Transcript margin of error - Greer Middle College || Building the Future

Warm Up
For each of the following please state the
population, sample, parameter, and statistic.
1. The local school board posts average SAT Math scores for
each high school in the district. For one particular high
school, the average math score is given as 516. A statistics
teacher at that high school thinks that this is too high. He
takes a random sample of 100 students and finds that
their average SAT Math score is 486.
2. Students in a statistics class weighed 30 bags of regular
size M&M’s and found that the average weight of the 30
bags excluding the wrapper was 1.75 ounces. The Mars
Company gives the net weight as 1.69 ounces.
Homework Check
Errors in Sampling
The margin of error is a statement of how
much confidence we can have in the
results of a sample in representing the
population.
Also, We say we are 95% confident that
the truth lies within the margin of error.
This is called the confidence level.
Margin of Error
For example, if we said the margin of error
is 3%...(confidence interval)
“Margin of error ±3%” means if we took
many samples using the same method,
95% of the samples would give a result
within plus or minus 3 percentage points of
the true population parameter.
Example
For example, you took a sample of Reese’s
Pieces and found the proportion of orange
was 52%.
Let’s say the margin of error in this
situation is 4%.
Then the confidence interval is…
Margin of Error
1. A sample will not estimate the
parameter exactly.
2. We need a margin of error (i.e. ±3%) to
tell us how close we are to the
parameter
3. We can’t be certain that the parameter
differs from the sample by no more than
the margin of error.
Margin of Error
4. So, we say 95% of all samples come this
close to the truth. 5% miss by more
than the margin of error.
5. We say we are 95% confident that the
truth lies within the margin of error.
Back to the Reese’s Pieces
Lets say we found that the average proportion
of orange candies was 52% in 100 samples.
We stated before that the margin of error was
4% and that the confidence interval would be a
48% to 56% proportion of orange candies.
95% confidence says that 95 of the 100 samples
fell between 48% and 56% and 5 of the 100
samples fell outside the confidence interval.
Finding the margin or error
Finding the EXACT margin of error is a job
for statisticians.
A simple formula to get an idea of the size
of the margin of error for 95% confidence
roughly equal to…
1
𝑛
Examples
When constructing a 95% confidence
interval for proportions, what would be the
approximate margin of error for a sample
size of…
1. 300 people?
2. 900 people?
Example
If we take a SRS of sample size of 1523
people. The margin of error for 95%
confidence will be about what?
If we take a SRS of sample size of 100
people from the same population, what
will the margin of error for 95% confidence
be?
Conclusion
Because the formula uses the square root
of the sample size, to cut the margin of
error in half, we must use a sample four
times as large.
Test this hypothesis.
Margin of Error/Confidence
Interval
Because the sample size 𝑛 appears in the
denominator of the fraction, larger
samples have smaller margins of error.
By increasing the sample size, the margin
of error becomes smaller and the length of
the confidence interval narrows.
Confidence Statement
A confidence statement combines the
sample example (i.e. lottery ticket sales),
margin of error, and the level of confidence
into one coherent statement.
Confidence Statement Details
1. Always includes the confidence level
and the confidence interval.
2. The conclusion always applies to the
populations, not the sample.
3. The conclusion is never completely
certain (100%)
4. It is usual to report the margin of error
for 95% confidence.
Confidence Statement
A survey’s conclusion about buying lottery
tickets found that 57% of adults in the sample
bought a lottery ticket in the past 12 months
with a margin of error of 3%.
Confidence Statement
“We are 95% confident that between 54% and
60% of all adults bought a lottery ticket in the
last 12 months.”
Confidence Statement
“We are 95% confident that between 54%
and 60% of all adults bought a lottery ticket
in the last 12 months.”
Examples
Write a confidence statement.
After taking a random sample of juniors
and seniors the proportion of students who
have completed Algebra 2 at GMC with a
grade above mastery is 93% with a margin
of error of 5%.
Examples
Write a confidence statement.
After taking a random sample of
households in South Carolina the average
number of households (with a margin of
error of 4%) that were not originally from
South Carolina is 66%.
Finding the Margin of Error
Sometimes you will have to find the margin
of error given a sample size in order to
write a confidence statement.
What was the equation to find the
approximate margin or error?
Examples
A statistics teacher at a local high school
takes a random sample of 230 students
throughout the district and finds that
their average SAT Math score is 486.
1. What is the margin of error?
2. What is the confidence interval?
3. Write a confidence statement.
Examples
A doctor notices that many patients that
want to lose weight have a hormone
imbalance. She chooses a random sample
of 112 patients and finds that 79 have a
hormone imbalance.
1. What is the margin of error?
2. What is the confidence interval?
3. Write a confidence statement.
Examples
The doctor is not satisfied with the
confidence level. It is too wide.
1. What could she do to find to narrow
the confidence level?
2. How many patients does she need to
sample to cut her original confidence
interval in half?
3. If she increases her sample size to 600
patients, what is her margin of error?
Homework
Margin of Error Worksheet
Due Monday!