04/20/23IENG 486 Statistical Quality & Process

Control 1

IENG 486 - Lecture 06

Hypothesis Testing & Excel Lab

04/20/23 IENG 486 Statistical Quality & Process Control 2

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

04/20/23 TM 720: Statistical Process Control 3

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.

04/20/23 TM 720: Statistical Process Control 4

04/20/23 IENG 486 Statistical Quality & Process Control 5

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.

04/20/23 IENG 486 Statistical Quality & Process Control 6

Hypothesis Tests

An Hypothesis 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 wrong beyond 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.

04/20/23 IENG 486 Statistical Quality & Process Control 7

Hypothesis Testing Errors

Type I Errors occur 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 .

04/20/23 IENG 486 Statistical Quality & Process Control 8

Testing Example

Single Sample, Two-Sided t-Test: H0: µ = µ0 versus HA: µ µ0

Test Statistic:

Critical Region: reject H0 if |t| > t/2,n-1

P-Value: 2 x P(X |t|), where the random variable X has a t-distribution with n _ 1 degrees of freedom

,0

s

xnt

04/20/23 IENG 486 Statistical Quality & Process Control 9

Hypothesis Testing

H0: = 0 versus HA: 0

tn-1 distribution

0-|t| |t|

P-value = P(X-|t|) + P(X|t|)

Critical Region: if our test statistic value falls into the region (shown in orange), we reject H0 and accept HA

04/20/23 IENG 486 Statistical Quality & Process Control 10

2

2

θ0θ

One-Sided TestStatistic < Rejection Criterion

H0: θ ≥ θ0

HA: θ < θ0

Types of Hypothesis Tests

Hypothesis Tests & Rejection Criteria

0 0

Two-Sided TestStatistic < -½ Rejection Criterion

orStatistic > +½ Rejection Criterion

H0: θ = θ0

HA: θ ≠ θ0

One-Sided TestStatistic > Rejection Criterion

H0: θ ≤ θ0

HA: θ > θ0

θθ0θ0θ θθ00

04/20/23 IENG 486 Statistical Quality & Process Control 11

Hypothesis Testing Steps

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

that the test statistic , ≥, or ≤.

2. Choose the alternative hypothesis (HA) from the alternatives:

, , or . (Respectively!)

Choose a significance level of the test (.

Select the appropriate test statistic and establish a critical region ().

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

region)

1. Compute the value of the test statistic () from the sample data.

2. 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 ); otherwise, do not reject H0.

04/20/23 IENG 486 Statistical Quality & Process Control 12

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 .

True Situation

Tes

t C

on

clu

sio

n

CORRECTType I Error

()H0 is False

Type II Error ()

CORRECTH0 is True

H0 is FalseH0 is True

04/20/23 IENG 486 Statistical Quality & Process Control 13

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.

H0

not plausibleH0

plausibleIntermediate

area

0 0.01 0.10 1

P-Value

04/20/23 IENG 486 Statistical Quality & Process Control 14

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

n

iixn

x1

1

04/20/23 IENG 486 Statistical Quality & Process Control 15

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

04/20/23 IENG 486 Statistical Quality & Process Control 16

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

x

x

n

n

NNxx

,),(~

),(~ Nx

04/20/23 IENG 486 Statistical Quality & Process Control 17

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 liquid

Ans: since the samples come from a normal distribution, and are added together in the process of computing the mean:

5

5,10~ Nx

04/20/23 IENG 486 Statistical Quality & Process Control 18

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?

?)24.2()(

24.236.2

5

5

51015

0

00

z

n

xz

04/20/23 IENG 486 Statistical Quality & Process Control 19

04/20/23 IENG 486 Statistical Quality & Process Control 20

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?

%3.101255.098745.01

98745.0)24.2()(

24.236.2

5

5

51015

0

00

or

z

n

xz