Inference for the Mean of a Population Section 11.1 AP Exam Registration Deadline: March 17 th Late Fee ($50): March 18 th – March 24 th Financial Aid.

Inference for the Mean of a PopulationSection 11.1

AP Exam Registration Deadline: March 17th

Late Fee ($50): March 18th – March 24th

Financial Aid Application Due: TODAY!!!!!!

Get out homework!

Conditions for Inference about a Mean

SRS of size n from the population of interest

Observations must be independent

Observations must have a normal distribution with mean µ and standard deviation σ

What if σ isn’t given? Because σ is usually unknown, we

estimate it by the sample deviation s.

Standard Error The standard error of the sample mean is

When we know the value σ, we base confidence intervals and tests for µ on the z - statistic:

The z – statistic has a normal distribution.

When we do not know σ, we substitute the standard error . This statistic does not have a normal distribution…

The t distributionDraw an SRS of size n from a population that has the normal distribution with mean µ and standard deviation σ. The one-sample t statistic

has the t distribution with n – 1 degrees of freedom.

Different t distributions? There is a different t distribution for

each sample size.

So how do we determine which one we use?

Degrees of freedom

n -1 , where n is the sample size.

Example 11.1, p. 619Using the “t Table”

What critical value t* from Table C (t Table) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t*?

Example 11.1, p. 619Using the “t Table”

What critical value t* from Table C (t Table) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t*?

Before we start, we will need the tail probability.

.90 probability

Tail area = .10

Df = 18, Tail Area = .10

Example 11.1, p. 619Using the “t Table”

What critical value t* from Table C (t Table) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t*?

So the desired critical value is t* = 1.330.

Example 11.1, p. 619Using the “t Table”

Now suppose you want to construct a 95% confidence interval for the mean µ of a population based on an SRS of size n = 12. What critical value of t* should you use?

Df = 12 – 1 = 11

Tail Area?

We don’t need it…look at the bottom of the chart.

Example 11.1, p. 619Using the “t Table”

Now suppose you want to construct a 95% confidence interval for the mean µ of a population based on an SRS of size n = 12. What critical value of t* should you use?

Example 11.1, p. 619Using the “t Table”

Now suppose you want to construct a 95% confidence interval for the mean µ of a population based on an SRS of size n = 12. What critical value of t* should you use?

So the desired critical value is t* = 2.201.

Homework: P. 619: 11.1 a, 11.2, 11.4

Due: Tuesday

FYI: I have a meeting Friday morning, 3/8. I will have to cancel tutoring that day.