MK

DASC 120 Week 3 Lecture Ch. 5.2, 5.3

Confidence Intervals

Confidence intervals are ranges that encompass where we believe the "true” value of the population mean resides.

We determine a "Confidence Level" associated with how much risk we are willing to take that we have "got it wrong": an (alpha) level of 0.05 means that we are wi ing to take a 5% chance of getting a false positive. The corresponding "Confidence Level" to this is 95%

"We are 95% sure that the mean lies in this range”

A confidence interval estimates a population mean based on a sample.

Sometimes the point estimate is not as helpful as we'd like. To check the plausibility of our point estimate, we can create confidence intervals around that point.

A confidence interval gives us a range of values describing where we expect that point estimate to fall, given the potential for error in our calculations.

For instance, if we calculate a 95% confidence interval around the mean of our distribution, we are saying that we are 95% sure that the true value falls within the values in the range

“95% sure that the true value falls within the values in the range.”

CONFIDENCE INTERVALS - EXAMPLES

We take a poll of people and ask what they think the most fair price for an ice cream cone is. The mean of responses is $5.

But because of error, a more appropriate representation might be between 3 and 7 do ars. If s=1, then 3 and 7 are both 2 sd from the mean of 5, which is 95% of the distribution (assuming it is normal).

So we can say that we are 95% sure that the true value of the poll result is [3,7].

  • Confidence intervals give a range of plausible values for the true proportion

Confidence Intervals - Calculating

For Quantitative Data, the confidence interval at 95% may be calculated as
μ ± 2sd = 1.96 x n

For Qualitative Data, the confidence interval at 95% may be calculated as
phat ± 2sd = 1.96 × n

phat (proportion) ± standard error (proportion) = 1.96 × n

  • A 95% confidence level leaves 5% total in the two tails of the normal distribution.

    Each tail has 2.5%, so you look for the z-value where the cumulative area = 0.975 (since 1 − 0.025 = 0.975).

  • The standard error measures the variability of the sample proportion estimate

If we repeated a study 1,000 times and constructed a 95% confidence interval for each study, then approximately 950 of those confidence intervals would contain the true fraction of U.S. adults who suffer from chronic illnesses

A 90% confidence interval is narrower, not wider, because it uses a smaller critical value (z = 1.645) than the 99% interval (z = 2.58). Higher confidence requires a wider range.

  • Simply put, you are more sure of a wider range being true

Four Steps:

  1. Prepare. Identify phat and n, determine CL

  2. Check. Verify phat is nearly normal. For one-proportion CIs, use to check the success-failure condition.

  3. Calculate: compute SE using phat, find z⋆, and construct CI

  4. Conclude. Interpret CI in Context

Confidence Intervals - Confidence Level

What level should you use? — depends on industry!

For instance, we used 99.9% for diagnostic assays.

Pharma uses 99.99 or 99.999.

Logging? 85% might be fine!

Confidence Intervals - Checking Condition

Remember the Central Limit Theorem for proportions says that we must meet the following:

np ≥ 10

n(1 − p) ≥ 10

This means that both groups must be ≥ 10. We can use phat

Confidence Intervals - Standard Error

Find the standard error of the sampling distribution

Find the Z* for this sample —> if 95%, 1.96; if other, check z-table

(Remember, Z relates to standard deviations!)

Margin of Error: Z* (σ of phat)

For proportions:

For means:

Confidence interval: ± 1.96 × n

Confidence Intervals - In Context

"In Context" means to describe the information in layperson's terms, casual conversational terms.

IE: CI=(2.4,4.5) at 95% related to the price of beans in Chicago:

We are 95% sure that the price of beans in Chicago is between $2.50 and $4.50