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6.2 Large Sample Significance Tests for a Mean

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“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.

- 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

- 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 …

- 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

- 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

- 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

- 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

- 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.

- 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)

- 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)

- 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)

- 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?

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

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

- 5 steps to set up a statistical hypothesis test

- 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: 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
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.
- 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.

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.

1.

2.

More evidence against H0 is smaller values of z

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