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```Problem Solving and Data Analysis
13 – Data Inferences
Calculator Required
Day 2
From earlier:
A large shoe company wants to know the average shoe size for every person currently living in the
United States. To approximate this value, the company randomly selects 10,000 US residents and
records their shoe size. The average shoe size recorded is 9.7 with a margin of error of 1.1 at the
95% confidence level.
Why did this situation mention a confidence level?
What is a confidence level?
Because all of these situations are trying to approximate a larger group, there is room for
mistakes.
A confidence level actually measures how well the survey/experiment approximates the value for
the large group.
Essentially, it says how sure they are of their answer.
For this situation, the shoe company is 95% sure that the actual average shoe size for US
residents is between 8.6 and 10.8.
A large shoe company wants to know the average shoe size for every person currently living in the
United States. To approximate this value, the company randomly selects 10,000 US residents and
records their shoe size. The average shoe size recorded is 9.7 with a margin of error of 1.1 at the
95% confidence level.
So what if the shoe company wanted to be more confident? Maybe they want to be 98% confident.
Increasing your confidence level with also increase the margin of error (a wider range of values).
So what if the shoe company wanted to be less confident? Maybe they want to be 90% confident.
Decreasing your confidence level with also decrease the margin of error (a smaller range of values).
So how can the shoe company maintain a high level of confidence but have a smaller margin of
error?
By sampling more people, the confidence level can remain 95% but the margin of error will
decrease.
In general:
Increasing the confidence level will increase the margin of error (larger confidence interval).
The shoe company can be 100% confident that the average shoe size of US residents is between
zero and infinity.
Decreasing the confidence level will decrease the margin of error (smaller confidence interval).
The shoe company can be 0% confident that the average shoe size of US residents is exactly 9.4.
Increasing the sample size will decrease the margin of error (smaller confidence interval). This is best if
By sampling more people, the company is getting closer to the actual US population.
Decreasing the sample size will increase the margin of error (larger confidence interval).
By sampling less people, the company is getting farther from the actual US population.
resources
allow.
In a poll of 1,578 randomly selected American adults, 44.8% of the respondents said that airlines
should allow in-flight calls on airplanes. The poll reported a margin of error of 2.5% at a 95%
confidence level.
1. What would happen to the margin of error if the poll reported a 99% confidence interval?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
2. What would happen to the margin of error if the poll reported a 68% confidence interval?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
3. What would happen to the margin of error if the poll interviewed 5,000 American adults?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
A random sample of 500 residents of a town included 173 residents who primarily
spoke a language other than English at home, with a margin of error of 25
residents and a confidence level of 98%. If the town has 25,000 residents, how
many residents primarily speak a language other than English at home, with a 98%
confidence level?
1. What would happen to the margin of error if the reported confidence interval was 80%?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
2. What would happen to the margin of error if 5,000 residents were surveyed?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
3. What would happen to the margin of error if 50 residents were surveyed?
A. Increase
B. Decrease
C. Stay the Same
D. Can’t Tell
An archaeologist uses an accelerator mass spectrometer to find the age of a buried
branch. At the 68% confidence level, the spectrometer estimates that the branch
was 10,000 years old with a margin of error of 200 years. Which of the following
could be the spectrometer estimate as the age of the branch at the 95% confidence
level?
A. 9,500 years old, with a margin of error of 500 years
Increasing the confidence level
increases the margin of error.
B. 10,000 years old, with a margin of error of 400 years
C. 9,500 years old, with a margin of error of 50 years
D. 10,000 years old, with a margin of error of 40 years
Changing the confidence level
does not change the estimation
(sample statistic).
We can also use a given confidence interval to work backwards.
Researchers measured the heart rates of several randomly selected astronauts exercising on
stationary bicycles during long-term space missions. The researchers found the mean heart rate of
the astronauts to be between 143 and 167 beats per minute at the 90% confidence level.
1. What was the mean heart rate of the randomly selected astronauts?
The sample statistic is in the middle of the given interval. Find the average.
143  167
 155 beats per minute
2
2. What is the margin of error for this situation?
Three ways to find this value:
167 – 155 = 12 beats per minute
155 – 143 = 12 beats per minute
167  143
 12 beats per minute
2
A random sample of 35 four-door passenger vehicles had a mean gas mileage
between 23.3 and 28.5 miles per gallon (mpg). The estimate was obtained at a
98% confidence level.
1. What was the mean gas mileage for the randomly selected vehicles?
28.5  23.3
 25.9 miles per gallon
2
2. What is the margin of error for this situation?
28.5  23.3
 2.6 miles per gallon
2
```
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