Transcript Section 8.3 First Day and Second Days Z

+
Z-Interval for µ
So, the formula for a Confidence Interval for a population mean is
xz
*

n
To be honest, σ is never known. So, this formula isn’t used very
much. The only homework problem regarding this formula is #55,
a problem where you are asked to find the sample size for a
desired margin of error for a CI for µ. Let’s practice that first.
+
Finding n for a desired MOE for a CI
for μ

High school students who take the SAT Math exam a second
time generally score higher than on their first try. Past data
suggest that the score increase has a standard deviation of 50
points. How large a sample of high school students would be
needed to estimate the mean change in SAT score to within 2
points with 95% confidence?
Standard Error
 We
don’t know σ. Therefore, we will estimate σ
based on s, the standard deviation of the sample.
σ is estimated from the data, the result is
called the standard error of the statistic.
 When
s
The standard error of x is
.
n
Finding the standard error of the mean
(like #59 in tonight’s homework)

Standard error of the mean is
often abbreviated SEM.

Suppose that a SRS of 20
people yields an average IQ of
107 with a standard deviation of
14.8. What is the SEM?
+
Characteristics of the t-distributions
 They

are similar to the normal distribution.
They are symmetric, bell-shaped, and are centered
around 0.
 The
t-distributions have more spread than a
normal distribution. They have more area in the
tails and less in the center than the normal
distribution.

That’s because using s to estimate σ introduces more
variation.
 As
the degrees of freedom increase, the tdistribution more closely resembles the normal
curve.

As n increases, s becomes a better estimator of σ.
+
Degrees of freedom

Unlike the z-statistic, the shape of the t-distribution changes
based on the sample size.

t(k) stands for a t-distribution with k degrees of freedom.

So, if our sample size is 30, and σ is unknown, the distribution is a tdistribution with 29 degrees of freedom. This would be written t(29).
+
The t-distribution
How many degrees of
freedom are there if
n=3? If n=10?
What happens to the tdistribution as n
increases?
What do I mean by n=∞ (normal)?
+
Using Table B
 Now
that we are studying a different distribution,
we’ll use a different table.
 Table
B gives the t-distribution critical values.
 Let’s
look for the t* value for an upper tail area of
.05 with n = 15.
 What
is the t* value for 18 degrees of freedom
with probability 0.90 to the left of t*?
 Suppose
you want to construct a 95%
confidence interval for the mean of a population
based on a SRS of size n = 12. What critical
value t* should you use?
Table B to Find Critical t* Values
Upper-tail probability p
df
.05
.025
.02
.01
10
1.812
2.228
2.359
2.764
11
1.796
2.201
2.328
2.718
12
1.782
2.179
2.303
2.681
z*
1.645
1.960
2.054
2.326
90%
95%
96%
98%
Confidence level C
In Table B, we consult the row
corresponding to df = n – 1 = 11.
We move across that row to the
entry that is directly above 95%
confidence level.
The desired critical value is t * = 2.201.
Estimating a Population Mean
Suppose you want to construct a 95% confidence interval for the
mean µ of a Normal population based on an SRS of size n =
12. What critical t* should you use?
+
 Using
t Interval for a Population Mean
Conditions
The One-Sample
for Inference
t Interval
about
for aaPopulation
PopulationMean
Mean
•Random:
Choose
an The
SRSdata
of size
come
n from
fromaapopulation
random sample
havingofunknown
size n from
mean
theµ.population
A level C
confidence
of
interest orinterval
a randomized
for µ is experiment.s
x  t*
x
• Normal: The population has a Normal distribution
or the sample size is large
n
(n ≥ 30).
where t* is the critical value for the tn – 1 distribution.
• Independent: The method for calculating a confidence interval assumes that
Use this interval only when:
individual observations are independent. To keep the calculations

accurate
wheniswe
sample
replacement
from(na ≥finite
(1) reasonably
the population
distribution
Normal
orwithout
the sample
size is large
30),
population, we should check the 10% condition: verify that the sample size
(2) the
at least
10population
times as large
is nopopulation
more thanis1/10
of the
size.as the sample.
Estimating a Population Mean
The one-sample t interval for a population mean is similar in both
reasoning and computational detail to the one-sample z interval for a
population proportion. As before, we have to verify three important
conditions before we estimate a population mean.
+
 One-Sample
Video Screen Tension Read the Example

Parameter: µ = the true mean tension of all the video terminals
produced this day
Conditions:
Random: We are told that the data come from a random sample of 20
screens from the population of all screens produced that day.
Normal: Since the sample size is small (n < 30), we must check whether it’s
reasonable to believe that the population distribution is Normal. Examine the
distribution of the sample data.
These graphs give no reason to doubt the Normality of the population
Independent: Because we are sampling without replacement, we must
check the 10% condition: we must assume that at least 10(20) = 200 video
terminals were produced this day.
Estimating a Population Mean
on page 508.
+
 Example:
Video Screen Tension
DO: Using our calculator, we find that the mean and standard deviation of
the 20 screens in the sample are:
x  306.32 mV
.10
sx  36.21 mV
.05
.025
Since n = 20, we use the t distribution with df = 19
to find the critical value.
From Table B, we find t* = 1.729.
Upper-tail probability p
df
and
18
1.130
1.734
2.101
19
1.328
1.729
2.093
20
1.325
1.725
2.086
90%
95%
96%
Confidence level C
Estimating a Population Mean
Example on page 508. We want to estimate the true mean
tension µ of all the video terminals produced this day at a
90% confidence level.
+
 Example:
Therefore, the 90% confidence interval for µ is:
sx
36.21
x  t*
 306.32  1.729
n
20
 306.32  14
 (292.32, 320.32)
CONCLUDE: We are 90% 
confident that the interval from 292.32 to 320.32 mV captures the
true mean tension in the entire batch of video terminals produced that day.
t Procedures Wisely
Using One-Sample t Procedures: The Normal Condition
• Sample size less than 15: Use t procedures if the data appear close to
Normal (roughly symmetric, single peak, no outliers). If the data are clearly
skewed or if outliers are present, do not use t.
• Sample size at least 15: The t procedures can be used except in the
presence of outliers or strong skewness.
• Large samples: The t procedures can be used even for clearly skewed
distributions when the sample is large, roughly n ≥ 30.
Estimating a Population Mean
Except in the case of small samples, the condition that the data
come from a random sample or randomized experiment is more
important than the condition that the population distribution is
Normal. Here are practical guidelines for the Normal condition
when performing inference about a population mean.
+
 Using
+
Confidence Intervals –
puzzles for math nerds
statistic  critical value standard deviation of statistic
Parameter
Mean
Proportion
Statistic

x
p
p
Standard
Deviation
of the
Statistic

n
p(1  p)
n
Standard
Error of
the
Statistic
s
n
p(1  p)
n