Cookies were selling at a mean rate of
$1323 per week. Suppose a random sample
of 30 weeks in 1995 in the same stores
showed that the cookies were selling at
the average rate of $1208 with standard
deviation of $275. Does this indicate that
the sales of the cookies is lower than the
earlier figure?
Assume:
•Have an SRS of weeks
•Distribution of sales is approximately normal due to
large sample size
• s unknown
H0: m = 1323
where m is the true mean cookie sales
error in context?
Ha: m < 1323What is the
per potential
week
What is a consequence of that error?
1208  1323
t 
 2.29 p value  .0147
275
30
Since p-value < a of 0.05, I reject the null hypothesis.
There is sufficient evidence to suggest that the sales of
cookies are lower than the earlier figure.
Example 3 Continued: President’s Choice
Chocolate Chip Cookies were selling at a mean
rate of $1323 per week. Suppose a random
sample of 30 weeks in 1995 in the same
stores showed that the cookies were selling
at the average rate of $1208 with standard
deviation of $275. Compute a 90%
confidence interval for the mean weekly
sales rate.
CI = ($1122.70, $1293.30)
Based on this interval, is the mean weekly
sales rate statistically less than the
reported $1323?