AP Statistics
 If
our data comes from a simple random sample (SRS)
and the sample size is sufficiently large, then we know
that the sampling distribution of the sample means is
approximately normal with mean μ and standard
deviation  .
n
 The
spread of the sampling distribution depends on n
and σ. σ is generally unknown and must be estimated.
 NOW…THEORY
ASIDE AND ONTO PRACTICE !
AP Statistics, Section 11.1
2
 SRS
– size n
 Normal distribution of a population
 μ and σ are unknown
 To estimate σ – use “S” in its place
Then the standard error of the sample mean is
s
n
AP Statistics, Section 11.1
3
z
x

n
 The
z statistic has N (0,1)
 When s is substituted the distribution is no longer
normal
AP Statistics, Section 11.1
4


The t statistic is used when
we don’t know the standard
deviation of the population,
and instead we use the
standard deviation of the
sample distribution as an
estimation.
The t statistic has n-1
degrees of freedom (df).
x 
t
s/ n
AP Statistics, Section 11.1
5

Interpret the t statistic in the
same way as the z statistic

There is a different distribution
for every sample size.

The t statistic has n-1 degrees
of freedom.

Write t (k) to represent the t
distribution with k degrees of
freedom.


Density curves for the t distribution
are similar to the normal curve
(symmetrical and bell shaped)
The spread is greater and there is
more probability in the tails and less
in the center.

Using s introduces more variability
than sigma.

As d.f. increase, t(k) gets more
normal
AP Statistics, Section 11.1
6



In statistical tests of
significance, we still have H0
and Ha.
We need to provide the mu
in the calculation of the t
statistic.
Looking at the t table is
fundamentally different than
the z table.
x 
t
s/ n
AP Statistics, Section 11.1
7


Assume SRS size n with population mean μ
Confidence interval will be correct for normal
populations and approx. correct for large n.
estimate  t * (SE estimate)
(1  C)
t* 
for t(n-1)
2
CI  x  t * (
s
n
)
AP Statistics, Section 11.1
8



Let’s suppose that Mr. Young has been
told that he should mop the floor by 1:25
p.m. each day.
We collect 12 sample times with an
average of 27.58 minutes after 1 p.m.
and with a standard deviation of 3.848
minutes.
Find a 95% confidence interval for Mr.
Young’s mopping times.
AP Statistics, Section 11.1
9
x  27.58 min
s  3.848
n  12
df  11
CL : 95%
From table C:
t* = 2.201
 3.848 
CI  27.58  2.201 

 12 
CI : (25.135, 30.025)
AP Statistics, Section 11.1
10
Step 1:
 Population of interest:
◦ Mr. Young’s mopping time

Parameter of interest:

Hypotheses
◦ average time of arrival to
mop
◦ H0: µ=25 min past 1:00
◦ Ha: µ>25 min past 1:00
x 
t
s/ n
AP Statistics, Section 11.1
11




We are using 1 sample t-test?
Bias?
◦ SRS not stated. Proceed with caution.
Independence?
◦ Population size is at least 10 times the
sample size?
◦ We assume that Mr. Young has mopped
on a lot of days
Normality?
◦ Big sample size (> 30). No
◦ Sample is somewhat normal because the
sample distribution is single peaked, no
obvious outliers.
AP Statistics, Section 11.1
x 
t
s/ n
12

Calculate the test statistic,
and calculate the p-value
from Table C
27.58  25
t
3.848 / 12
 2.322
P(t  2.322) is between .025 and .02
AP Statistics, Section 11.1
13


Is the t-value of 2.322 statistically significant at the 5%
level? At the 1% level?
Does this test provide strong evidence that Mr. Young
arrives on time to complete his mopping?
Try this exercise on your calculator using:
STAT TESTS Tinterval
STAT TESTS T-Test
AP Statistics, Section 11.1
14



Wednesday:
Thursday:
Friday:
11.6 – 11.11
11.13 – 11.20
T-Test Worksheet
AP Statistics, Section 11.1
15