IXL - Find confidence intervals for population proportions (Algebra 2 practice)

The president of an audio technology company is alarmed by the declining sales of the company's MP3 players. To better understand market trends, company employees asked 50 randomly chosen teenagers whether they currently own MP3 players.

Owns an mp3 player?	Teenagers
yes	8
no	42

Find a

90%

confidence interval for

the proportion of teenagers who do not own MP3 players.

Round your answers to the nearest thousandth.

ref_doc_title.

Learn with an example

question

Isaac is conducting a study in his cognitive psychology lab about people's ability to remember rhythms. He played a short rhythm to 125 randomly chosen people. One minute later, he asked them to repeat it by clapping. He counted the number of people who could repeat the rhythm correctly.

Repeated rhythm correctly	People
yes	81
no	44

Find a

95%

confidence interval for

the proportion of people who can repeat the rhythm after one minute.

Round your answers to the nearest thousandth.

key idea

You can use a random sample to calculate a

confidence interval for the population proportion. The population proportion is the proportion of items in a population with a specified characteristic.

A confidence interval may or may not contain the population proportion. A

confidence interval will contain the population proportion for

of random samples whose elements were chosen randomly and independently.

The

confidence interval for the population proportion,

has endpoints

where:

p
is the sample
proportion,
the proportion of items in the sample with the specified characteristic.
q
=
1
–
p
n
is the sample size.
z
is the critical value for the confidence level
c
%
.
It is the
z
-value
for which the interval from
–
z
to
z
is the middle
c
%
of the standard normal distribution.
c
%
–
z
z
Here are some common critical values:
Confidence level,
c
%
Critical value,
z
90%
1.645
95%
1.960
99%
2.576

The formula for the endpoints of a confidence interval can be used when all of the following inequalities hold:

n
≥
30
n
p
≥
5
n
q
≥
5

solution

The population proportion,

is the proportion of people who can repeat the rhythm after one minute.

To find a

95%

confidence interval for the population proportion, start by listing the given information:

The sample size is
n
=
125
.
In this sample, 81 could repeat the rhythm correctly.
Repeated rhythm correctly People
yes 81
no 44
The critical value for a
95%
confidence level is
z
=
1.960
.

Find

and

Use these values to find sample proportion,

and its complement,

125

0.648

–

125

0.352

Verify that the formula for the endpoints of a confidence interval can be used.

n
=
125
≥
30
n
p
=
125
81
125
=
81
≥
5
n
q
=
125
44
125
=
44
≥
5

Notice that

is the number of people in the sample who could repeat the rhythm, and

is the number who could not.

Since all three inequalities are true, the formula for the endpoints of a confidence interval can be used.

Find the endpoints of a confidence interval.

The sample size is

125

0.648

0.352

and the critical value for a

95%

confidence interval is

1.960

Use these values to find the endpoints of a confidence level for the population proportion.

0.648

1.96

0.648

0.352

125

0.648

0.083725…

The left endpoint is:

0.648

–

0.083725…

0.564274…

Round to the nearest thousandth

≈

0.564

Round to the nearest thousandth

The right endpoint is:

0.648

0.083725…

0.731725…

Round to the nearest thousandth

≈

0.732

Round to the nearest thousandth

State the confidence interval.

So, a

95%

confidence interval for the proportion of people who can repeat the rhythm after one minute is:

0.564

0.732

Looking for more? Try this:

$Confetti$

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RR.5Find confidence intervals for population means

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Confidence level, c %	Critical value, z
90%	1.645
95%	1.960
99%	2.576