Transcript 11.E Confidence Intervals and Hypothesis Testing

Confidence Intervals
In a recent Gallup Poll, 1514 teens polled that had taken a summer job saved an
average of $1033 over the summer. The poll had the following disclaimer: “For results
based on the total sample of national teens, one can say with 95% confidence that the
margin of sampling error is ±$31.”
Inferential statistics are used to draw conclusions or statistical inferences about a
population using a sample.
For example, the sample mean of $1033 can be used to estimate the population mean.
A confidence interval is an estimate of a population parameter stated as a range with a
specific degree of certainty.
Typically, statisticians use 90%, 95%, and 99% confidence intervals, but any percentage can
be considered. In the opening example, we are 95% confident that the population mean is
within $31 of $1033 (from $1002 to $1064)
• The margin of error for a population mean is: M  z 

n
where z is the z-value that corresponds to a particular confidence level, σ is the
standard deviation, and n is the sample size.
confidence level:
90%
95%
99%
z-value:
1.645
1.96
2.576
• The confidence interval for a population mean is: CI  x  M
where
x is the sample mean and M is the margin of error.
1. A sample of 400 adults was asked the average time spent watching television each
weeknight. The mean time was 68.4 minutes with a standard deviation of 20.9
minutes.
a. Determine a 90% confidence interval for the population mean.
b. Determine a 95% confidence interval for the population mean.
c. Determine a 99% confidence interval for the population mean.
a. M  z 
z-value for

M

z

b.
90%
n
confidence

n
M  1 . 645 
20 . 9 level
M  1 . 96 
c.
M  z

n
20 . 9
M  2 . 576 
400
400
20 . 9
400
M  1 . 72
M  2 . 05
M  2 . 69
CI  x  M
CI  x  M
CI  x  M
CI  68 . 4  2 . 05
CI = 68.4 ± 2.69
We are 95% confident the
population mean (μ) is
between 66.35 and 70.45
minutes.
We are 99% confident the
population mean (μ) is
between 65.71 and 71.09
minutes.
CI = 68.4 ±1.72
We are 90% confident the
population mean (μ) is
between 66.68 and 70.12
minutes.
2. A sample of 224 students showed that they attend an average of 2.6 school athletic
events per year with a standard deviation of 0.7. Determine a 95% confidence interval
for the population mean.
M  z

n
M  1 . 96 
0 .7
224
M  0 . 09
CI  x  M
CI  2 . 6  0 . 09
We are 95% confident the population mean (μ) is
between 2.51 and 2.69 events.
Differences between the formulas for margin of error:
𝑀=
1
𝑛
𝑀=𝑧
𝜎
𝑛
• Use when finding the margin
of error of a proportion.
• The sample size and a
proportion is given
• Use when finding the margin
of error of a mean.
• The sample size, mean, and
standard deviation is given.
example:
600 students were polled and 57%
of them are voting for student A for
ASB president. What is the margin
of error and what is the interval that
represents the percent of the
population that are likely to vote for
student A?
example:
600 were asked how many hours
they spend doing homework each
night. The mean was 1.5 hours with
a standard deviation of 0.9. Use a
95% confidence level to determine
the margin of error and the
confidence interval for the
population mean.
margin of error: 𝑀 =
1
600
≈ 0.04 or 4%
Between 53% and 61% of the population will
likely vote for student A.
margin of error: 𝑀 = 1.96
0.9
600
≈ 0.07
We are 95% confident that the population
spends between 1.43 and 1.57 hours on
homework each night.