Warm up

The mean salt content of a certain type of potato chips is supposed to be 2.0mg. The salt content of these chips varies normally with standard deviation σ = 0.1mg. An inspector takes a sample of 50 chips and finds the mean salt content of these chips to be 1.97mg. The inspector should reject the entire batch if the sample mean salt content is significantly different from 2mg at the 5% significance level.

-Should this batch be rejected? -Describe a type 1 and type 2 error. -What is the probability of a type 1 error?

Statistics

Descriptive Statistics

Inferential Statistics

Confidence Intervals

Hypothesis Tests

One-Sided Confidence Intervals

Two-Sided Confidence Intervals

Two-Sided Hypothesis

Tests

One-Sided Hypothesis

Tests

There is a relationship between Confidence Intervals and Hypothesis Tests

How to use Table B

If you have a two-sided graph and want an α of .05, what is our cut off z-score? How does this relate to Table B.

However, if your graph is two sided, then both tails could not exceed α, so you would look at 1- α to find your cut off z-score.

Duality

A level α two-sided significance test rejects a hypothesis H0: μ = μ0when the value μ0 falls outside a level 1 – α confidence interval for μ. Ex. If your α = .1, then you reject your null

hypothesis if your μ0falls outside of a 90% confidence interval

Special note: we are talking about TWO-sided hypothesis tests (μ ≠ μ0)!!

Some cautions

This information is for you to understand a link between the two. However, if you just use a confidence interval, you do not have a physical p-value calculated, so you can not complete your sentence.

You may very well see this type of relationship on your AP exam.

Section 9.3 2nd Day

Matched pairs t-proceduresAnd

Robustness for t-procedures

Comparative studies vs. single-sample investigations

Comparative studies give us more information and are more convincing than single-sample investigations. Why?

What is a comparative study?

Matched Pairs t-procedures

To compare the responses to two treatments in a matched pairs design, just subtract the data to make one list (if you’re given all of the data). Perform the t-procedures on the list of the differences.

Stating Ho and Ha.

When conducting a matched-pairs t-test, you have to be careful when you state Ho and Ha.

In a matched pairs t-test, the null and alternate hypotheses should have a subscript of a D to indicate that you’re subtracting. Then you have to define which order you subtracted.

Matched-Pairs Design

Recall from Chapter 5 what a matched pairs design is.

I know we have actually already studied a matched pairs procedure or two. Remember investigating whether the diet cola loses its sweetness?

Let’s look at our homework example from yesterday We hear that listening to Mozart improves

student’s performance on tests. Perhaps pleasant odors have a similar effect. To test this idea, 21 subjects worked a paper-and-pencil maze while wearing a mask. The mask was either unscented or carried a floral scent. The response variable is their average time on three trials. Each subject worked the maze with both masks, in a random order. The randomization is important is important because subjects tend to improve their times as they work a maze repeatedly.

How the hypotheses are written

Ho: μD=0, where μD=unscented – scented

Ha: μD > 0Which symbol (< or >) should go here if the claim is that pleasant odors would improve the time it takes to complete a maze?

Try this one on your own. Read the problem on page 581. Use

the data on this page. Define what difference will be here. Conduct a 99% confidence interval on

the difference in pressure lost. Is there evidence that nitrogen

reduces the tire pressure loss in tires?

Assumptions

Recall the assumptions for the t-procedures (t-intervals AND t-tests)

The sample is a SRS from the population of interest Observations from the population are distributed

normally. We don’t know this so we verify whether this is plausible.

Is n large? If it is, then the CLT verifies that the distribution of x-bar is approximately normal.

If n is not large, then we need to examine the data. If the histogram or boxplot of the data is symmetric with no outliers (or the normal probability plot is linear), it is plausible to assume that the data are from a normal population.

If the question stem tells us that the population is distributed normally, then we do not need to verify the assumption.

Robust?

A confidence interval or hypothesis test is called robust if the confidence level or p-value doesn’t change very much when the assumptions of a procedure are violated.

Are the t-procedures robust?

In reality, very few populations are exactly normal. How does that affect our t-procedures?

The t-procedures are actually quite robust against non-normality of the population IF there are no outliers.

What do I mean by “check the data?”

Always make a plot (boxplot, histogram, normal probability plot) to check for skewness and outliers.

If n is large, this step isn’t AS necessary. WHY?

Because the CLT theorem tells us that x-bar is distributed approximately normally if n is large.

Some practical guidelines

Except in the case of small samples, the assumption that the data are an SRS from the population of interest is more important than the assumption that the population distribution is normal.

Homework

Chapter 9 # 80, 82, 89, 92, 93