powerpoint presentation · 2017-09-22 · 4-3 hypothesis testing 4-3.1 statistical hypotheses test...

4-1 Statistical Inference

• The field of statistical inference consists of those

methods used to make decisions or draw

conclusions about a population.

•These methods utilize the information contained in

a random sample from the population in drawing

conclusions.

4-1 Statistical Inference

4-2 Point Estimation

4-3 Hypothesis Testing

We like to think of statistical hypothesis testing as the

data analysis stage of a comparative experiment, in which the engineer is interested, for example, in

comparing the mean of a population to a specified

value (e.g. mean pull strength).

4-3.1 Statistical Hypotheses



Two-sided Alternative Hypothesis

One-sided Alternative Hypotheses



Test of a Hypothesis

• A procedure leading to a decision about a particular

hypothesis

• Hypothesis-testing procedures rely on using the information

in a random sample from the population of interest.

• If this information is consistent with the hypothesis, then we

will conclude that the hypothesis is true; if this information is

inconsistent with the hypothesis, we will conclude that the

hypothesis is false.


4-3.2 Testing Statistical Hypotheses



Sometimes the type I error probability is called the

significance level, or the -error, or the size of the test.



Reduce α

1. push critical regions further

2. Increasing sample size

increasing Z value



error when μ =52 and n=16.

Increasing the sample size results

in a decrease in β.



error when μ =50.5 and n=10.

β increases as the true value of μ

approaches the hypothesized value.



• The power is computed as 1 - b, and power can be interpreted as

the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power

properties.

• For example, consider the propellant burning rate problem when

we are testing H 0 : m = 50 centimeters per second against H 1 : m not equal 50 centimeters per second . Suppose that the true value of the

mean is m = 52. When n = 10, we found that b = 0.2643, so the power of this test is 1 - b = 1 - 0.2643 = 0.7357 when m = 52.


4-3.3 P-Values in Hypothesis Testing

P-value carries info about the weight of evidence against H0.

The smaller the p-value, the greater the evidence against H0.


4-3.4 One-Sided and Two-Sided Hypotheses

Two-Sided Test:

One-Sided Tests:


4-3.5 General Procedure for Hypothesis Testing

OPTIONS NODATE NONUMBER;

DATA EX415;

mu_h0=12; mu_h1=11.25; sigma=0.5; n=4; alpha=0.05;

xbar=11.7; sem=sigma/sqrt(n); /* sem = mean standard error */

Z=(xbar-mu_h0)/sem; z_beta= probit(alpha);

xbar_beta=mu_h0+z_beta*sem;

z_beta=(xbar_beta-mu_h1)/sem;

PVAL=PROBNORM(Z);

beta=1-probnorm(z_beta);

power=1-beta;

PROC PRINT;

VAR PVAL beta power;

TITLE '4-15 (page 168)';

RUN; QUIT;

4-15 (page 168)

OBS PVAL beta power

1 0.11507 0.087685 0.91231


4-4 Inference on the Mean of a Population,

Variance Known

Assumptions


Variance Known

4-4.1 Hypothesis Testing on the Mean

We wish to test:

The test statistic is:


Variance Known



Variance Known


Reject H0 if the observed value of the test statistic z0 is

either:

or

Fail to reject H0 if


Variance Known



Variance KnownEx. 4-3

X – burning rate

μ = 50, s = 2 a = 0.05, n = 25, ҧ𝑥 = 51.3

i) Hypothesis

H0: μ = 50

H1: μ ≠ 50

ii) Test Statistic

z0 =ഥ𝑋 − 𝜇0

ൗ𝜎

𝑛

=51.3 −50

ൗ2 25= 1.3/0.4 = 3.25

iii) p-value = 2[1 – Φ(3.25)] = 0.0012

Critical Region (CR) 또는 Region of Rejection (ROR)z0 > z0.025= 1.96 or z0< - z0.025= -1.96

iv) Conclusion

p-value=0.0012 값이유의수준 a = 0.05보다작기때문에(또는, 통계량 z0=3.25 값이기각역 (z0 = z0.025> 1.96) 에포함됨으로) H0를기각함. 즉,평균 burning rate이 50m/sec 이아니다라고결론내림.


Variance Known


DATA EX43;

MU=50; SD=2; N=25; ALPHA=0.05; XBAR=51.3;

SEM=SD/SQRT(N); /*MEAN STANDARD ERROR*/

Z=(XBAR-MU)/SEM;

PVAL=2*(1-PROBNORM(Z));

PROC PRINT;

VAR Z PVAL ALPHA;

TITLE 'TEST OF MU=50 VS NOT=50';

RUN; QUIT;

TEST OF MU=50 VS NOT=50

OBS Z PVAL ALPHA

1 3.25 .001154050 0.05


Variance Known

4-4.2 Type II Error and Choice of Sample Size

Finding The Probability of Type II Error b


Variance Known


Sample Size Formulas


Variance Known



Variance Known

4-4.3 Large Sample Test

In general, if n 30, the sample variance s2

will be close to σ2 for most samples, and so s

can be substituted for σ in the test procedures

with little harmful effect.


Variance Known

4-4.4 Some Practical Comments on Hypothesis

Testing

The Seven-Step Procedure

Only three steps are really required:


Variance Known

4-4.4 Some Practical Comments on Hypothesis

Testing

Statistical versus Practical Significance


Variance Known

4-4.5 Confidence Interval on the Mean

Two-sided confidence interval:

One-sided confidence intervals:

Confidence coefficient:


Variance Known



Variance Known


Relationship between Tests of Hypotheses and

Confidence Intervals

If [l,u] is a 100(1 - ) percent confidence interval for the

parameter, then the test of significance level of the

hypothesis

will lead to rejection of H0 if and only if the hypothesized

value is not in the 100(1 - ) percent confidence interval

[l, u].


Variance Known


Confidence Level and Precision of Estimation

The length of the two-sided 95% confidence interval is

whereas the length of the two-sided 99% confidence

interval is


Variance Known


Choice of Sample Size


Variance Known


One-Sided Confidence Bounds


Variance Known

4-4.6 General Method for Deriving a Confidence

Interval


Variance Unknown



Variance Unknown


Calculating the P-value


Variance Unknown



Variance Unknown


Fixed Significance Level Approach


Variance Unknown



Variance Unknown

Ex. 4-7ത𝑋 – mean coefficient of restitution n = 15

i) Hypothesis

H0: μ = 0.82

H1: μ > 0.82

ii) Normality Check!

iii) Test Statistic

t0 =ഥ𝑋 − 𝜇0

ൗ𝑠

𝑛

=0.83725 −0.82

ൗ0.02456 15= 2.72

iv) Fixed significance level approach

Critical Region (CR) 또는 Region of Rejection (ROR): t0 > t0.05= 1.761P-value approach

t0= 2.72는 t0.01;14= 2.624 값과 t0.005;14= 2.977 사이에있으므로0.005


Variance Unknown


DATA ex47;

xbar=0.83725; sd=0.02456;

mu=0.82; n=15;alpha=0.05; df=n-1;

t0=(xbar-mu)/(sd/sqrt(n));

pval = 1- PROBT(t0, df); /* PROBT function returns the probability from a t distribution

less than or equal to x */

cv=tinv(1-alpha, df); /* critical value */

/* TINV function returns a quantile from the t distribution */

PROC PRINT;

var t0 pval cv;

title 'Test of mu=0.82 vs mu>0.82';

title2 'Ex. 4-7 in pp193';

RUN; QUIT;


Variance Unknown

OPTIONS NOOVP NODATE NONUMBER;

DATA ex47;

input coeffs @@;

cards;

0.8411 0.8191 0.8182 0.8125 0.8750

0.8580 0.8532 0.8483 0.8276 0.7983

0.8042 0.8730 0.8282 0.8359 0.8660

proc univariate data=ex47 mu0=0.82 plot normal;

var coeffs;

title 'using proc uinivariate for t-test for one population mean';

title2 “Example 4-7 in page 191”;

RUN; QUIT;


Variance Unknown


Variance Unknown

OPTIONS NOOVP NODATE NONUMBER LS=80;

DATA ex47;

input coeffs @@;

cards;

0.8411 0.8191 0.8182 0.8125 0.8750

0.8580 0.8532 0.8483 0.8276 0.7983

0.8042 0.8730 0.8282 0.8359 0.8660

ods graphics on;

proc ttest data=ex47 h0=0.82 sides=u;

/* h0는귀무가설, sides는양측일경우 2, lower 단측검정일경우 L, upper 단측검정일경우 u*/var coeffs;

title 'using t-test for one population mean';

title “Example 4-7 in page 191”;

RUN; ods graphics off;

QUIT;


Variance Unknown


Variance Unknown


Fortunately, this unpleasant task has already been done,

and the results are summarized in a series of graphs in

Appendix A Charts Va, Vb, Vc, and Vd that plot for the

t-test against a parameter d for various sample sizes n.


Variance Unknown


These graphics are called operating characteristic

(or OC) curves. Curves are provided for two-sided

alternatives on Charts Va and Vb. The abscissa scale

factor d on these charts is defined as


Variance Unknown


Chart V (c) in pp. 496


Variance Unknown



Variance Unknown

OPTIONS NOOVP NODATE NONUMBER LS=80;

DATA ex60;

input diam @@;

cards;

8.23 8.31 8.42 8.29 8.19 8.24

8.19 8.29 8.30 8.14 8.32 8.40

ods graphics on;

proc univariate plot normal mu0=8.2;

var diam;

title 'univariate for t-test with H0: mu=8.2';

proc ttest data=ex60 h0=8.2;

var diam;

title 't-test for one population mean';

title “Example 4-60 in page 198”;

RUN; ods graphics off;

QUIT;


Variance Unknown

4-6 Inference on the Variance of a

Normal Population

4-6.1 Hypothesis Testing on the Variance of a

Normal Population


Normal Population

4-6.2 Confidence Interval on the Variance of a

Normal Population


Normal Population


DATA ex410;

p_var=0.01; s_var=0.0153;

n=20;alpha=0.05; df=n-1;

chisq=(n-1)*s_var/p_var;

p_value=1-probchi(chisq, df);

cv1=cinv(1-alpha, df); /* critical value */

cv2=cinv(alpha, df);

lcl=sqrt((n-1)*s_var/cv1); /* ucl= upper confidence limit */

PROC PRINT;

var chisq p_value cv1 cv2 lcl;

title 'Test of H0: var=0.01 vs H0: var>0.01';

title2 'Ex. 4-10 in pp202';

RUN; QUIT;

4-7 Inference on Population Proportion

4-7.1 Hypothesis Testing on a Binomial Proportion

We will consider testing:


4-7.1 Hypothesis Testing on a Binomial Proportion


4-7.3 Confidence Interval on a Binomial Proportion


4-7.3 Confidence Interval on a Binomial Proportion

Choice of Sample Size

4-8 Other Interval Estimates for a

Single Sample

4-8.1 Prediction Interval

4-8 Other Interval Estimates for a

Single Sample

4-8.2 Tolerance Intervals for a Normal Distribution

4-10 Testing for Goodness of Fit

• So far, we have assumed the population or probability

distribution for a particular problem is known.

• There are many instances where the underlying

distribution is not known, and we wish to test a particular

distribution.

• Use a goodness-of-fit test procedure based on the chi-

square distribution.

powerpoint presentation · 2017-09-22 · 4-3 hypothesis testing 4-3.1 statistical hypotheses test...

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