Teacher notes:
These notes are accompanied
with
1) 8.3 hand out involving
Monkeys and sample size
2) Table B
Confidence Intervals:
Estimating Population Mean
𝜇 𝑏𝑦 𝑢𝑠𝑖𝑛𝑔 𝑥
Section 8.3
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Objectives
• Confidence Intervals: when σ is known: The OneSample Z-Interval for a Population Mean
• Confidence Intervals: when σ is Unknown: The tDistribution
•
•
•
•
Calculator
Conditions for Calculating Confidence Intervals
Following 4 Step Process
Robust
So…
• Inference about a population proportion usually
arises when we study categorical variables. We
learned how to construct and interpret confidence
intervals for an unknown parameter p in section 8.2.
• To estimate a population mean, we have to record
values of a quantitative variable for a sample of
individuals. It makes sense to try to estimate the
mean amount of sleep that students at a large high
school got last night but not their mean eye color!
In this section, we’ll examine confidence intervals for a
population mean μ.
One Sample Z Interval for a Population Mean:
When σ is Known
As I’ve said multiple times, confidence intervals have the same form, just
replacing mean and standard deviation calculations depending on the
conditions.
Statistic ± (critical value)* (standard deviation of statistic)
Draw a SRS of size n from a population having unknown mean µ
and known standard deviation σ. As long as the Normal and
Independent conditions are met, a level C confidence interval for
µ is:
∗ σ
𝑥± 𝑧
𝑛
The critical value 𝑧 ∗ is found from the standard Normal
distribution.
Choosing the Sample Size
• To determine the sample size n that will yield a level C
confidence interval for a population mean with a specified
margin of error ME, solve the following inequality for n:
σ
∗
𝑧
≤ 𝑀𝐸
𝑛
• Lets look at an example from a hand out
Check Your Understanding
To assess the accuracy of a laboratory scale, a standard
weight known to weigh 10 grams is weighed repeatedly. The
scale readings are Normally distributed with unknown mean
(this mean is 10 grams if the scale has no bias). In previous
studies, the standard deviation of the scale readings has
been about 0.0002 gram. How many measurements must be
averaged to get a margin of error of 0.0001 with 98%
confidence? Show your work.
Solution:
When σ is unknown:
The t-distribution
• When we knew all of the information to find a
z score with our formula of:
𝑥−µ
𝑧= σ
𝑛
But what IF we don’t know the standard deviation anymore?
??=
𝑥−µ
??
𝑛
When we don’t know σ, we estimate it using the sample
standard deviation 𝑠𝑥 . This is known as t-score of a t-distribution
t=
𝑥−µ
𝑠𝑥
𝑛
The t-distribution for a Population Mean:
When σ is Unknown
• Draw a SRS of size n from a large population that has a
Normal distribution with mean µ and standard deviation σ.
t=
𝑥−µ
𝑠𝑥 / 𝑛
The statistic has the t distribution with degrees of
freedom, 𝑑𝑓 = 𝑛 − 1.
This statistic will have approximately a normal
distributions as long as the sampling distribution of 𝑥 is
close to Normal.
Degrees of Freedom
• The figure below compares the density curves of the standard
Normal distribution and the t distributions with 2 and 9 degrees
of freedom. The figure illustrates these facts about the t
distributions:
Degree of Freedom
• Density curves for the t distributions with 2 and 9 degrees of freedom and
the standard Normal distribution. All are symmetric with center 0. The t
distributions are somewhat more spread out.
• The density curves of the t distributions are similar in shape to the
standard Normal curve. They are symmetric about 0, single-peaked, and
bell-shaped.
• The spread of the t distributions is a bit greater than that of the standard
Normal distribution. The t distributions in the figure have more probability
in the tails and less in the center than does the standard Normal. This is
true because substituting the estimate sx for the fixed parameter σ
introduces more variation into the statistic.
• As the degrees of freedom increase, the t density curve approaches the
standard Normal curve ever more closely. This happens because sx
estimates σ more accurately as the sample size increases. So using sx in
place of σ causes little extra variation when the sample is large.
s is Unknown: The t Distributions
It has a different shape than the
standard Normal curve:
• When
Like any standardized statistic, t tells us how far x is from its mean m
in standard deviation units.
Estimating a Population Mean
It is symmetric with a single peak at 0,
However, it has much more area in the
tails.
However, there is a different t distribution for each sample size, specified by its
degrees of freedom (df).
•
Using Table B to Find Critical t* Values
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?
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
Upper-tail probability p
Calculator in Finding 𝑡
∗
• Most newer TI-84 and TI-89 calculators allow you to
find critical values t* using the inverse t command. As
with the calculator’s inverse Normal command, you
have to enter the area to the left of the desired critical
value.
TI 84:
2nd VARS (DIST) > 4:invT( > complete the command
invT(.975,11) > Enter
TI 89:
In Statistics/List Editor > F5> 2:Inverse>2:Inverse t… in
the dialog box, enter Area: .975 and Deg of Freedom df
: 11> Enter
Check For Understanding
• Use Table B to find the critical value t* that you would use for a
confidence interval for a population mean μ in each of the following
situations. If possible, check your answer with technology.
• (a) A 98% confidence interval based on n = 22
observations.
• Correct Answer ….t* = 2.518
• (b) A 90% confidence interval from an SRS of 10
observations.
• Correct Answer…. t* = 1.833
• (c) A 95% confidence interval from a sample of
size 7.
• Correct Answer….t* = 2.447
AP Tip
On the AP exam, if the desired degree of
freedom is not included in the table, it is
acceptable to use the next smaller df in the
table or technology.
The One Sample t Interval for a
Population Mean
Choose an SRS of size n from a population having unknown
mean μ. A level C confidence interval for μ is
statistic ± (critical value) · (standard deviation of statistic)
𝒔𝒙
𝒙±𝒕
𝒏
∗
where t* is the critical value for the tn−1 distribution. Use
this interval only when (1) the population distribution is
Normal or the sample size is large (n ≥ 30), and (2) the
population is at least 10 times as large as the sample.
Conditions
• Random: The data come from a random sample of size n
from the population of interest or a randomized experiment.
This condition is very important.
• Normal:
1. The population has a Normal distribution
2. OR by the CLT, (n ≥ 30), where n= 45, assume approx.
normal.
3. OR if sample is less than 30 and you have data you MUST
show a graph / histogram
• Independent: The method for calculating a confidence
interval assumes that individual observations are
independent. To keep the calculations reasonably accurate
when we sample without replacement from a finite population,
we should check the 10% condition: verify that the sample
size is no more than 1/10 of the population size.
Standard Error
• The Standard error of the sample mean 𝑥 is
𝑠𝑥
, where 𝑠𝑥 is the sample standard
𝑛
deviation. It describes how far 𝑥 will be from µ,
on average, in repeating SRSs of size n.
𝑆𝐸𝑥 =
𝑠𝑥
𝑛
Calculator: One Sample t Interval
TI 84:
STAT> TESTS> 8:Tinterval… from here adjust
your settings to your problem > Enter
TI 89:
In Statistics/List Editor > 2nd F2 (F7) > interval
(ints) >2:Tinterval… Choose “data” as the data
input method> adjust your settings as needed>
Enter
AP Exam Common Error
• When students are constructing a one-sample
t interval for a population mean based on a
small sample, many students neglect to graph
the sample data and to comment on the
Normal conditions. Simply saying that we
“assume normality” will not earn full credit
when the data are provided. Students must
show a graph and give an appropriate
comment that addresses the normality of the
population
Remember to follow the
A Four-Step Process
• State: what parameter do you want to
estimate, and at what confidence level?
• Plan: Identify the appropriate inference
method. Check conditions.
• Do: If the conditions are met, preform
calculations.
• Conclude: Interpret your interval in the context
of the problem.
• AP expects ALL FOUR, do not skip step 1!
Video Screen Tension
A manufacturer of high resolution video
terminals must control the tension on the mesh
of fine wires that lies behind the surface of the
viewing screen. Too much tension will tear the
mesh, and too little will allow wrinkles. The
tension is measured by an electrical device with
output readings in millivolts (mV) some variation
is inherent in the production process. Here are
the tension readings from a random sample of
20 screens from a single days production.
Video Screen Tension
Construct a 90% confidence interval for the
mean tension 𝜇 of all the screens produced on
this day.
269.5
297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1
Another Example
Biologists study the healing of skin wounds measured
the rate at which new cells closed a cut made in the
skin of an anesthetized newt. Here are the data from a
random sample of 18 newts, measured in micrometers
(millionths of a meter) per hour.
29
27
34
40
22
28
14
35
26
35
12
30
23
18
11
22
23
33
Calculate and interpret a 95% confidence interval for
the mean healing rate 𝜇
•
Using t Procedures Wisely
The stated confidence level of a one-sample t interval for µ is exactly correct
when the population distribution is exactly Normal. No population of real
data is exactly Normal. The usefulness of the t procedures in practice
therefore depends on how strongly they are affected by lack of Normality.
An inference procedure is called robust if the
probability calculations involved in the procedure
remain fairly accurate when a condition for using the
procedures is violated.
Estimating a Population Mean
Definition:
Fortunately, the t procedures are quite robust against non-Normality of
the population except when outliers or strong skewness are present.
Larger samples improve the accuracy of critical values from the t
distributions when the population is not Normal.
Objectives
• Confidence Intervals: when σ is known: The OneSample Z-Interval for a Population Mean
• Confidence Intervals: when σ is Unknown: The tDistribution
•
•
•
•
Calculator
Conditions for Calculating Confidence Intervals
Following 4 Step Process
Robust
Homework
8.3 Homework Worksheet
Continue working on Ch. 8 Reading Guide