Chapter 8: Confidence Intervals

Confidence level

considered the probability that the calculated confidence interval estimate will contain the true population parameter.

Alpha level

is the probability that the interval does not contain the unknown population parameter.

standard error of the mean

𝜎 / √n

Confidence interval

an interval estimate for an unknown population parameter. This depends on

Confidence interval form

(point estimate – margin of error, point estimate + margin of error)

Empirical rule

Around 68% of values are within 1 standard deviation of the mean. Around 95% of values are within 2 standard deviations of the mean.

The margin of error

how many percentages points your results will differ from the real population value

Inferential statistics

We use sample data to make generalizations about an unknown population

Sample data

help us to make an estimate of a population parameter.

Point estimate

a single number computed from a sample and used to estimate a population parameter

x¯

is a point estimate for μ

p′

is a point estimate for ρ

s

is a point estimate for σ

Calculating the Error Bound

EBM = (𝑧 𝛼/2)(𝜎/√n)

Confidence level interpretation

"We estimate with _% confidence that the true population mean (include the context of the problem) is between _ and ___ (include appropriate units)."

Increasing, wider

_______ the confidence level increases the error bound, making the confidence interval ______.

Decreasing, narrower

_______ the confidence level decreases the error bound, making the confidence interval _______.

Sample size

Increasing the _________ causes the error bound to decrease, making the confidence interval narrower.

Confidence interval

Decreasing the sample size causes the error bound to increase, making the ________ wider.

Student's t-distribution

a type of probability distribution that is similar to the normal distribution with its bell shape but has heavier tails

Standard deviation

a number that is equal to the square root of the variance and measures how far data values are from their mean; notation

Normal distribution

continuous random variable (RV) with pdf 𝑓(𝑥)=(1 / 𝜎√2𝜋) 𝑒^–(𝑥–𝜇)^2/2𝜎^2, where μ is the mean of the distribution and σ is the standard deviation, notation

Degrees of freedom

the number of objects in a sample that is free to vary

df = n - 1

the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

The invT command requires two inputs

invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specified.

Z-score formula

If 𝑃′~𝑁(𝑝 , √𝑝𝑞/𝑛) then the z-score formula is 𝑧=𝑝′−𝑝/√𝑝𝑞/𝑛