Ex St 801Statistical Methods

Inference about aSingle Population

Mean

TYPES OF STATISTICAL INFERENCE

• ESTIMATION Answers the question

What is the value of the population parameter?

• HYPOTHESIS TESTING Answers the question

Is the parameter equal to a specific value?

PARAMETER ESTIMATION

TERMS AND DEFINITIONS

• An ESTIMATOR is a rule that tells us how to calculate the estimate based on sample information.

• A POINT ESTIMATOR of a parameter is a rule that estimates the parameter with a single value.

TERMS AND DEFINITIONS(contd.)

• An ESTIMATE is a number calculated using an estimator.

• An estimator is called UNBIASED if its average value is equal to the parameter being estimated. Otherwise, the estimator is called BIASED.

BIASED AND LARGE STANDARDERROR

BIASED AND SMALL STANDARD ERROR

UNBIASED AND LARGE STANDARD ERROR

UNBIASED AND SMALL STANDARD ERROR

SOME EXAMPLES OF UNBIASED ESTIMATORS

• The SAMPLE MEAN is ALWAYS an unbiased estimator of the population mean.

• The SAMPLE MEDIAN is an unbiased estimator of the population mean if the distribution being sampled is symmetric about the population mean.

SOME EXAMPLES OF UNBIASED ESTIMATORS (contd.)

• The SAMPLE VARIANCE is ALWAYS an unbiased estimator of the population variance.

SOME EXAMPLES OF BIASED ESTIMATORS

• The SAMPLE MEDIAN is a biased estimator of the population mean if the distribution being sampled is not symmetric about the population mean.

• The SAMPLE VARIANCE is a biased estimator of the population variance if we use n in the denominator of the formula rather than n-1.

TERMS AND DEFINITIONS

• An INTERVAL ESTIMATOR of a parameter is a rule that provides two numbers that form an interval that is likely to contain the parameter. The interval is called a CONFIDENCE INTERVAL.

(Smaller Value, Larger Value)

TERMS AND DEFINITIONS (contd.)

• The smaller value is called the LOWER CONFIDENCE LIMIT (LCL) and the larger value is called the UPPER CONFIDENCE LIMIT (UCL).

• The CONFIDENCE COEFFICIENT is the probability that a confidence interval will capture the parameter being estimated.

WHAT MIGHT BE EXPECTED TO HAPPEN IF 90% CONFIDENCE

INTERVALS ARE CALCULATED FOR THE MEAN

SAMPLE 10SAMPLE 9SAMPLE 8SAMPLE 7SAMPLE 6SAMPLE 5SAMPLE 4SAMPLE 3SAMPLE 2SAMPLE 1

CRITICAL VALUE Z

Z.025 = 1.96

LARGE SAMPLE CONFIDENCE INTERVAL FOR A SINGLE

POPULATION MEAN

FACTORS AFFECTING THE “MARGIN OF ERROR”

• The value of the standard deviation (s).

• The value of the confidence coefficient (1-).

• The value of the sample size (n).

DETERMINING THE SAMPLE SIZE FOR A CONFIDENCE INTERVAL

NORMAL AND T-DISTRIBUTIONS

Z

-4 -3 -2 -1 0 1 2 3 4

NORMAL

T-DISTRIBUTION

SMALL SAMPLE CONFIDENCE INTERVAL FOR A SINGLE

POPULATION MEAN

HYPOTHESIS TESTING

94

10298

10495

X_

S2

STATISTICALEVIDENCE

STATING THE HYPOTHESES

• The NULL HYPOTHESIS (H0)

Is the statement that is assumed to be true unless sufficient evidence is gathered to reject the hypothesis.

• The ALTERNATIVE HYPOTHESIS (HA)

Is the statement that one wishes to support as being true. This is done by gathering evidence against the null hypothesis.

TYPES OF ERRORS

• A TYPE I ERRORoccurs by rejecting the null hypothesis when it is true.

• is the probability of making a Type I Error.

• A TYPE II ERRORoccurs by failing to reject the null hypothesis when it is false.

• ß is the probability of making a Type II Error.

HYPOTHESIS TESTING TABLE

Actual Situation

Decision H0 Is True H0 Is FalseReject H0 Type I Error

(Prob. )Good

Decision(Prob. 1-)

Fail toReject H0

GoodDecision

(Prob. 1-)

Type IIError

(Prob. )

OZONE DEPLETION EXAMPLE

Many scientists have become concerned that there has been a depletion in the ozone level.

The mean concentration should be 100 PPM.

Suppose we wanted to test for a depletion in the mean ozone level.

What would be the null and alternative hypotheses?

H0: = 100 HA: < 100

OZONE DEPLETION EXAMPLE

What would be the TYPE I ERROR?

To conclude there was a depletion in the mean ozone level, when actually there is no depletion.

What would be the TYPE II ERROR?

To conclude there was no depletion in the mean ozone level, when actually there has been a depletion.

OZONE DEPLETION PROBLEM

If you were given a choice between the following probabilities, which would you choose? Why?

A. = .025 AND = .10, ORB. = .10 AND = .025.

BUNGEE JUMPING EXAMPLE

Suppose you were considering going bungee jumping, but would only go if the mean breaking strength of the cord was greater than 250 lbs.

What would be your null and alternative hypotheses?

H0: = 250HA: > 250

BUNGEE JUMPING EXAMPLE

What would be the TYPE I ERROR?

To conclude that it is safe to go bungee jumping when it is actually dangerous.

What would be the TYPE II ERROR?

To conclude that it is dangerous to go bungee jumping when it is actually safe.

BUNGEE JUMPING EXAMPLE

If you were given a choice between the following probabilities, which would you choose? Why?

A. = .025 AND = .10, ORB. = .10 AND = .025.

HYPOTHESIS TESTING METHODS

• ORIGINAL SCALE

• STANDARDIZED SCALE

• P-VALUE

PROCEDURE FOR ORIGINAL AND STANDARDIZED SCALE METHODS

• State the alternative hypothesis (HA) and the null hypothesis (H0).

• State the TYPE I and II ERRORS.

• State level of significance (maximum acceptable ).

• Determine the rejection region.• Compute the test statistic.

• Compare the test statistic and rejection region, then make a decision.

• Draw a conclusion.

PROCEDURE FOR THE P-VALUE METHOD

• State the alternative hypothesis (HA) and the null hypothesis (H0).

• State the TYPE I and II ERRORS.

• State level of significance (maximum acceptable ).• Compute the test statistic.• Calculate the p-value.

• Compare the p-value with the level of significance, then make a decision.

• Draw a conclusion.

Y

92 94 96 98 100 102 104 106 108

ORIGINAL SCALE METHODSampling distribution if the null hypothesis is true

Z

-4 -3 -2 -1 0 1 2 3 4

STANDARDIZED SCALE METHODSampling distribution if the null hypothesis is true

Z

-4 -3 -2 -1 0 1 2 3 4

P-VALUE METHODSampling distribution if the null hypothesis is true

HYPOTHESES

LOWERTAIL

UPPERTAIL

TWOTAIL

TEST STATISTIC

LARGE SAMPLE SMALL SAMPLE

REJECTION REGION

• LARGE SAMPLE Standard

normal distribution

• SMALL SAMPLE t-distribution

with df=n-1

TYPE II ERROR AND THE POWER OF A HYPOTHESIS TEST

• The TYPE II ERROR occurs by failing to reject a null hypothesis that is false.

• is the probability of making a TYPE II ERROR.

• The power of a hypothesis test is the probability of not making a TYPE II ERROR. (1-)

DISTRIBUTIONWHEN H0 IS TRUE

Y

90 92 94 96 98 100 102 104 106 108

DISTRIBUTIONWHEN HA IS TRUE