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6.2 Large Sample Significance Tests for a MeanPowerPoint Presentation

6.2 Large Sample Significance Tests for a Mean

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6.2 Large Sample Significance Tests for a Mean

“The reason students have trouble understanding hypothesis testing may be that they are trying to think.” Deming

- In a law case, there are 2 possibilities for the truth—innocent or guilty
- Evidence is gathered to decide whether to convict the defendant. The defendant is considered innocent unless “proven” to be guilty “beyond a reasonable doubt.” Just because a defendant is not found to be guilty doesn’t prove the defendant is innocent. If there is not much evidence one way or the other the defendant is not found to be guilty.

For Statistical Hypothesis truth—innocent or guiltyTesting, We have 2 possibilities to choose from,

- H0=Null hypothesis (innocent)
Held on to unless there is sufficient evidence to the contrary

- Ha=Alternative hypothesis (guilty)
We reject H0 in favor of Haif there is enough evidence favoring Ha

Tests truth—innocent or guiltyof Hypotheses

- Distribution(s) or population(s):
- Parameter(s) such as mean and variance
- Assertion or conjecture about the population(s) – statistical hypotheses
1. About parameter(s): means or variances

2. About the type of populations: normal ,

binomial, or …

Example truth—innocent or guilty

- Is a coin balanced?
This is the same as to ask if p=0.5

- Is the average lifetime of a light bulb equal to 1000 hours?
The assertion is μ=1000

Null Hypotheses and Alternatives truth—innocent or guilty

- We call the above two assertions
Null Hypotheses

Notation: H0: p=0.5 and H0:μ=1000

If we reject the above null hypotheses,

the appropriate conclusions we arrive are

called alternative hypotheses

Ha: p0.5 Ha: μ1000

Null Hypothesis vs Alternative truth—innocent or guilty

- H0: p=0.5 vsHa: p0.5
- H0:μ=1000vsHa: μ1000
- It is possible for you to specify other alternatives
- Ha: p>0.5 or Ha: p<0.5
- Ha: μ>1000 or Ha: μ<1000

Significance Testing /Hypothesis Testing truth—innocent or guilty

- A company claims its light bulbs last on average 1000 hours. We are going to test that claim.
- We might take the null and alternative hypotheses to be
H0:μ=1000vsHa: μ1000

or may be

H0:μ=1000vsHa: μ<1000

Mistakes or errors: truth—innocent or guilty

- Law case—convict an innocent defendant; or fail to convict a guilty defendant.
- The law system is set up so that the chance of convicting an innocent person is small. Innocent until “proven guilty” beyond a reasonable doubt.

Two Types of Errors in statistical testing truth—innocent or guilty

- Type I error -- reject H0 when it is true (convict innocent person)
- Type II error -- accept H0 when it is not true (find guilty person innocent)

Statistical hypotheses are set up to truth—innocent or guilty

- Control type I error
=P(type I error)

=P(reject H0 when H0 true)

(a small number)

- Minimize type II error
=P(type II error)

=P(accept H0 when H0 false)

Control Types of Errors truth—innocent or guilty

- In practice, is set at some small values, usually 0.05
- If you want to control at some small values, you need to figure out how large a sample size (n) is required to have a small also.
- 1- is called the power of the test
- 1- =Power=P(reject H0 when H0 false)

Example truth—innocent or guilty

- X=breaking strength of a fish line, normal distributed with σ=0.10.
- Claim: mean is =10
- H0: =10 vs HA: 10
A random sample of size n=10 is taken,

and sample mean is calculated

- Accept H0 if
- Type I error?
- Type II error when =10.10?

Solution truth—innocent or guilty

- Type I error=P(reject H0 when =10)

Solution truth—innocent or guilty

- Type II error=P(accept H0 when H0 false)
- Power=1-0.0571=0.9429

Tests truth—innocent or guiltyconcerning Means

- 5 steps to set up a statistical hypothesis test

Steps (p. 350) truth—innocent or guilty

- Steps 1 and 2: State the null and alternative hypothesis.
- Step 3: State the test criteria. That is, give the formula for the test statistic (plugging in only the hypothesized value from the null hypothesis but not any sample information) and the reference distribution. Then state in general terms what observed values of the test statistic constitute evidence against the null hypothesis.

- Step 4: truth—innocent or guilty Show the sample based calculations.
- Step 5: Report an observed level of significance, p-value, and (to the extent possible) state its implications in the context of the real engineering problem.

- Interpret the Results truth—innocent or guilty
If the p-value is small,

This type of data are unlikely if H0 is true.

The fact that we are looking at this data set right now indicates that H0 is likely not true.

The null hypothesis looks bad reject H0 .

- The p-value is the probability of a result at least as extreme (away from what the null hypothesis would have predicted) if in fact the null hypothesis is true.
- So if the data are extremely unlikely when the null hypothesis is true,
- The p-value is small and
- The null hypothesis looks bad.

- P-values and hypothesis testing are widely used. extreme (away from what the null hypothesis would have predicted) if in fact the null hypothesis is true.
- However, in my opinion and some others’ opinions (see author’s comments later in the chapter), more often than not, such significance tests are not useful summaries. See Deming quote earlier.
- Generally, confidence intervals are more useful summaries.

Exercise extreme (away from what the null hypothesis would have predicted) if in fact the null hypothesis is true.

Given that n=25, s=100, and sample mean is 1050,

1. Test the hypotheses H0: m=1000 vs HA: m<1000 at level a=0.05.

2. Test the hypotheses H0: m=1000 vs HA: m≠1000 at level a=0.05.

Solution extreme (away from what the null hypothesis would have predicted) if in fact the null hypothesis is true.

1.

2.

More evidence against H0 is smaller values of z

Evidence against H0 is z values away from 0 in either direction

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