Chapter 8 Sections 1 and 2

Reminders

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Confidence Intervals

►A confidence interval provides a range of values, derived from a
sample, that is likely to contain the population parameter (e.g., mean,
proportion) with a specified level of confidence.

►Informs Decision-Making

►if a hypothesized population mean lies outside a 95% confidence interval, it
suggests that the hypothesized mean may not be an accurate reflection of the
population.

►Still following a normal distribution

Confidence Interval For the Population
Mean When σ Is Known

►A confidence interval is the mean of your estimate plus and minus the
variation in that estimate. This is the range of values you expect your
estimate to fall between if you redo your test, within a certain level of
confidence.

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For a z statistic : most common values
are shown in this table:

Confidence level

90%

95%

99%

alpha

0.1

0.05

0.01

alpha/2 CI

0.05

0.025

0.005

z statistic

1.645

1.96

2.576

The confidence level and how it relates to the alpha and Z- score.

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Ex: A sample of 25
cereal boxes yields a
mean weight of 1.02
pounds of cereal per
box.

Construct the 95%
confidence interval
for the mean weight
of all cereal boxes.

Assume that the
weight is normally
distributed with a
population standard
deviation of 0.03
pound.

►Note: 95% confidence does not imply
there is a 95% probability the interval captures the population mean.

►It can imply that if numerous samples of
size n are taken, 95% of the intervals would contain the population mean.

Ex: A sample of 25
cereal boxes yields a
mean weight of 1.02
pounds of cereal per
box.

Construct the 95%
confidence interval
for the mean weight
of all cereal boxes.

Assume that the
weight is normally
distributed with a
population standard
deviation of 0.03
pound.

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Ex: A sample of 25
cereal boxes yields a
mean weight of 1.02
pounds of cereal per
box.

Construct the 95%
confidence interval
for the mean weight
of all cereal boxes.

Assume that the
weight is normally
distributed with a
population standard
deviation of 0.03
pound.

►

How Sample Size Effects the CI Width

►CI width decreases(becomes more narrow):

►as the sample size increases, the standard error decreases and the margin of
error decreases

►These will cause the CI width to decrease/become more narrow

►CI width increases:

►As the sample size decreases, the standard error increases and the margin of
error increases

►These will cause the CI width to increase

How the confidence level (CL) effects
the size/width of the CI

►When you increase the CL, going from 95% to 99% as an example, the Z
score increases, the margin of error increases making the CI width to
increase.

►When you decrease the CL, going from 99% to 90% as an example, the
Z score decreases, the margin of error decreases making the CI width to
decrease.

How the Standard Deviation Effects
the Confidence Width

►For a given confidence level and sample size, the larger the population
standard deviation, the wider the confidence interval.

►If the standard deviation increases, then the confidence interval
becomes wider.

How the width changes with standard
deviation

►

How the width changes with the
sample size

►

How the width changes with the
confidence level

►

Ex: IQ tests are
approximately
normally
distributed with a
standard deviation
of 15. A sample of
22 employees
gives a sample
mean of 106.

►Compute 90% and 99% confidence
intervals for the average IQ at this firm.

►Use the results to infer if the mean IQ in
this firm is significantly different from the
national average of 100.

Ex: IQ tests are
approximately normally
distributed with a
standard deviation of
15. A sample of 22
employees gives a
sample mean of 106.

Compute 90%
confidence intervals for
the average IQ at this
firm.

► 90% confidence interval:

Ex: IQ tests are
approximately
normally distributed
with a standard
deviation of 15. A
sample of 22
employees gives a
sample mean of 106

Compute 99%
confidence intervals
for the average IQ
at this firm.

99% confidence interval:

Chapter 8

SECTION 2: CONFIDENCE INTERVAL FOR THE
POPULATION MEAN WHEN POPULATION
STANDARD DEVIATION IS UNKNOWN

Confidence Interval for the Population
Mean When σ Is Unknown

►

Confidence Interval for the Population
Mean When σ Is Unknown

►

t distribution

►Family of distributions similar to z

►Bell-shaped and symmetric around zero

►Slightly broader tails than the z distribution

►Identified by the degrees of freedom, df, that determine the broadness
of the tails

►Fewer df, broader the tails

►df increases, t becomes more like z

Degrees of Freedom

►

T-Table

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Confidence Interval

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Confidence Interval

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Ex: A car manufacturer
advertises that its new
“ultra-green” car obtains
an average of 100 MPG.
For a sample of 25 cars,
each car is driven the
same distance in
identical conditions in
order to obtain the car’s
MPG.

►

Ex: A car manufacturer
advertises that its new
“ultra-green” car obtains
an average of 100 MPG.
For a sample of 25 cars,
each car is driven the
same distance in identical
conditions in order to
obtain the car’s MPG.

Examples

At 99% confidence, what is the margin of
error?

Many U.S. consumers are
using debit cards to avoid
accruing debt Based on a
sample of 100 U.S.
consumers, a researcher
finds that the average
amount spent annually on
a debit card is $7,790.
Assume that the
population standard
deviation is $500.

Construct the 90% confidence interval for the average
number of customers served on weekdays.

The manager of The
Cheesecake Factory in
Boston reports that on six
randomly selected
weekdays, the number of
customers served was 120,
130, 100, 205, 185, and
220. She believes that the
number of customers
served on weekdays
follows a normal
distribution

►Use the Z-Table when the population standard deviation
is known.

►Use the T-Table when the population standard deviation
is unknown.

►Review the extra material in Moodle.

►Start chapter 8 Connect Assignments.

Final Notes