Statistical Confidence - Proportions

Math 145


  • We're going to deviate from the order in the book a little - chapter 6
  • Lab


What was a sampling distribution?

Let's consider a deck of cards

Suppose we want to know what percent of the deck is red

Grab A Deck

  • How many cards should we draw?
  • How should we draw them?
  • Draw that many, then calculate the percentage, we'll record this
  • Do this 10 times
  • What if we drew a different number?
  • How would this change the distribution?

Let's draw the distribution

What do we get

  • This is a sampling distribution for \( \hat{p} \)
  • It should look nearly normal
  • Conditions:
    • observations are independent
    • expect at least 10 successes and 10 failures in our sample (\( np \geq 10 \), \( n(1-p) \geq 10 \))
  • If this is true, the sampling distribution will be nearly normal with mean \( p \) and standard deviation \( \sqrt{\frac{p(1-p)}{n}} \).
    • This is called the standard error (any standard deviation of a sampling distribution)
  • We write this as: \[ SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]


  • Point estimates vs interval estimates
  • Interval estimates are usually 95%, we'll make 80% now


Margin of error - this is the distance from the point estimate to the ends of the confidence interval

What is the formula for the margin of error?

The "Usual" Defintion

Confidence interval: \[ \hat{p} \pm z^* S.E. = \hat{p} \pm z^* \sqrt{\frac{p(1-p)}{n}} \]


\[ (\hat{p}-z^* S.E.,\hat{p}+z^* S.E.) = (\hat{p}-z^* \sqrt{\frac{p(1-p)}{n}},\hat{p}+z^* \sqrt{\frac{p(1-p)}{n}}) \]

How do we make sense of the \( z^* \)?

What can we change

  • \( n \)?
  • \( p \)?
  • \( \hat{p} \)?
  • \( z^* \)?

Sample Size Calculations

What if we want a specific margin of error. Can we select \( n \) before we conduct the experiment?

Confidence Intervals for Testing Claims

  • Suppose you want to test a claim that 52% of students like the food on campus.
  • You sample 150 people, and find that 48% of the sample likes the food.
  • What can we conclude?

A Quote


Tara Golshan: Right now, these numbers are showing about 53 percent of Republicans either want [the corporate tax rate] to be raised or stay the same. When selling this idea, do you see that as becoming a problem?

Lee Zeldin: What I have come in contact with would reflect different numbers. So it would be interesting to see an accurate poll of 100 million Americans. But sometimes the polls get done of 1,000 [people].