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## IENG 486 - Lecture 06

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Assignment:

- Preparation:
- Print Hypothesis Test Tables from Materials page
- Have this available in class …or exam!
- Reading:
- Chapter 4:
- 4.1.1 through 4.3.4; (skip 4.3.5); 4.3.6 through 4.4.3; (skip rest)
- HW 2:
- CH 4: # 1a,b; 5a,c; 9a,c,f; 11a,b,d,g; 17a,b; 18, 21a,c; 22* *uses Fig.4.7, p. 126

IENG 486 Statistical Quality & Process Control

Relationship with Hypothesis Tests

- Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield?
- Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail.
- To get 1 – .0025 = 99.75% yield before the right tail requires the upper specification limit to be set at

+ 2.81.

TM 720: Statistical Process Control

Relationship with Hypothesis Tests

- Assuming that our process is Normally Distributed and centered at the mean, how far apart should our specification limits be to obtain 99. 5% yield?
- Proportion defective will be 1 – .995 = .005, and if the process is centered, half of those defectives will occur on the right tail (.0025), and half on the left tail.
- To get 1 – .0025 = 99.75% yield before the right tail requires the upper specification limit to be set at + 2.81.
- By symmetry, the remaining .25% defective should occur at the left side, with the lower specification limit set at – 2.81
- If we specify our process in this manner and made a lot of parts, we would only produce bad parts .5% of the time.

IENG 486 Statistical Quality & Process Control

Hypothesis Tests

- AnHypothesis is a guess about a situation, that can be tested and can be either true or false.
- The Null Hypothesis has a symbol H0, and is always the default situation that must be proven wrongbeyond a reasonable doubt.
- The Alternative Hypothesis is denoted by the symbol HA and can be thought of as the opposite of the Null Hypothesis - it can also be either true or false, but it is always false when H0 is true and vice-versa.

IENG 486 Statistical Quality & Process Control

Hypothesis Testing Errors

- Type I Errorsoccur when a test statistic leads us to reject the Null Hypothesis when the Null Hypothesis is true in reality.
- The chance of making a Type I Error is estimated by the parameter (or level of significance), which quantifies the reasonable doubt.
- Type II Errors occur when a test statistic leads us to fail to reject the Null Hypothesis when the Null Hypothesis is actually false in reality.
- The probability of making a Type II Error is estimated by the parameter .

IENG 486 Statistical Quality & Process Control

Testing Example

- Single Sample, Two-Sided t-Test:
- H0: µ = µ0 versus HA: µ ¹ µ0
- Test Statistic:
- Critical Region: reject H0 if |t| > t/2,n-1
- P-Value: 2 x P(X ³ |t|), where the random variable X has a t-distribution with n_1 degrees of freedom

IENG 486 Statistical Quality & Process Control

Hypothesis Testing

H0: m = m0 versus HA: mm0

P-value = P(X£-|t|) + P(X³|t|)

tn-1 distribution

Critical Region: if our test statistic value falls into the region (shown in orange), we rejectH0and accept HA

0

|t|

-|t|

IENG 486 Statistical Quality & Process Control

2

2

θ0

θ

θ

θ0

θ0

θ

θ

θ0

0

0

Types of Hypothesis TestsHypothesis Tests & Rejection Criteria

0

One-Sided Test

Statistic < Rejection Criterion

H0: θ≥θ0

HA: θ< θ0

Two-Sided Test

Statistic < -½ Rejection Criterion

or

Statistic > +½ Rejection Criterion

H0: θ = θ0

HA: θ≠θ0

One-Sided Test

Statistic > Rejection Criterion

H0: θ≤θ0

HA: θ>θ0

IENG 486 Statistical Quality & Process Control

Hypothesis Testing Steps

- State the null hypothesis (H0) from one of the alternatives:

that the test statistic q = q0 ,q ≥ q0 , or q ≤ q0 .

- Choose the alternative hypothesis (HA) from the alternatives:

q ¹ q0 ,q < q0,or q > q0 . (Respectively!)

- Choose a significance level of the test (a).
- Select the appropriate test statistic and establish a critical region (q0).

(If the decision is to be based on a P-value, it is not necessary to have a critical region)

- Compute the value of the test statistic () from the sample data.
- Decision: Reject H0 if the test statistic has a value in the critical region (or if the computed P-value is less than or equal to the desired significance level a); otherwise, do not reject H0.

IENG 486 Statistical Quality & Process Control

H0 is True

H0 is False

H0 is True

CORRECT

Type II Error ()

Test Conclusion

H0 is False

Type I Error ()

CORRECT

Hypothesis Testing- Significance Level of a Hypothesis Test:A hypothesis test with a significance level or size rejects the null hypothesis H0 if a p-value smaller than is obtained, and accepts the null hypothesis H0 if a p-value larger than is obtained. In this case, the probability of a Type I error (the probability of rejecting the null hypothesis when it is true) is equal to .

IENG 486 Statistical Quality & Process Control

Hypothesis Testing

- P-Value:One way to think of the P-value for a particular H0 is: given the observed data set, what is the probability of obtaining this data set or worse when the null hypothesis is true. A “worse” data set is one which is less similar to the distribution for the null hypothesis.

P-Value

1

0

0.10

0.01

H0not plausible

Intermediate

area

H0plausible

IENG 486 Statistical Quality & Process Control

Statistics and Sampling

- Objective of statistical inference:
- Draw conclusions/make decisions about a population based on a sample selected from the population
- Random sample – a sample, x1, x2, …, xn , selected so that observations are independently and identically distributed (iid).
- Statistic – function of the sample data
- Quantities computed from observations in sample and used to make statistical inferences
- e.g. measures central tendency

IENG 486 Statistical Quality & Process Control

Sampling Distribution

- Sampling Distribution – Probability distribution of a statistic
- If we know the distribution of the population from which sample was taken,
- we can often determine the distribution of various statistics computed from a sample

IENG 486 Statistical Quality & Process Control

e.g. Sampling Distribution of the Average from the Normal Distribution

- Take a random sample, x1, x2, …, xn, from a normal population with mean and standard deviation , i.e.,
- Compute the sample average
- Then will be normally distributed with mean and std deviation
- That is

IENG 486 Statistical Quality & Process Control

Ex. Sampling Distribution of x

- When a process is operating properly, the mean density of a liquid is 10 with standard deviation 5. Five observations are taken and the average density is 15.
- What is the distribution of the sample average?
- r.v. x = density of liquidAns: since the samples come from a normal distribution, and are added together in the process of computing the mean:

IENG 486 Statistical Quality & Process Control

Ex. Sampling Distribution of x (cont'd)

- What is the probability the sample average is greater than 15?
- Would you conclude the process is operating properly?

IENG 486 Statistical Quality & Process Control

Ex. Sampling Distribution of x (cont'd)

- What is the probability the sample average is greater than 15?
- Would you conclude the process is operating properly?

IENG 486 Statistical Quality & Process Control

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