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EVEN Homework Lesson 3 ANSWERS
p. 496 #34, 35, 37, 41, 43, 46, 47
34.) (a) Population: undergraduates at a large
university. Parameter: true proportion of students
who would be willing to report cheating.
(b) Random: The sample was an SRS. Normal:
(172)(.11) and (172)(.89) are both greater or equal to
10. Independent: Since this is a large university, it is
most likely less than 10% of the undergraduate student
population
(c) (0.049, 0.172)
(d) We are 99% confident that the interval from 0.049
and 0.172 captures the true proportion of students who
would be willing to report cheating.
46.) 356
Unit 8 Lessons 4 and 5
ESTIMATING A POPULATION MEAN
SECTION 8.3
OBJECTIVES
 Construct and interpret a confidence interval for a




population mean.
Determine the sample size required to obtain a level C
confidence interval for a population mean with a
specified margin of error.
Carry out the steps in constructing a confidence interval
for a population mean: define the parameter; check
conditions; perform calculations; interpret results in
context.
Determine sample statistics from a confidence interval
Understand why each of the three inference conditions –
Random, Normal, and Independent – is important.
One-Sample z Interval for a Population Mean
 Statistic ± (critical value) • (standard deviation of statistic)
 Let’s rewrite this specifically for the population mean:
x  z *

n
 This is from an SRS of size n from a large population that
contains an unknown mean µ and known standard
deviation σ. As long as the Normal and
Independent conditions are met, z* is the critical value
for the standard Normal curve with area C between –z*
and z*.
 This interval is sometimes called a one-sample z
interval for a population mean.
Choosing the Sample Size
 When you start planning a study, you might be
unsure how large of a sample you want, but you
might have an idea of the margin of error you want it
to be/keep below.
This part
 Recall the confidence intervalrepresents
for estimating a
your margin
population’s mean
of error

x  z*
n
 If we want to keep it within a specific margin of
error, we can set up an inequality that would look

like this:
z*
n
 ME
Choosing the Sample Size
z*
This critical value will be
determined by the Confidence
level (given to you in the
problem)

n
 ME
Wait, how do we know the
population’s standard
deviation?
This n is what we want to
solve for!
 We will have to guess the  .
 Get a reasonable value by using a standard deviation
from a pilot study/past experience with similar
studies .
EXAMPLE
 Administrators at your school want to estimate how much time
students spend on homework, on average, during a typical week. They
want to estimate µ at the 90% confidence level with a margin of error of
at most 30 minutes. A pilot study indicated that the standard deviation
of time spent on homework per week is about 154 minutes.
 How many students need to be surveyed to estimate the mean number
of minutes spent on homework per week with 90% confidence and a
margin of error of at most 30 minutes?
154
1.645(154)
1.645
 15
 15
n
nneed
The administrators
1.645
)  15atnleast
1.645
(154)
to(154
survey
286
 n
2
15
645(154) 
 1.students.

 n
285.2  n
15


What if the standard deviation is unknown???
 Most of the times, if you don’t know the population
mean, you’re not going to know the population
standard deviation either!
 Recall that if the sampling distribution of x is close to
Normal, we can find probabilities involving x by
standardizing:
x
z
( / n )
 Since we typically don’t know the standard deviation,
we can use the standard deviation of our sample:
x

(sx / n )
ACTIVITY: BINGO!
 If we are doing inference about a population mean µ, what happens when we use
the sample standard deviation sx to estimate the population standard deviation
σ?
 Before we look into this, let’s look at a population we might know more
information about:
 Let’s start with a Normal population with mean µ=100 and standard deviation
σ = 5.
 We are going to take an SRS of size 4 from the population;
 We are going to compute the sample mean
 Standardize the value of the sample mean using the “known” value σ = 5.
 In your calculator, this is what you will type:
randNorm(100, 5, 4)L1:
1-Var Stats L1:
x
( -100)/(5/√4)
Math  PRB  option 5
STAT  Calc  option 1
STO>  LIST (2nd STAT)  L1
: (ALPHA .)
: (ALPHA .)
VARS  5: Statistics 
option 2
ACTIVITY: BINGO!
 Hit Enter 100 times and say BINGO every time
your z-score is above 3 or below -3 standard
deviations
 Write down the value every time you say BINGO
 According to the 68-95-99.7 rule, about how often
should a “Bingo!” occur?
ACTIVITY: BINGO!
 Now let’s see what happens when you standardize the value
of x using the sample’s standard deviation sx instead
of the “known” σ.
 In your calculator, this is what you will type:



ENTRY (2nd ENTER) … this pulls up what you previously typed
You want to change the standard deviation of 5 to the sample’s standard
deviation sx .
To do this: Using the arrow keys, scroll to the last command and put your
cursor on 5, then hit VARS option 5: Statistics option 3:Sx
 AGAIN, Hit Enter 100 times and say BINGO every time your z-score
is above 3 or below -3 standard deviations
 Write down the value every time you say BINGO.
ACTIVITY: BINGO!
 What did you notice the difference between when
we used the POPULATION’S standard deviation σ
versus when you used the SAMPLE’S standard
deviation sx?
 There were more Bingos the 2nd time – meaning
that there were more z-scores outside of 3 standard
deviations. – way more than 0.03%.
 What does this mean?!?
t distribution
 When we used our sample’s standard deviation,
what happened?

More values were outside 3 standard deviations!
 This is representing a NEW distribution: the t
distribution:
x
t
(sx / n )
 It has a different shape than the standard
Normal curve - still symmetric with single peak,
but with much more area in the tails.

See page 504 in your textbook to see a picture of the shape differences
William S. Gosset (1876 – 1937)
 This distribution was
discovered when William S.
Gosset worked for the
Guinness Brewery – his goal
in life was to make better
beer.
 He used his new t
procedures to find the best
combination of barley and
hops, which got him the job
of head brewer.
 Gosset used the penname
“Student” when publishing
this mathematical work, so
often the t distribution is
referred as the “Student’s t”.
Degrees of Freedom
 The statistic t has the SAME interpretation as any




standardized statistic: it says how far x is from its
mean in standard deviation units.
There is a different t distribution for each sample
size.
Because of this, we need to identify a particular t
distribution by number of degrees of freedom (df )
df = n – 1 (subtract 1 from the sample size)
The notation to identify a t distribution with a
particular degrees of freedom is t n-1
The t distributions, Degrees of Freedom
 Draw an SRS of size n from a large population that
has Normal distribution with mean µ and
standard deviation σ. The statistic:
x
t
(sx / n )
Has the t distribution with degrees of freedom df =
n-1. The statistic will have approximately a tn-1
distribution as long as the sampling distribution x
is close to Normal.
More about the Degrees of Freedom
 The density curves of t distributions are similar in
shape to the standard Normal curve
 The spread of t distributions is a bit greater than
that of the standard Normal distribution.
 As the degrees of freedom increase, the t density
curve approaches the standard Normal curve
more closely.

This is because the larger the sample size, the closer your
sx gets to σ.
Using TABLE B
 Table B shows the critical values t* for the t
distributions.
 The left column represent the degrees of freedom
 Common confidence levels are given at the bottom
of the table.
 By looking down any column, you can check that
the t critical values approach the Normal critical
values z* as the degrees of freedom increase.
EXAMPLE - Using TABLE B
 Suppose you wanted to construct a 90% confidence
interval for the mean µ of a Normal population
based on an SRS of size 10. What critical value t*
should you use?
 Using the line for df = 10 -1 = 9 and the column
with a tail probability of .05 (10%/2), the desired
critical value is t* = 1.833.
YOUR TURN
 Use Table B to find the critical value t* that you
would use for confidence interval for a population
mean µ for a 98% confidence interval based on
n=22 observations.
 Using the line for df = 22 -1 = 21 and the column
with a tail probability of .01 (2%/2), the desired
critical value is t* = 2.518.
In your calculator
 For TI-84:
 DISTRIBUTION (2nd VARS), option 4: invT(
 In the parentheses, type in the area to the left of the desired
critical value, the degrees of freedom.
 From the last problem, you would type in
invT(.01,21)
CONTRUCTING A CONFIDENCE INTERVAL FOR µ
 First, check your conditions!
 RANDOM: The data comes from a random sample of
size n from the population of interest of a randomized
experiment.
 NORMAL: The population has a Normal distribution of
the sample size is large (n≥30).
 INDEPENDENT: 10% rule: The sample size is no more
than 1/10 of the population.
The One-Sample t Interval for a Population Mean
 Estimate ± (critical value)(standard deviation of statistic)
sx
x t*
n
 Remember, technically we call
sx
n
the
standard error of the sample mean – which
describes how far x will be from µ on average, in
repeated SRSs of size n.
EXAMPLE
 As part of their final project in AP Statistics, Christina and
Rachel randomly selected 18 rolls of generic brand of toilet
paper to measure how well this brand could absorb water.
To do this, they poured ¼ cup of water onto a hard surface
and counted how many squares it took to completely
absorb the water. Here are the results from their 18 rolls:
29
20
25
29
21
24
27
25
24
29
24
27
28
21
25
26
22
23
 Construct and interpret a 99% confidence interval for µ =
the mean number of squares of generic toilet paper needed
to absorb ¼ cup of water.
EXAMPLE
 STATE: We want to estimate µ = the mean of number of
squares of generic toilet paper needed to absorb ¼ cup of
water with 99% confidence.
EXAMPLE
 PLAN: We will construct a one-sample t interval, provided
the following conditions are met:



RANDOM: The students selected the rolls of generic toilet paper at
random.
NORMAL: Since the sample size is small (n=18), and we aren’t told that
the population is Normally distributed, we need to check whether it is
reasonable to believe that the population has Normal distribution.
(draw a dotplot to see that there are no outliers and it roughly follows a
Normal distribution)
INDEPENDENT: Since we are sampling without replacement, we must
check the 10% condition. It is reasonable to believe that there are at
least 10(18) = 180 rolls of generic toilet paper.
EXAMPLE
x  24.94
and the sample standard deviation is s x  2.86 .
 DO: The sample mean for these data is
Since there are 18 – 1 = 17 degrees of freedom and we want
99% confidence, we will use a critical value of t* = 2.898
(from Table B).
sx
2.86
x t*
 24.94  2.898
n
18
 24.94 1.95
 (22.99,26.89)
EXAMPLE
 CONCLUDE: We are 99% confident that the interval from
22.99 squares to 26.89 squares captures the true mean
number of squares of generic toilet paper needed to absorb
¼ cup of water.
Using t Procedures Wisely
 What happens when a condition for using a t procedure is
violated??

If your result is still pretty accurate, then we call that procedure
ROBUST.
 According to the Merriam Webster Dictionary, a definition for
robust “is having or showing vigor, strength, or firmness”….or
another one is “capable of performing without failure under a
wide range of conditions “
 According to Statistics, an inference procedure is robust if the
probability calculations involved in that procedure
remain fairly accurate when a condition for using the
procedure is violated.
Using t Procedures Wisely
 Good news for us!

The t procedures are quite robust against non-Normality of the
population EXCEPT when outliers or strong skewness is
present.
 Larger samples improve the accuracy of critical values from
the t distributions when the population is not Normal
because:


x
The sampling distribution of
is close to Normal if the
sample size is large enough (CLT!)
As the sample n grows, the sample standard deviation sx will
be an accurate estimate of σ whether or not the population has
Normal distribution.
Using t Procedures Wisely
 Always plot the data to check if it’s roughly Normal – but
more importantly, that there’s no outliers or major
skewness.
 And it’s definitely MORE important to make sure that it
comes from RANDOM data, rather than being picky about
how Normal it looks.
 Follow these procedures to help using sample size n:



If n<15: Use t procedures if the data appears close to Normal
(roughly symmetric, single peak, no outliers). If the data are
clearly skewed or there are outliers, DO NOT USE t.
If n≥15: The t procedures can be used except in the presence
of outliers or strong skewness.
Large samples (n≥30): The t procedures can be used even for
clearly skewed distributions when the sample is large!
EXAMPLE – can we use t?
 Don’t use t for these situations either:


If your sample data gives a biased estimate for some reason
And if you have all the data for all your population of interest
 Determine whether we can safely use a one-sample t
interval to estimate the population mean in each of the
following settings:

A.) Below is a histogram of the total number of students in class and
their heights.
NO. We have data for the entire number of students,
so we do NOT need inference. Remember, you only
use inference when you are ESTIMATING something
about the population (because that proportion or
mean is unknown!)
EXAMPLE – can we use t?
 Determine whether we can safely use a one-sample t
interval to estimate the population mean in each of the
following settings:

B.) The dot plot below shows expenditure costs for 6 of the employees at
a company
NO. This is a sample of 6 (less than 15!), so we can
only use t procedures if the data appears close to
Normal. It does not appear that way.
EXAMPLE – can we use t?
 Determine whether we can safely use a one-sample t
interval to estimate the population mean in each of the
following settings:

C.) The boxplot below shows the SAT Math scores for a random sample
of 20 students at your high school.
YES. The sample size is 20 (greater than 15!)
Although slightly skewed, there doesn’t seem to be
strong skewness or the presence of any outliers.
For more examples with these, see p. 513 in book
In your calculator:
 One-sample t intervals for µ on your calculator:


STAT  TESTS  option 8:Tinterval…
If the problem gives you actual data:


(Make sure your data is in L1) Choose Data option, List:L1, Freq:1, Clevel: (type in your confidence level out of percent form), Highlight
Calculate and press ENTER
If the problem just gives you summary statistics:

Choose Stats option, and type in your sample mean, sample standard
deviation, sample size n, and C-level (out of percent form), Highlight
Calculate and press ENTER
 Calculators aren’t always right! Be careful because sometimes
there are called “parallel solutions” where your calculator and
your calculations might give you two different answers. That’s
why you always need to show your work and explanation on
exams!
EXAMPLE – for practice!
 The principal at a large high school claims that students
spend at least 10 hours per week doing homework, on
average. To investigate this claim, an AP Statistics class
selected a random sample of 250 students from their school
and asked them how long they spent doing homework
during the last week. The sample mean was 10.2 hours and
the sample standard deviation was 4.2 hours.
 (a) Construct and interpret a 95% confidence interval for
the mean time that students at this school spent doing
homework in the last week.
 (b) Based on your interval in part (a), what can you
conclude about the principal’s claim?
EXAMPLE
 STATE: We want to estimate µ = the mean time spent
doing homework in the last week for students at this school
with 95% confidence.
EXAMPLE
 PLAN: We will construct a one-sample t interval, provided
the following conditions are met:



RANDOM: The students were randomly selected.
NORMAL: We are not told if the population is normal. However, since
the sample size is large (n=250), we are safe using t procedures.
INDEPENDENT: Since we are sampling without replacement, we must
check the 10% condition. It is reasonable to believe that there are at
least 10(250) = 2500 students since it is a large high school.
EXAMPLE
x  10.2
and the sample standard deviation is s x  4.2 .
 DO: The sample mean for these data is
Since there are 250 – 1 = 249 degrees of freedom and we
want 95% confidence, we will use a critical value of t* =
1.984 (from Table B).
sx
4.2
x t*
 10.2  1.984
n
250
 10.2  0.53
 (9.67,10.73)
EXAMPLE
 CONCLUDE: We are 95% confident that the interval from
9.67 hours to 10.73 hours captures the true mean of hours
that students at this school spent doing homework in the
last week.
 (b) Since the interval of plausible values for µ includes
values less than 10, the interval does not provide
convincing evidence to support the principal’s claim that
students spend at least 10 hours on homework per week, on
average.
Lesson 4: Homework problems
 Read textbook pages: p. 499-511
 Complete exercises:
 p. 498 #49-52
 p. 518 #55, 57, 59, 60, 63
 Check answers to odd problems
Lesson 5: Homework problems
 Read textbook pages: p. 511-517
 Complete exercises: p. 519 #65-67, 71, 73-78
 Check answers to odd problems