Margin of Error and Standard Deviation

3.4 Estimating a Margin of Error

The margin of error of an estimate describes how far, at most, we expect the estimate to vary from the true population value. 

Learning Targets

• Use simulation to approximate the margin of error for a sample proportion and interpret the margin of error.

• Use simulation to approximate the margin of error for a sample mean and interpret the margin of error.

Multiple Choice

What kind of data is this, categorical or quantitative?

• categorical

• quantitative

Fill in the Blank

What is the mean of the data?

Multiple Choice

Do you expect that this mean of 5 is the same as the true population mean?

• Yes

• No

Do you expect that this mean of 5 is the same as the true population mean?

No, because of sampling variability we know that if we continue to sample students in groups of 25 we will get different means. Some will be above and some will be below our sample mean of 5.

One Quantitative Variable

• Enter the data and begin analysis.

• Scroll down to Perform Inference and choose Simulate sample mean.

Draw and label

• The mean (5.051)

• One SD (0.292) above and below

• Two SD (0.584) above and below

Fill in the Blank

How many sample means (dots) are outside the two standard deviations above and below the mean?

Fill in the Blank

What percentage of the sample means are within two standard deviations above or below the mean?

Interval that includes all the data two SD above and below the mean.

• mean: 5.051 and SD: 0.292

• 5.051 + 2(0.292) = 5.635

• 5.051 - 2(0.292) = 4.467

• Confidence Interval: (4.467, 5.635)

Margin of Error

We multiply the standard deviation by 2 in order to get the margin of error. The reason is because a majority of our estimates will be within 2 standard deviations away from the mean (should be around 95%). 

In this example, the margin of error = 2(0.292) = 0.584.

Multiple Choice

Is this a categorical or quantitative variable?

• categorical

• quantitative

Fill in the Blank

How many students responded yes?

Fill in the Blank

How many students responded no?

One Categorical Variable

• Scroll down to Perform Inference

Margin of Error

• = 2(Standard Deviation)

• = 2(0.064)

• = 0.128

• We expect the true proportion of students who text during class to be at most 0.128 from the estimate of 0.639

Fill in the Blank

Use the results of the simulation to approximate the margin of error.

Open Ended

Interpret the margin of error (0.096). What does it mean?

Is a claim that 75% of people can roll their tongue plausible?

Yes, the true proportion can be as low as 0.701 - 0.096 = 0.605 or as high as 0.701 + 0.096 = 0.797 The confidence interval is (0.605, 0.797) and .75 falls within that interval. So 75% is plausible.

Multiple Choice

How could Javier decrease the margin of error?

• Increase the sample size

• Decrease the sample size