Test Topics1)Notation and symbols 2)Determining if CLT applies.3)Using CLT to find mean/mean

proportion and standard error of sampling distribution

4)Finding confidence intervals (means & proportions)

5)Finding sample size to get specific margin of error

CI = statistics ± z* · standard error

margin of error

What if you want the margin of error to have a certain value because you want to estimate the TRUE mean or proportion within a specific amount?

Example: You want to estimate your candidates approval rating (%) within 3% of the ACTUAL approval rating across the country.

Often a high confidence level (95 or 99%), means that your interval must be very large (high margin of error). Ultimately, we would like to create a confidence interval with a high confidence level and very small margin of error.

How can we control that???Make the z* value

smaller this means a lower

confidence level.

Make the s value smaller

this does make it

easier to get a more

accurate m, but is difficult

to control.

Make the n (sample size)

larger

dividing by a larger number makes the standard error smaller and in turn the margin of error smaller.Best Option!

The one part that would have the power to change the margin of error is the

sample size (n). Margin of error

z* will be determined by the confidence levels or will be determined by the dataGiven a certain margin of error (E),

we can solve for n.

𝝈𝒙=𝝈√𝒏

𝝈 �̂�=√ �̂� (𝟏− �̂� )𝒏

Example: We want to estimate the average number of college games attended by all football fans per season within 2 games based upon a 95% confidence level. We know that s = 3.5 games.

E 2≥1.96 ∙3.5

√𝑛

2√𝑛≥1.96 ∙3.5

Multiply by

√𝑛≥ 6.862

Divide by 2

n≥ (3.43 )2=11.76 ≈12

Square both sides

Example: We want to estimate the average number of college games attended by all football fans per season within 2 games based upon a 95% confidence level. We know that s = 3.5 games.

E

Divide by E

Square both sides

*We could solve the formula for “n” and use it each time we need to compute a sample size.

Multiply by

n

Example: You want to estimate your candidate’s approval rating (%) within 3% of the ACTUAL approval rating across the country. You want to be 99% confident that your estimate is accurate and know that = 0.43. What is the minimum sample size needed to make this happen?

This is a proportion.

0.03≥2.575 ∙√ 0.43(1−0.43)𝑛Divide by 2.575

0.032.575

≥√ 0.2451𝑛

(0.116504854 )2≥ 0.2451𝑛Square both sides

Multiply by n and then divide

𝑛≥0.2451

(0.116504854 )2=1805.74≈1806

𝐸≥ 𝑧∗ ∙√ �̂� (1− �̂�)𝑛

𝐸≥ 𝑧∗ ∙√ �̂� (1− �̂�)𝑛

Divide by z*

Square both sides

Multiply by n

OR )

*We could solve the formula for “n” and use it each time we need to compute a sample size.

Then divide