S - 8.1 Confidence Interval

Lesson 8.1
The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

8

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

After this lesson you should be able to:

Interpret a confidence interval.

Determine the point estimate and margin of error
from a confidence interval.

Use confidence intervals to make decisions.

Learning Targets

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

How long does a battery last on
the newest iPhone, on average?

What proportion of college undergraduates
attended all of their classes last week?

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 iPhone battery, to ask all
undergraduates about their attendance, or to weigh every burger after cooking.

Instead, we choose a random sample of individuals (batteries, undergraduates,
burgers) to represent the population and collect data from those individuals.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Chapter 8 begins the formal study of statistical inference—

using information from a sample

to draw conclusions about a population parameter such as p or µ.

This is an important transition from Chapter 7, where you were given
information about a population and were asked questions about the
distribution of a sample statistic such as or ̅𝑥.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In this activity, each team of three or four students will use data from a random
sample to estimate the mean µ of a normally distributed population with a
standard deviation of σ = 20.

1. Suppose your teacher has selected a “Mystery Mean” value µ and

stored it as “M” in their calculator.

2. The following command was executed on their calculator
mean(randNorm(mean: M, SD: 20, n: 16))

This tells the calculator to choose an SRS of 16 values from a normal
population with mean µ and standard deviation 20, and then compute the
mean ̅𝑥 of those 16 sample values.

Activity: What’s the mystery mean?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In this activity, each team of three or four students will use data from a random
sample to estimate the mean µ of a normally distributed population with a
standard deviation of σ = 20.

The result was 240.79. This tells us the calculator chose a SRS of 16 observations

from a Normal population with mean M and standard deviation 20. The resulting sample mean of those 16 values was 240.79.

3. Is the sample mean shown likely to be equal to the population mean µ?

Explain your answer.

Activity: What’s the mystery mean?

Multiple Choice

Is the sample mean likely to be equal to the population mean?

• Yes

• No

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

4. How much will the sample mean typically vary from the population

mean in SRSs of size 16 from a population with σ = 20? Hint: You’ll need
to use what you learned in Chapter 7.

5. Use your answer from Question 4 to create an interval of plausible

values for the population mean µ.

6. How confident are you that your interval of plausible values contains the

population mean? Explain your reasoning.

7. Share your team’s results with the class.

Activity: What’s the mystery mean?

Open Ended

Share your interval of plausible values.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

This activity illustrates one of the most common tasks of a statistician—
using a sample statistic to estimate the value of a population parameter.

When Mr. Girard’s class did the Mystery Mean
activity, they obtained a sample mean of ̅𝑥 = 23.8
and used this estimate to create an interval
of plausible values from 13.8 to 33.8.

When the estimate of a parameter is reported as an interval of
plausible values, it is called a confidence interval.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

“Plausible” means that we shouldn’t be surprised if any one of the
values in the interval is equal to the value of the parameter.

The Idea of a Confidence Interval

DEFINITION

Confidence interval

A confidence interval gives an interval of plausible values for a
parameter based on sample data.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

We use an interval of plausible values rather than a single-value estimate to
increase our confidence that we have a correct value for the parameter.

Of course, there is a trade-off between the amount of confidence we have and
how much information the interval provides.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Confidence intervals are constructed so that we know how much
confidence we have that the interval successfully captures the parameter.

The most common confidence level is 95%.

The Idea of a Confidence Interval

DEFINITION

Confidence level

The confidence level C gives the long-run success rate of
confidence intervals calculated with C% confidence.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The Gallup polling organization recently asked a random
sample of U.S. adults about the most important financial
problem facing their family today.

There were many different options, with “health care
costs” leading the way with 14% of responses.

Based on this sample, the 95% confidence interval for the population proportion
is

That is, we are 95% confident that the interval from 0.10 to 0.18 captures the
proportion of all U.S. adults who would say that health care costs are the biggest
financial problem facing their family today.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The Idea of a Confidence Interval

How to Interpret a Confidence Interval

To interpret a C% confidence interval for an unknown
parameter, say,

“We are C% confident that the interval from ______ to ______
captures the [parameter in context].”

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

When interpreting a confidence interval, make sure that
you are describing the parameter and not the statistic.

It’s wrong to say that we are 95% confident that the interval from 0.10 to 0.18
captures the proportion of U.S. adults who said health care costs were the biggest
financial problem facing their family today.

Recall from Chapter 7 that parameter descriptions should include the word all,
true, or population.

The Idea of a Confidence Interval

The “proportion who said health
care” is the sample proportion,
which is known to be 0.14.

The interval gives plausible values for
the proportion of all U.S. adults
who would say health care if asked.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

A large company is concerned that many of its
employees are in poor physical condition, which
can result in decreased productivity. To estimate
how many steps employees take per day, on
average, the company provides a fitness tracker
to 50 randomly selected employees to use for
one month. After collecting the data, the
company statistician reports a 95% confidence
interval of 4547 steps to 8473 steps. Interpret
this confidence interval.

Example: How many steps?

Multiple Choice

A large company is concerned that many of its

employees are in poor physical condition, which

can result in decreased productivity. To estimate

how many steps employees take per day, on

average, the company provides a fitness tracker

to 50 randomly selected employees to use for

one month. After collecting the data, the

company statistician reports a 95% confidence

interval of 4547 steps to 8473 steps. Interpret

this confidence interval.

• We are 95% confident that the interval from 4547 steps to 8473
  steps captures the mean number of steps per day for all employees at
  this company.

• We are 95% confident that the interval from 4547 steps to 8473
  steps captures the number of steps per day for the 50 randomly selected employees at
  this company.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

A large company is concerned that many of its employees are in poor physical
condition, which can result in decreased productivity. To estimate how many steps
employees take per day, on average, the company provides a fitness tracker to 50
randomly selected employees to use for one month. After collecting the data, the
company statistician reports a 95% confidence interval of 4547 steps to 8473
steps. Interpret this confidence interval.

SOLUTION:

We are 95% confident that the interval from 4547 steps to 8473
steps captures the mean number of steps per day for all employees at
this company.

Example: How many steps?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Sometimes, confidence intervals are reported in the form

(lower boundary, upper boundary)

For example, the confidence interval in the
preceding example could be reported as
(4547, 8473) instead of “4547 to 8473.”

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

To create an interval of plausible values for a parameter, we need two
components:

a point estimate to use as the midpoint of the interval

and

a margin of error to account for sampling variability.

The Idea of a Confidence Interval

DEFINITION

Point estimate, Margin of error

A point estimate is a single-value estimate of a population
parameter.

The margin of error of an estimate describes how far, at most, we
expect the point estimate to vary from the true population mean.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Unfortunately, the point estimate is unlikely
to be correct because of the variability
introduced by random sampling.

We include the margin of error to
account for this sampling variability.

In a C% confidence interval, the distance between the point estimate and the true
parameter value will be less than the margin of error in C% of all samples.

The Idea of a Confidence Interval

(“Dotplot” of the simulated sampling distribution of ̂𝑝
from the Penny for Your Thoughts activity in Chapter 7)

The point estimate is our best guess for the value of the parameter based on data from a sample

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The structure of a confidence interval is

point estimate ± margin of error

We can visualize a C% confidence interval like this:

The Idea of a Confidence Interval

[ ]

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Earlier, you learned that the 95% confidence interval for the proportion of
all U.S. adults who would say that health care costs are the biggest
financial problem facing their family today is 0.10 to 0.18.

This interval could also be expressed as

0.14 ± 0.04

Confidence intervals reported in the media are often presented in this
format.

The Idea of a Confidence Interval

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the previous example, you learned that the 95% confidence interval for
the mean number of steps per day for all employees at the large company
is 4547 to 8473. Calculate the point estimate and margin of error used to
create this confidence interval.

Example: How many more steps?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the previous example, you learned that the 95% confidence interval for the
mean number of steps per day for all employees at the large company is 4547 to
8473. Calculate the point estimate and margin of error used to create this
confidence interval.

SOLUTION:

Point estimate = 4547+8473

2

= 6510 steps

Margin of error = 8473 – 6510

= 1963 steps

Example: How many more steps?

Multiple Choice

According to the U.S. Census Bureau, the 90% confidence interval for Alabama's median household income in 2018 is $49,078 to $50,644. Calculate the point estimate.

• $49906

• $49078

• $50644

• $49861

Multiple Choice

According to the U.S. Census Bureau, the 90% confidence interval for Alabama's median household income in 2018 is $49,078 to $50,644. Calculate the margin of error.

• 783

• 1655

• 1765

• 1543

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Besides estimating the value of a population parameter, we can also
use confidence intervals to make decisions about a parameter.

In the preceding examples, you learned that the 95% confidence interval for the
mean number of steps per day for all employees at the large company is 4547 to
8473. Recent health guidelines suggest that people aim for at least 10,000 steps
per day. Based on this interval, is there convincing evidence that the employees of
this company are not meeting the guideline, on average? Explain your reasoning.

Example: Do they need to step it up?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Besides estimating the value of a population parameter, we can also
use confidence intervals to make decisions about a parameter.

In the preceding examples, you learned that the 95% confidence interval for the
mean number of steps per day for all employees at the large company is 4547 to
8473. Recent health guidelines suggest that people aim for at least 10,000 steps
per day. Based on this interval, is there convincing evidence that the employees of
this company are not meeting the guideline, on average? Explain your reasoning.

SOLUTION:

Because all of the plausible values for the population mean are less than
10,000, there is convincing evidence that the employees of this
company are taking fewer than 10,000 steps per day, on average.

Example: Do they need to step it up?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

What if the health guidelines suggested that people aim for at least 8000 steps
per day?

Because there are values in the confidence interval that are 8000 and greater, it is plausible that the employees at the company are meeting (or exceeding) this goal, on average.

In other words, there wouldn’t be convincing evidence that the employees of this
company are taking fewer than 8000 steps, on average.

The Idea of a Confidence Interval

[

]

4547

8473

4000

5000

6000

7000

8000

9000

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

A recent survey asked a random sample of U.S.
adults if they had ever traveled internationally.
Based on the sample, the 95% confidence interval
for the true proportion of U.S. adults who have
traveled internationally is 0.436 to 0.478.

1. Interpret the confidence interval.

2. Calculate the point estimate and margin of

error used to create this confidence interval.

3. Is it plausible that a majority of U.S. adults have

traveled internationally? Explain your reasoning.

LESSON APP 8.1

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

What did you learn?

Interpret a confidence interval.

Determine the point estimate and margin of error
from a confidence interval.

Use confidence intervals to make decisions.

Learning Targets

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

8.1 - 2, 3, 5, 14, 18

Homework