Confidence Interval Estimation of Population Mean, μ, when σ is Unknown

Chapter 9 Section 2

The BIG Idea!

Most of the time, the value for the populations standard deviation is NOT known. For example, what’s the likelihood anyone know the average number of kids in each family in this school or the standard deviation for that population?

How do we develop a confidence interval when the standard deviation for the population is unknown????

Confidence Interval Estimation of Population Mean, μ, when σ is Unknown

• If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S

• This introduces extra uncertainty, since S varies from sample to sample

• So we use the student’s t distribution instead of the normal Z distribution

t-distribution

The t – distribution is actually a family of curves based on the concept of degrees of freedom, which relate to sample size.

As the sample increases, the t-distribution approaches the standard normal distribution.

• Student’s t distribution• Note: t Z as n increases

t0

t (df = 5)

t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal

Standard Normal

Degrees of Freedom

The degrees of freedom (denoted d.f.) are the number of values that are free to vary after a sample statistic has been computed, and tell the researcher which specific curve to use when a distribution consists of a family of curves.

For example: If the mean of 5 values is 10, then 4 of the 5 values are free to vary. But once 4 values are selected, the fifth value must be a specific number to get the sum of 50, since 50/5 = 10.

Hence, the degrees of freedom are 5 – 1 = 4, and this tells the researcher which curve to use.

d.f. Formula

• The formula for finding the degrees of freedom for the confidence interval of a mean is simply

d.f. = n – 1

• Confidence Interval Estimate Use Student’s t Distribution :

(where t is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail)

• t distribution is symmetrical around its mean of zero, like Z dist.• Compare to Z dist., a larger portion of the probability areas are in the

tails.• As n increases, the t dist. approached the Z dist.• t values depends on the degree of freedom.

n

StX 1-n

Example #1

A sample of 14 commuters in Chicago showed the average of the commuting time was 33.2 minutes. If the standard deviation of the sample was 8.3 minutes, fine the 95% confidence interval of the true mean.

Example #2

Poll Everywhere poll on sleep last night.

Link to Poll

What is the mean and standard deviation for our sample?

What is the degrees of freedom for our sample?Find a 90% confidence interval of the mean time.

Assume the situation is normally distributed.