Confidence Intervals for the Population Mean

Because of sampling error, the point estimate is not likely to equal the population parameter in any given sample.

A confidence interval is a range for which one can assert with a given probability 1 − α, called the degree of confidence, that it will contain the parameter it is intended to estimate. This interval is often referred to as the 100(1 − α)% confidence interval for the parameter.

A 100(1 − α)% confidence interval for a parameter:

Point Estimate± Reliability factor× Standard error

Confidence Intervals for the Population Mean (Normally Distributed Population with KnownVariance). A 100(1 − α)% confidence interval is

90 percent confidence intervals: Use z_0.05 = 1.65

95 percent confidence intervals: Use z_0.025 = 1.96

99 percent confidence intervals: Use z_0.005 = 2.58

Confidence Intervals for the Population Mean—The z-Alternative (Large (≥30) Sample, Population Variance Unknown).

A 100(1 − α)% confidence interval is

The t-distribution is a symmetrical probability distribution defined by a single parameter degreesof freedom (df).

Symmetrically distributed around its mean value of zero.

df = n − 1

Fatter tails.

As the df increase, approach the standard normal.

Confidence Intervals for the Population Mean (Population Variance Unknown)—t-Distribution.

the sample is large (≥30), or

the sample is small (<30) but the population is normally distributed, or approximately normally distributed

the number of degrees of freedom for t_α/2 is n − 1