Confidence Intervals with a proportion

Schwab

Practice Making CI with proportions

CI mathematically

We build the CI around the sample proportion.

Assume

\[ \hat{p} \sim N(p,se) \]

  1. We’ll need \(\hat{p}\) and \(\text{se}= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

  2. We also need the z score that goes with the confidence interval we are looking for.

  3. \(\hat{p} \pm z_{score} \times se\)

Remember these z-scores

z-scores you should know
CI Z-score
90% \(z_{0.05}=1.645\)
95% \(z_{0.025}=1.96\)
99.7% \(z_{0.005}=2.968\)

How to find zscore with qnorm()

qnorm() finds the quantile that goes with a pvalue (probability)

[1] 1.644854

pnorm() finds a pvalue from a quantile.

[1] 0.975

Let’s make a 95% CI

Calculate these: \(\hat{p}\) and \(\text{se}= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

\(\hat{p} \pm z_{score} \times se\)

Assumptions for a single parameter

Independent and “large” sample.

Success-Failure Condition

Why do we need to assume this:

\[ \hat{p} \sim N(p,se) \]

The Central Limit Theroem

Notation

\[ \hat{p} \sim N(p,SE)\]

With \(SE= \sqrt{\frac{p(1-p)}{n}}\)

or

\(SE= \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)

Rules of thumb

  • The operations need to be independent.

  • n should be large

  • Success-Failure Condition

    • \(np >=10\) and

    • \(n(1-p) >= 10\)

Quarantine again

We ask 1042 New Yorkers if they are for quarantining people who have been exposed to ebola. The results are below.

# A tibble: 2 × 2
  quarantine     n
  <fct>      <int>
1 against      188
2 favor        854

  • Write the notation for the theoretical sampling distribution.

  • Make a 90, 95 and 99.7% CI for those against and interpret.

Exclusive relationships

What is the proportion of college students that have had more than 1 exclusive_relationships?

# A tibble: 3 × 2
  `num > 1`     n
  <lgl>     <int>
1 FALSE        51
2 TRUE        152
3 NA           15

  • Check Conditions.

  • Write the notation for the theoretical sampling distribution.

  • Make a 90, 95 and 99.7% CI for those against and interpret.

Worse off?

Every week the Yougov/The Economist do a survey from a representative sample of about 1500 adults. methodology On March 4, 2024 people were asked about their personal finances. See the results here.

42% of respondants said they were worse off than a year ago. Find a CI for the true proportion of Americans who feel they are worse off financially than a year ago.