STAT 5372:Experimental Statistics

STAT 5372:Experimental Statistics

Wayne WoodwardWayne Woodward

Office:Office: 143 Heroy Phone:Phone: (214)768-2457 e-mail:e-mail: waynew@smu.edu URL:URL: faculty.smu.edu/waynew Hours:Hours: 2:00 - 3:00 MWF 3:00 - 4:00 Th - others by appointment

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• Name

• Major (undergraduate/graduate)

• Previous stat courses:– STAT 5371?

– STAT/CSE/EMIS 4340?

– other – describe briefly

• Have you used SAS?

On a sheet of paper:

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Review Review

• Sampling Distributions

• Statistical Inference– Confidence Intervals

– Hypothesis Tests

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Sampling / Sampling Distributions

Sampling / Sampling Distributions

• Population -- totality of all observations of interest

• Random Variable (rv) -- a characteristic that can take on different values from object to object

• Sample -- subset of a population

– random sample:random sample: observations made independently and at random

Parameter – a characteristic of a population

Y1, Y2, … , Yn – typical notation for a random sample

-- population mean (), standard deviation (), …

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• Discrete – you can count the possible outcomes

–Discrete distributions: binomial, Poisson, …

• Continuous – possible values fall along a continuum

–Continuous distributions: normal (Gaussian), chi-square, t, F, …

Random VariablesRandom Variables

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Normal Curve:

-- symmetric, bell-shaped

-- for this particular distribution:

- data concentrated about 60

- very few data values above 100 or less than 20

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XZ

Standard NormalStandard Normal (Z-score)

Normal table gives P[Z ≤ z]

Has mean zero and standard deviation 1

Graph of standard normal is symmetric about 0

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9

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Find: P[Z≤ 2.5]

Suppose = 50 and = 10.

Find: P[X≤ 45]

P[Z > 1.6]

P[X > 70]

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Examples of Statistics:Examples of Statistics:

1

1 n

ii

Y Yn

2

2 1

( )

1

n

ii

Y Y

Sn

sample mean

sample variance

Statistic - function of random variables

- typically used to estimate parameters

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Statistics are random variables and have their own distributions

Key ConceptKey ConceptKey ConceptKey Concept

- called sampling distributions

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Sampling Distribution of the Sample Mean

Sampling Distribution of the Sample Mean

IF:• Data are Normally distributed

• Observations are independent

Then:

/

YZ

n

The Sample Mean has aNormal Probability Distribution with

-- Mean

-- Standard Errorn

has a standard normal distribution

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Suppose = 50 and = 10 for a normal population and suppose further that a random sample of size n = 25 is taken.

Find: [ 45]P X

[ 47.5]P X

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Central Limit TheoremCentral Limit TheoremIF:• Independent Observations

• Sample Size is Sufficiently Large

Then:

( ) The Sample Mean is with

Y Approximately Normally Distributed

-- Mean

-- Standard Error / n

/

YZ

n

has a an approximateapproximateStandard Normal distribution

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Suppose = 50 and = 10 for a non-normal population and suppose further that a random sample of size n = 50 is taken.

Find: [ 52]P X

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Distribution of Sample Mean - Unknown

Distribution of Sample Mean - Unknown

/

Yt

S n

IF:IF:• Data Values are Normally Distributed

• Observations are Independent

Then:Then:

has a Student’s tStudent’s t distributionwith n - 1 df

18t-distribution -- Figure 5.16, page 229

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20

t

tNotation

.05,20t

.01,15t

.05z

.025z

z is obtained from bottom (inf.) row of t-table

.9,18t

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(1-)x100% Confidence Intervals for (1-)x100% Confidence Intervals for Setting:• Data are Normally Distributed

• Observations are Independent

• We want an interval that probably contains the population mean

Case 1: known

/ 2 / 2X z X zn n

Case 2: unknown

/ 2 / 2s s

X t X tn n

( 1n df )

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CI Example CI Example An insurance company is concerned about the number and magnitude of hail damage claims it received this year. A random sample 20 of the thousands of claims it received this year showed an average claim amount of $6,500 and a standard deviation of $1,500. (You can assume that claims have a normal distribution.

Find a a 95% confidence interval on the mean claim damage amount. 

Suppose that company actuaries believe the company does not need to increase insurance rates for hail damage if the mean claim damage amount is no greater than $7,000. Use the above information to make a recommendation regarding whether rates should be raised.  

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Last time we found 95% CI to be:

($5798, $7202)

What does this mean?

“There is a .95 probability that the population mean ( is between $5798 and $7202”?

Not exactly.

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Interpretation of 95% Confidence Interval

Interpretation of 95% Confidence Interval

100 different 95% CI plotted in the case for which true mean is 80

i.e. about 95% of these confidence intervals should “cover” the true mean

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Last time we found 95% CI to be:

($5798, $7202)

What does this mean?

“There is a .95 probability that the population mean () is between $5798 and $7202”?

Not exactly.

“About 95% of confidence intervals obtained in this manner will cover the true mean.”

A better statement;

We say:“we are 95% confident that the mean falls in the interval … ”

Concern has been mounting that SAT scores are falling.

• 3 years ago -- National AVG = 955

• Random Sample of 200 graduating high school students this year (sample average = 935) (each year the standard deviation is about 100)

Question: Have SAT scores dropped ?

Procedure: Determine how “extreme” or “rare” our sample AVG of 935 is if population AVG really is 955.

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If Population average = 955, what is the probability of getting a sample average (from a sample of size 200) that is less than or equal to 935?

We must decide:

• The sample came from population with population AVG = 955 and just by chance the sample AVG is “small.”

OR

• We are not willing to believe that the pop. AVG this year is really 955. (Conclude SAT scores have fallen.)

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Statistical HypothesisStatistical Hypothesis- statement about the parameters of one or more populations

Null Hypothesis (Null Hypothesis (HH00)- hypothesis to be “tested”

(standard, traditional, claimed, etc.)- hypothesis of no change, effect, or difference

(usually what the investigator wants to disprove)

 

Alternative Hypothesis (Alternative Hypothesis (HHaa)- null is not correct

(usually what the hypothesis the investigator suspects or wants to show)

Hypothesis Testing TerminologyHypothesis Testing TerminologyHypothesis Testing TerminologyHypothesis Testing Terminology

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Basic Hypothesis Testing Question:Basic Hypothesis Testing Question:

Do the Data provide sufficient evidence to refute the Null Hypothesis?

Test StatisticTest Statistic - measures how far the observed statistic is from the hypothesized parameter (under H0)

Example: H0: = 50

50

/

Xt

s n

Test statistic:

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Critical Region (Rejection Region)Critical Region (Rejection Region)- region of test statistic that leads to

rejection of null (i.e. t > c, etc.)

Critical ValueCritical Value- endpoint of critical region

Significance LevelSignificance Level - probability that the test statistic will

be in the critical region if null is true - probability of rejecting H0 when it is true

Hypothesis Testing (cont.)Hypothesis Testing (cont.)Hypothesis Testing (cont.)Hypothesis Testing (cont.)

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Types of HypothesesTypes of Hypotheses

0 0

0

:

:a

H

H

0 0

0

:

:a

H

H

One-Sided TestsOne-Sided Tests

Two-sided TestsTwo-sided Tests

0 0

0

:

:a

H

H

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Rejection Regions for One- and Two-Sided Alternatives

Rejection Regions for One- and Two-Sided Alternatives

-t

Critical Value

0 0 0 : : vs. aH H

0 0 0 : : vs. aH H

0 0 0 : : vs. aH H

0H t t Reject if

0H t tReject if

0 / 2|H t tReject if |

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A Standard Hypothesis Test Write-up

A Standard Hypothesis Test Write-up

1. State the null and alternative

2. Give significance level, test statistic,and the rejection region

3. Show calculations

4. State the conclusion- statistical decision

- give conclusion in language of the problem

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Hypothesis Testing Example 1Hypothesis Testing Example 1A solar cell requires a special crystal. If properly manufactured, the mean weight of these crystals is .4g. Suppose that 25 crystals are selected at random from a batch of crystals and it is calculated that for these crystals, the average is .41g with a standard deviation of .02g. At the = .01 level of significance, can we conclude that the batch is bad?

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Hypothesis Testing Example 2Hypothesis Testing Example 2A box of detergent is designed to weigh on the average 3.25 lbs per box. A random sample of 18 boxes taken from the production line on a single day has a sample average of 3.238 lbs and a standard deviation of 0.037 lbs. Test whether the boxes seem to be underfilled.

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Actual Situation

Errors in Hypothesis TestingErrors in Hypothesis Testing

Null is True Null is False

Do NotReject Ho

Reject Ho

Conclusion

CorrectDecision

CorrectDecision

( )

( )( 1 - )

( 1 - )(Power)

Type IIError

Type IError

Power

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Note:Note: There are many ways that H0 can be false

Example:Example:H0: 50This null hypothesis is “false” if:

(a) (b) (c)

If (c) is the actual situation, then the “power” of the test will probably be large

In the case of (a), the “power” will likely be small

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p-Value p-Value

(observed value of t)

-2.39

p-value

0 0 0 : : vs. aH H

0H t t Reject if

Suppose t = - 2.39 is observed from data for test above

Note: “Large negative values” of t make us believe alternative is true

the probability of an observation as extreme or more extreme than the one observed when the null is true

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Note:Note:-- if p-value is less than or equal to then we reject null at the significance level 

-- the p-value is the smallest level of significance at which the null hypothesis would be rejected

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Find the p-values for Examples 1 and 2