Warm UpFor 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. ) 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…

ExamplesWhen 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 IntervalBecause the sample size 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 Details1. 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?

ExamplesA 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.

ExamplesA 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.

ExamplesThe 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 WorksheetDue Monday!