Statistics Chapter 8 (Bentley

The Required Sample Size when Estimating the Population Mean

For a desired margin of error E, the minimum sample size n required to estimate a 100(1-α)% confidence interval for the population mean μ is (look-up formula), where σ (with ^ over it) is a reasonable estimate of σ in the planning stage

The Required Sample Size when Estimating the Population Proportion

For a desired margin of error E, the minimum sample size n required to estimate a 100(1-α)% confidence interval for the population proportion p is (look-up formula) where p (with ^ over it) is a reasonable estimate of p in the planning stage

Confidence Interval (or "Interval Estimate")

Provides a range of values that, with a certain level of confidence, contains the population parameter of interest Common to construct as Point Estimate + or - Margin of Error

Confidence Coefficient Equation

1-α

Three Factors that Influence Confidence Interval

1. For a given confidence level 100(1-α)% and sample size n, the larger the population standard deviation σ, the wider the confidence interval. 2. For a given confidence level 100(1-α)% and population standard deviation σ, the smaller the sample size n, the wider the confidence interval. 3. For a given sample size n and population standard deviation σ, the greater the confidence level 100(1-α)%, the wider the confidence interval.

Three Points to Summarize the tdf Distribution

1. Like the z distribution, the tdf distribution is bell-shaped and symmetric around 0 with asymptotic tails (the tails get closer and closer to the horizontal axis but never touch it) 2. The tdf distribution has slightly broader tails than the z distribution 3. The tdf distribution consists of a family of distributions where the actual shape of each one depends on the degrees of freedom df. As df increases, the tdf distribution becomes similar to the z distribution; it is identical to the z distribution when df approaches infinity

Confidence Interval for μ when α is Known

A 100 (1-α)% confidence interval for the population mean μ when the population standard deviation σ is known is computed as (look up formula)

Confidence Interval for μ when σ is Not Known

A 100(1-α)% confidence interval for the population mean μ when the population standard deviation σ is not known is computed as (look up formula) where s is the sample standard deviation. This formula is valid only if X (line over it) (approximately) follows a normal distribution

Confidence Interval for p

A 100(1-α)% confidence interval for the population proportion p is computed as (look-up formula). This formula is valid only if P (with a line over it) (approximately) follows a normal distribution

Interval

A range of values

The t Distribution

If a random sample of size n is taken from a normal population with a finite variance, then the statistics T=(X( with a line over it)-μ)/(S/sqr(n)) follows the t distribution with (n-1) degrees of freedom, df (degrees of freedom)

Interpreting a 95% Confidence Interval

Technically, a 95% confidence interval for the popular mean μ implies that for 95% of the samples, the procedure (formula) produces an interval that contains μ. Informally, we can report with 95% confidence that μ lies in the given interval. It is not correct to say that there is a 95% chance that μ lies in the given interval.