Lesson 8.1 - Confidence Intervals: The Basics
Remember from Unit 7:
 Sample mean and sample proportion
 Population parameters vs. sample statistics
̅ has
 The sampling distribution of 𝒙
o mean 𝝁
o standard deviation 𝝈⁄√𝒏
o an approximately Normal shape if the parent population is normal or 𝒏 ≥ 𝟑𝟎 CLT
̂ has
 The sampling distribution of 𝒑
o mean 𝒑
o standard deviation √𝒑(𝟏 − 𝒑)⁄𝒏
o an approximately Normal shape if 𝒏𝒑 ≥ 𝟏𝟎 and 𝒏(𝟏 − 𝒑) ≥ 𝟏𝟎 Large Count Condition
 Normal probability calculations
*Section 8.1 is all about putting these ↑ ideas together to produce a useful statistical procedure.
Introduction
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How long does the new iPad battery last?
What proportion of college undergraduates have engaged in binge drinking?
How much does the weight of a quarter-pound hamburger at a fast food restaurant vary after cooking?
It wouldn’t be practical to determine the lifetime of every iPad battery, to ask all college undergraduates about their
drinking habits, or to weigh every burger after cooking. Instead, we choose a sample of individuals (batteries, college
students, burgers) to represent the population and collect data from those individuals. Our goal in each case is to use a
sample statistic to estimate an unknown population parameter. If we randomly select the sample, we should be able to
generalize our results to the population of interest.
But we cannot be certain that our conclusions are correct – a different sample would probably yield a different estimate.
Statistical inference uses the language of probability to express the strength of our conclusions. Probability allows us to
take chance variation due to random selection or random assignment into account.
We are making a big transition as we move from Unit 7 to Unit 8. In Unit 7, we assumed that we knew the true value of a
parameter and then asked questions about the distribution of the statistic used to estimate that parameter. For
example, we asked questions like “Assuming 𝜇 = 10, what is the probability that 𝑥̅ is greater than 12?” In Unit 8, we no
longer pretend to know the true value of the parameter. We start with a more realistic situation where we know only
the value of a sample statistic. Then we use the value of the statistic to estimate the value of the population parameter.
For example, we ask questions like “If 𝑥̅ is 123.8, what are the plausible values of 𝜇?”
**What will be the BIGGEST difference in the problems that we will complete in Unit 8, compared to the way they were
given to us in Unit 7??**
Activity: The mystery mean
Let’s try to estimate the mystery mean- it is the population mean 𝜇 (represented by M) that Mrs. Lakey stored in her
calculator before class.
_________________________________________________ = ________________
↑ tells the calculator to choose a SRS of 16 observations from a Normal population with mean M and standard deviation
of 20 and then compute the mean of those 16 sample values. Will the sample mean shown above be EQUAL to the
mystery mean M? Why or why not?
Your group must determine an interval of reasonable values for the population mean µ with the following intervals:
Within 100:
How confident are you??
Within 50:
How confident are you??
Within 10:
How confident are you??
Exactly:
How confident are you??
Confidence Intervals: The Basics
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If you had to give one number to estimate an unknown population parameter, what would it be?
o
To estimate a population mean _____, you would probably use _____.
o
To estimate a population proportion _____, you would probably use _____.
o
In both cases, you would be providing a point estimate of the parameter of interest.
 Define point estimator and point estimate
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An ideal point estimator will have ___________ bias and ____________ variability.
The Idea of a Confidence Interval
Recall the “Mystery mean” Activity. Is the value of the population mean exactly what our estimate was? Probably not.
However, since our sample mean is _______________, we could guess that µ is “somewhere” around this value. How
can we get closer to the true value of µ?
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To estimate the Mystery Mean 𝜇, we can use 𝑥̅ = ___________ as a point estimate. We don’t expect 𝜇 to be exactly
equal to 𝑥̅ so we need to say just how accurate we think our estimate really is. (Basically, how confident we are)
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In repeated samples, the values of 𝑥̅ follows a Normal distribution with
mean of ____________ and standard deviation of ________________.
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The 68-95-99.7 Rule tells us:
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In ______ of all samples of this size, 𝑥̅ will be within______ of 𝜇
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In ______ of all samples of this size, 𝑥̅ will be within ______ of 𝜇
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In ______ of all samples of this size, 𝑥̅ will be within ______ of 𝜇
 If we estimate that µ lies somewhere in the interval _______________ to ____________, we’d be calculating an
interval using a method that CAPTURES the mean 𝜇 in about _________ of all possible samples of this size.
 If we estimate that µ lies somewhere in the interval _______________ to ____________, we’d be calculating an
interval using a method that CAPTURES the mean 𝜇 in about _________ of all possible samples of this size.
 If we estimate that µ lies somewhere in the interval _______________ to ____________, we’d be calculating an
interval using a method that CAPTURES the mean 𝜇 in about _________ of all possible samples of this size.
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The big idea: The sampling distribution of 𝑥̅ tells us how close to 𝜇 the sample mean 𝑥̅ is likely to be. All confidence
intervals we construct will have a form similar to this:
 A C% confidence interval gives an interval of ____________________ values for a parameter.
 The difference between the point estimate and the true parameter value will be less than the margin of error in C%
of all samples.
 The Confidence Level, C, gives the overall success rate of the method for calculating the confidence interval. That is,
in C% of all possible samples, the method would produce an interval that ____________________ the __________
parameter value.
Interpreting Confidence Levels and Intervals
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The confidence level is the overall ___________________ rate if the method is used many times. The sample mean
will vary from sample to sample, but when we use the method estimate ± margin of error to get an interval based
on each sample, C% of these intervals capture the unknown population mean 𝜇.
 Interpreting Confidence Intervals
To interpret a C% confidence interval for an unknown
parameter, say, “We are _______ confident that the
interval from _______ to _______ captures the actual
value of the [population parameter in context].”
 Interpreting Confidence Levels
To say that we are 95% confident is shorthand for
“If we take __________ samples of the same size from
this population, about _________ of them will result in
an interval that captures the actual parameter value.”
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CAUTION: The confidence level _____________ ___________ tell us the _______________________ that a
particular confidence interval captures the population parameter.
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Instead, the confidence interval gives us a set of ______________________ values for the parameter.
Example: Do you use Twitter? The Pew Internet and American Life Project asked a random sample of 2253 U.S. adults,
“Do you ever… use Twitter or another service to share updates about yourself or to see updates about others?” Of the
sample, 19% said “Yes.” According to Pew, the resulting 95% confidence interval is (0.167, 0.213).
Interpret the confidence interval and the confidence level.
Confidence interval:
Confidence level:
Constructing Confidence Intervals
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Why settle for 95% confidence when estimating a parameter?
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When we calculated a 95% confidence interval for the mystery mean µ, we started with
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Our estimate came from the sample statistic 𝑥̅ .
Since the sampling distribution of 𝑥̅ is approximately
Normal, about ______ of the values of 𝑥̅ will lie within
2 standard deviations (________) of the mystery mean µ.
That is, our 95% confidence interval could be written as:
 This leads to a more general formula for confidence intervals:
Calculating a Confidence Interval
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The confidence interval for estimating a population parameter has the form
where the statistic we use is the point estimator for the parameter.
Properties of Confidence Intervals:
• The “margin of error” is the
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The user chooses the ___________________________ level, and the margin of error follows from this choice.
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The _____________________value depends on the confidence level and the sampling distribution of the statistic.
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_________________ confidence requires a ________________ critical value.
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The standard deviation of the ______________________ depends on the sample size _____.
Using Confidence Intervals Wisely
Here are two important cautions to keep in mind when constructing and interpreting confidence intervals.
 Our method of calculation assumes that the data come from an SRS of size 𝑛 from the population of interest.
 The margin of error in a confidence interval covers only chance variation due to random sampling or random
assignment.
Example: Explaining confidence
The admissions director from Big City University found that (107.8, 116.2) is a 95% confidence interval for the mean IQ
score of all freshmen. Discuss whether each of the following explanations is correct.
(a) There is a 95% probability that the interval from 107.8 to 116.2 contains 𝜇.
(b) There is a 95% chance that the interval (107.8, 116.2) contains 𝑥̅ .
(c) This interval was constructed using a method that produces intervals that capture the true mean in 95% of all
possible samples.
(d) If we take many samples, about 95% of them will contain the interval (107.8, 116.2).
(e) The probability that the interval (107.8, 116.2) captures 𝜇 is either 0 or 1, but we don’t know which.