One Sample
Hypothesis Tests for
Means
(t-test)
T-Test
Calculating r values
Formulas:
statistic - parameter
test statistic 
standard deviationof statistic
t=
x  m
s
n
Calculating p-values
• For t-test statistic – Using the calculator
[STAT]
[TESTS]
2. [T-Test]
Writing Conclusions:
1) A statement of the decision
being made (reject or fail to
reject H0) & why (linkage)
AND
2) A statement of the results in
context. (state in terms of Ha)
Example 1: Drinking water is
considered unsafe if the mean
H0: m = 15
concentration
of lead is 15 ppb (parts
Ha: m
> 15 or greater. Suppose a
per
billion)
t=2.1
Where
m
is
the
true
mean
concentration
community randomly
selects
of 25
of leadsamples
in drinking
water
water
and
computes
Since the p-value < a,
I reject Ha0. t-test
There is
statistic
2.1. Assume
that
lead
sufficient
evidence
to suggest
that
the
P-value =of
tcdf(2.1,E99,24)
concentrations
are of
normally
mean
concentration
lead in drinking
=.0232
water is greater
thanthe
15 ppb.
distributed.
Write
hypotheses,
calculate the p-value & write the
appropriate conclusion for a = 0.05.
Example 2: A certain type of frozen
dinners states that the dinner
H0: m240
= 240calories. A random
contains
Ha: of
m > 12
240of these frozen dinners
sample
t=1.9
Where
m
is
the
mean caloric
was selected fromtrue
production
to see
Since
the p-value
<frozen
a, I reject
H0. There is
content
of
the
dinners
if the caloric content was greater
sufficient evidence to suggest that the
than
stated
on
the
box.
The
t-test
P-value
=
tcdf(1.9,E99,11)
true mean caloric content of these frozen
statistic
was
calculated
to
be
1.9.
=.0420
dinners is greater than 240 calories.
Assume calories vary normally. Write
the hypotheses, calculate the p-value
& write the appropriate conclusion
for a = 0.05.
Example 3: The Degree of Reading Power
(DRP) is a test of the reading ability of
children. The national average of DRP scores
in 3rd grade is 34. The DRP scores for a
random sample of 44 third-grade students in
a suburban district had a mean score of
35.091 with a standard deviation of 11.189.
If 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
How do you
is approximately normally distributed (or)
know?
•Since the histogram is unimodal with no outliers, the
Do you
sampling distribution is approximately normally
know s?
What are your
distributed
hypothesis
• s is unknown
statements? Is
H0: m = 34 where m is the true mean reading
there a key word?
Ha: m = 34
ability of the district’s third-graders
Plug values
35
.
091

34
Data, m0: 34,
L1, Freq 1
tT-Test,

 .List:
6467
OR…STAT,
into formula.
1 1.1 89
TEST, T-Test
/ 0, Calculate
mm
44
rp-value
.5212 = tcdf(.6467,1E99,43)=.2606(2)=.5212
Use t-test 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.
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.
Conclusion:
Write conclusion in
context in terms of Ha.
Example 4: 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 different from
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
per week
1208  1323
 2.29 p  value  .0295
Ha: m ≠ 1323 t 
275
30
Since p-value < a of 0.05, I reject the null
hypothesis. There is sufficient to suggest that the
sales of cookies are different from the earlier
figure.
Review of Confidence Intervals: 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 95%
confidence interval for the mean weekly
sales rate.
CI = ($1105.30, $1310.70)
Based on this interval, is the mean weekly
sales rate statistically different from the
reported $1323?