CHAPTER 11Section 11.1 – Inference for the Mean of a Population

INFERENCE FOR THE MEAN OF A POPULATION

Confidence intervals and tests of significance for the mean μ of a normal population are based on the sample mean . The sampling distribution of has μ as its mean. That is an unbiased estimator of the unknown μ.

In the previous chapter we make the unrealistic assumption that we knew the value of σ. In practice, σ is unknown.

CONDITIONS FOR INFERENCE ABOUT A MEAN

Our data are a simple random sample (SRS) of size n from the population of interest. This condition is very important.

Observations from the population have a normal distribution with mean μ and standard deviation σ. In practice, it is enough that the distribution be symmetric and single-peaked unless the sample is very small.

Both μ and σ are unknown parameters.

STANDARD ERROR

When the standard deviation of a statistic is estimated from the data, the result is called the Standard Error of the statistic.

The standard error of the sample mean is .

THE T DISTRIBUTIONS When we know the value of σ, we base

confidence intervals and tests for μ on one-sample z statistics

When we do not know σ, we substitute the standard error of for its standard deviation .

The statistic that results does not have a normal distribution. It has a distribution that is new to us, called a t distribution.

T DISTRIBUTIONS (CONTINUED…) The density curves of the t distributions are

similar in shape to the standard normal curve. They are symmetric about zero, single-peaked, and bell shaped.

The spread of the t distribution is a bit greater than that of the standard normal distribution. The t have more probability in the tails and less in the center than does the standard normal.

As the degrees of freedom k increase, the t(k) density curve approached the N(0,1) curve ever more closely.

THE ONE-SAMPLE T PROCEDURES Draw an SRS of size n from a population having

unknown mean μ. A level C confidence interval for μ is

Where is the upper (1 - C)/2 critical value for the t(n – 1) distribution. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.

The test the hypothesis H0 : μ = μ0 based on an SRS of size n, computed the one-sample t statistic

DEGREES OF FREEDOM

There is a different t distribution for each sample size. We specify a particular t distribution by giving its degree of freedom.

The degree of freedom for the one-sided t statistic come from the sample standard deviation s in the denominator of t.

We will write the t distribution with k degrees of freedom as t(k) for short.

EXAMPLE 11.1 - USING THE “T TABLE”

What critical value t* from Table C (back cover of text book, often referred to as the “t table”) would you use for a t distribution with 18 degrees of freedom having probability 0.90 to the left of t?

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

THE ONE-SAMPLE T STATISTIC AND THE T DISTRIBUTION

Draw 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.

THE ONE-SAMPLE T PROCEDURE (CONTINUED…) In terms of a variable T having then t(n – 1) distribution, the P-value for a test of Ho against

These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases.

Ha: μ > μo is P( T ≥ t)

Ha: μ < μo is P( T ≤ t)

Ha: μ ≠ μo is 2P( T ≥ |t|)

EXAMPLE 11.2 - AUTO POLLUTION See example 11.2 on p.622

Minitab stemplot of the data (page 623)

The one-sample t confidence interval has the form:

(where SE stands for “standard error”)

Homework: P.619 #’s 1-4, 8 & 9