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Chapter 10. AP Statistics St. Francis High School Fr. Chris. Introduction to Inference. 1. Estimating with Confidence. 2. Tests of Significance. 3. Using Significance Tests. 4. Inference as a Decision. Confidence Interval. Two parts: the estimate and the confidence level

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Chapter 10

Chapter 10

AP Statistics

St. Francis High School

Fr. Chris


Introduction to inference
Introduction to Inference

1. Estimating with Confidence

2. Tests of Significance

3. Using Significance Tests

4. Inference as a Decision.


Confidence interval
Confidence Interval

  • Two parts: the estimate and the confidence level

  • Estimate±Margin of Error

  • If you have an SRS, you can assume that the mean is normally distributed

  • If you can estimate the spread, you can find the boundaries where 90%, 95% or 99% of the area under this normal curve lie.


A pharmaceutical manufacturer
A Pharmaceutical Manufacturer

To establish the concentration of an active ingredient of a drug, the same specimen of a particular batch is measured three times: 0.8403, 0.8363, and 0.8447. The standard deviation is known to be 0.0068 grams per liter. What is the 95% confidence interval?

So .8404 is a good estimate of the actual amount, but how big a margin of error do we need to be 95% confident the actual measurement is in our interval?


The 95 confidence interval
The 95% Confidence Interval

On a standard normal curve (N[0,1]), 95% is between -1.96 and +1.96 standard deviations.

Since we know sigma, we convert the drug’s distribution to the standard normal so the interval is

95%

+1.96

-1.96

Where z = 1.96 (95% Confidence)


Substituting the numbers
Substituting the numbers...

So we can be 95% sure that the actual measurement of the sample is between .8303 and .8505

We would have a more narrow interval if we are willing to drop our confidence interval to 90%. On a standard normal curve, 90% of the area is between -1.645 and +1.65...

Or (.8339, .8469)


Caution
Caution!!!!!

  • Data must be SRS from the population

  • Formulas for more complex designs are available

  • Fancy formulas cannot rescue badly produced data

  • Outliers can influence x-bar

  • If n is small (<15), population needs to be normal

  • Need to know the sigma of the population (If sample is huge, you can use s, otherwise, wait for more methods next chapter!)


10 2 tests of significance
10.2 Tests of Significance

  • H0 and Ha

  • One Sided vs. Two Sided Alternative

  • P-Value

  • Critical values and the z Statistic


Blood pressure
Blood Pressure

Males age 33-44 have a systolic blood pressure of 128 with a =15.

A researcher looks at the records of 72 male ATC’s in this age

group and found their average systolic blood pressure to be

131.07. Obviously this is higher than the average. It is likely thatthis is due to chance variation, or that ATC’s have a significantly higher blood pressure? It is SIGNIFICANT if the result is unlikelyto be caused by chance.


The null hypothesis
The Null Hypothesis

Males age 33-44 have a systolic blood pressure of 128 with a =15.

72 of our ATC’s average at over 131. The null Hypothesis is to say

There is no difference between ATC’s and males in general.

H0 :  = 0

One way to state the alternative is to say that the average of ATC’s is different, so the alternative hypothesis is

Ha :  ≠ 0

This is a “Two-tail” test, since the alternative includes two possibilities: Either the ATC’s have a higher blood pressure or they have a lower blood pressure.


Let s get graphic
Let’s get graphic

A two tailed test means that the sample mean falling on either extreme will be considered grounds to reject the H0

Knowing what we know about the Normal Distribution, we know the critical z score that partitions the inner 95% is where the cumulative area is at 2.5% on both ends (thus our =.05)


Are atc s average on the fringe
Are ATC’s average on the fringe?

If they belong to the same distribution as anyone else, we would expect the  of the sample mean of 72 randomly selected males to be

So to convert to the Standard Normal Curve...


Fail to reject the null hypothesis
Fail to reject the Null Hypothesis

Since 1.7366 is less than 1.96, we are in the 95% of sample means, and we cannot reject the H0 …

But what if we only consider the alternative that ATC’s, due to the stressful nature of their work, would have a HIGHER blood pressure than other males...


The one tailed test
The One Tailed Test

The one tailed test has the same H0 :  =0but a different alternative hypothesis, viz., Ha :  >0

The critical value is different, since 95% is on the left, and 5% is on the upper end.


Rejecting the null hypothesis
Rejecting the Null Hypothesis

Since our critical value is now 1.645, and the sample mean of 131.07 has a z value of 1.7366, it is in the extreme upper 5%, so we can reject the null hypothesis in favor of the alternative hypothesis: The mean of ATC’s systolic blood pressure is higher than other males of this age group.


Review of terms
Review of terms

Notice our  is still .05, but only when we consider the one-tailed test (which is appropriate in this case), are we able to reject the null hypothesis. Computers and statistical calculators can compute a p-value (probability) which allows to to see how unlikely the average is if the null hypothesis is true.


Using your calculator

First let’s select the two tailed alternative hypothesis then press Calculate

Using your calculator

Select STAT, then TESTS

Select “1:Z-Test”

Select “Stats” since we don’t have the raw data (72 sbp’s)

Notice that it shows the z-score, and the p-value (.08 is bigger than .05). So we cannot reject the null hypothesis.

Go back and instead of selecting “Calculate”, select “Draw”


Using your calculator for one tail test
Using your calculator for One Tail Test

Select STAT, then TESTS

Select “1:Z-Test”

Select “Stats” since we don’t have the raw data (72 sbp’s)

First let’s select the two tailed alternative hypothesis >0 then press Calculate

Notice that it shows the z-score, and the p-value (.04 is less than .05). So we can reject the null hypothesis.

Go back and instead of selecting “Calculate”, select “Draw”


10 3 using significance tests
10.3 Using Significance Tests

  • Choosing a level of significance

  • When is Statistical Inference Valid

  • Examples


Choosing a level
Choosing a level

  • P-values are more informative than Traditional  levels

  • Small effects can be highly significant

  • Lack of signifcance doesn’t imply H0 is true


Validity
Validity

  • Faulty data collection

  • Outliers

  • Testing a hypothesis on the same data that suggests a hypothesis can invalidate the test

  • Many tests at once can produce significance due to chance alone, even if the H0’s are true


Examples
Examples

Hawthorne Effect: Will background music increase productivity? Any short term change can effect, regard less of what it is (See Example 10.19, p 590)

A researcher looking for evidence of ESP tests 500 subjects. Four do significantly better (P<.01) than random guessing.

Can we conclude these 4 have ESP? What should the researcher do to test whether these 4 have ESP? (10.61 p. 592)


Kidney failure
Kidney Failure

Patients with kidney failure are often treated by dialysis, and can also cause the retention of phosphorous that must be corrected by changes in the diet. One patient in a study had these 6 measurements over time: 5.6, 5.3, 4.6, 4.8, 5.7, 6.4 (These are separated over time and can be considered an SRS of the patient’s blood phosphorous level). If this level varies normally with = 0.9 mg/dl, give a 90% confidence interval for the mean blood phosphorus level.

(4.7956, 6.0044)


Conclusions
Conclusions

The normal range of phosphorous in the blood is considered to be 2.6 to 4.8 mg/dl. Is there strong evidence that the patient has a mean phosphorous level that exceeds 4.8?

What are the hypotheses?

Conclusion?

H0: =4.8 Ha: >4.8

There is evidence that the patient’s mean phosphorous level is higher than 4.8 mg/dl, but it is not strong. Whether this is sufficient evidence is a judgement call. .05 is not sacred

What if the inference is use to determine a change in diet? Surgury?

What is the z-score/P-value?

Z=1.633

P-value is .051235