Transcript Inference for Means: The t Distribution

Warm Up
• For a sample of size 23 with an area to the
right=.05, what would the t value be?
• 90% confidence with n=24. What would
the t value be?
• If df=19 and t=.90 and it’s a right tailed
test, what would the p value be?
Part VI – Learning about the World
Ch. 23 – Inferences About Means
(Day 2: Confidence Intervals & Hypothesis Tests)
Conditions for Procedures
Involving Sample Means
To use the z-distribution
• Random sample
• n < 10%N
• Pop. approx. normal
or n ≥ 30
To use the t-distributions
• Random sample
• n < 10%N
• Pop. approx. normal is
still a condition, but it is
not as strict for t (we
will talk about this
condition more
tomorrow)
Using t for hypothesis testing…
• The average resting systolic blood pressure is
120 mm Hg. The director of the residency
program at a medical school believes that the
stress of the program causes higher blood
pressure in the new residents. The director took
a sample of 20 first year residents. He found that
their mean systolic blood pressure was 130.05
mm Hg, with a standard deviation of 9.96 mm
Hg. Can the director conclude that the average
blood pressure is higher for new residents?
x  130.05
s  9.96
n  20
df  19
μ = the mean systolic blood pressure for all first year residents
H0: μ = 120 mmHg
One-sample t-test, α = .05,
df = 19
Ha: μ > 120 mmHg
Condition
Check
Random
Sample
Assume
n < 10%N
Assume
20 < 10%
of all
residents
Pop
approx.
normal
Assume
(for now)
statistic  parameter
t
standard error
t
130.05  120
t
9.96
20
x 
s
n
 4.51
p < .0005
Since p < α, reject Ho. There is enough evidence to conclude
that the average blood pressure for all new residents is more
than 120 mmHg.
Confidence Intervals
• In the previous problem, construct a 98%
confidence interval for the mean blood pressure of
first year residents.
Statistic ± critical value · standard deviation
1-Sample t- interval, df = 19
Condition
Check
Random
Sample
Assume
n < 10%N
Assume
20 < 10%
of all
residents
Pop
approx.
normal
Assume
(for now)

xt  s

*


n


9.96
130.05  2.539 

20 

(124.395,135.705)
Don’t forget to interpret…
• We are 98% confident that the true mean
blood pressure for all first year residents in
the program is between 124.395 mmHg
and 135.705 mmHg.
Homework 23-2
• p. 554 #14, 17, 30
• Remember that if you
have a df that is not
on the table, use the
next smallest one