A.P. Statistics

Section 11.1

“Inference for the Mean of a Population”

In Chapter 10, we looked at two forms of FORMAL INFERENCE:

(i) Confidence Intervals

(ii) Tests of Significance

We had to make an unrealistic assumption, though, in order to first understand the basic process. It was unrealistic to assume we knew , the standard deviation of the population.

In Chapter 11, we are going to examine how to adjust for not really knowing .

When we looked at steps involved with performing a test of significance in Chapter 10, we said there were 4 steps:

(1) State the hypotheses.

(2) Calculate the test statistic.

(3) Find the p-value.

(4) Interpret the results in context.

WELL … there are actually 5 steps, and now that we are going to quit assuming we know , we are going to add in that additional step.

This step you can label as STEP 0 because it actually will occur prior to even stating your hypotheses:

(0) State and address assumptions!

This new step is CRITICAL for success on the AP Exam!! Without stating and addressing assumptions, you CAN NOT score a 4 on that free response question!!

For the inference we’ve examined so far (inference for the mean), there are 2 assumptions:

(1) The sample is an SRS.

(2) The observations from the population have a normal distribution.

At the start of your significance test, you need to write:

ASSUMPTIONS:

1. Sample is an SRS
2. Distribution of observations is normal. Check by sketching a plot.

Since we cannot know , the best we can do is use s (the sample standard deviation) to estimate it … just like you would use to estimate .

THEREFORE: (the standard error) is .

If we use this estimate in calculating the test statistic, then we get:

Using s introduces more uncertainty – more spread and variation – the resulting distribution is NOT the standard normal (or Z) distribution.

The resulting distribution is called a

t-distribution with n – 1 degrees of freedom

You can abbreviate this using the notation t(k), where k = df = n – 1

**Also referred to sometimes as the Student’s t distribution.

The t-distribution is actually a family of curves.

The accompanying test statistic is called the

One sample t test statistic =

The t-distribution is very similar to the z-distribution in two ways:

(a) Both are symmetric and centered about 0.

(b) Values along both distributions are standardized and expressed in units of standard deviations from the mean.

**The big difference between the two is in the spread. Look at the figure on p. 618.

The t-distribution, since it has more variation, is flatter with more observations in the tails.

Notice also that as the df (degrees of freedom) increases (meaning that the sample size is increasing), the t-distribution becomes more and more normal … closer and closer to the z-distribution, N(0,1).

If you allow the df to increase without bound (in other words, allow , you would get the z-distribution.

That’s why the z^* row of the t-table is labeled as ∞ … it’s referring to an ever-increasing degree of freedom.

How this will affect our confidence intervals and hypothesis tests …

We will be using critical values (now t^* instead of z^*) and p-values from the t(k) distribution.

For CI’s:

t^* is the upper critical value.

For Hypothesis Tests: no real difference

H_a:

Type I error, Type II error, Power:

No real difference

P(Type I) = α

P(Type II) = β

Power = 1 – β

In order to use t-procedures …

(1) We already know we need an SRS.

(2) Re: normality, though …

(i) If n < 15, data needs to be close to normal and NO OUTLIERS!! (Remember, since the calculation of t involves , which is non-resistant, then t also is non-resistant.)

(ii) If , OK to use t so long as data distribution isn’t strongly skewed and there are NO OUTLIERS!!

(iii) If n > 40, then t is fine regardless of the distribution.

**Look in your text at (ex) 11.7 on p. 636.

Even though the t-distribution ISN’T resistant to outliers, it is a ROBUST procedure. This means that even if you violate the assumptions, the confidence intervals and p-values are affected very little.

This is due to the fact that the distribution of is going to be normal regardless of the data’s distribution (Central Limit Theorem).

Before getting into any examples, let’s look at how the t-table is different from a z-table.

(1) Values in the body of the table are the t-scores … NOT the p-values!

(2) df are listed down the side

(3) Area to the right of the t-values are given (NOT the area to the left, like in the z-table). This is the upper probability value.

*See the bottom of the table for C.

Now some (ex)’s …

(ex.) 11.2 on p. 622 – 623

(ex.) 11.3 on p. 624-625

Let’s also look at the Minitab output for a t-test on p. 627

Everything we’ve talked about today has involved a single sample. More often than not, though, you will use a MATCHED-PAIRS DESIGN instead. Evidence is more convincing when a comparison or control is involved. (Think back to our 3 principles of good statistical experimental design: Randomize, CONTROL, and Replicate.)

A Matched-Pairs design can be …

2 subjects paired together … one gets one treatment, one gets another treatment, and the difference is examined.
Before & after … there is an original measure, a treatment, and then a new measure – then the “pre” and “post” difference is examined.

(ex) 11.4 on p. 629 – 630

Re: Power of a t-test … (ex) 11.8 on

p. 639-640

Assignment:

p. 619 – 620 (11.1 – 11.4)

p. 627 – 628 (11.7 – 11.10)

p. 633 – 635 (11.12 – 11.15)

p. 638 – 639 (11.17, 11.20)

p. 640 – 641 (11.21 – 11.23)