A textbook of Computer Based Numerical and Statiscal Techniques part 51 ppsx

12.2 SOME IMPORTANT DEFINITIONSTest of a Statistical Hypothesis: A test of a statistical hypothesis is a two action decision problem after the experimental sample values have been obtain

Negative LCL being taken as zero Also for drawing the control chart we mark the sample No.’s along the horizontal axis and control limits and central line marked along the

vertical axis Finally the number of defects (c i ) per inspection units are marked in the

c-chart

From the control chart, we observe that the point corresponding to 6th inspection unit goes beyond UCL showing a out-of-control situation So for computing revised control limits we omit this unit and use the remaining 15 inspection units for the purpose The average number of defects in the remaining 15 units is

15

1

2.27

15i i 15

=

so the revised limits for c-chart are:

UCL 3 2.27 3 2.27 2.27 4.52 6.79

CL 2.27 LCL 3 2.27 3 2.27 2.27 4.52 2.25 0

c

= =

Negative LCL being as zero

Example 12 A food company puts mango juice into cans advertised as containing 10 ounces of the juice The weights of the juice drained from cans immediately after filling for 20 samples are taken

by a random method (at an interval of every 30 minutes) Each of the samples includes 4 cans The samples are tabulated in the following table The weights in the table are given in units of 0.01 ounces

in excess of 10 ounces For example, the weight of juice drained from the first can of the sample is 10.15 ounces whch is in excess of 10 ounces being 0.15 ounces (10.15 – 10 = 0.15) since the unit in the table is 0.01 ounce, the excess is recorded as 15 units in the table Construct an x — -chart to control the weights of mango juice for the filling.

Weight of each can (4 cans in each sample, x, n = 4) Sample Number

15 15 15 6 16

Sol

Total Sample Sample Weight of each can weight of Mean Range (4 cans in each sample, x, n = 4) 4 cans

4

∑

R = x max – x min

= 263.0 ∑R = 211

UCL= +x A R2

= 13.15 + 0.729 × 10.55 (A2 = 0.729 for n = 4)

2

UCL 20.84095

CL 13.15 LCL

x

x A R

=

= =

= − = 13.15 – 0.729 × 10.55 = 5.46

Sample Number

The values in above computation are expressed in units of 0.01 ounces in excess of 10 ounces The actual value of UCL = 10.2084, and LCL = 10.0546 ounces Since all points are falling with in control limits the process is in a statistical control

Now since standards are not given calculating

1 The mean of the sample mean x is given by

263 13.15

20 20

x

2 The mean of the Range valuesR is given by

211 10.55

20 20

R

3 Trial control limits for x-chart

20

15

10

5

0

FIG 11.27

PROBLEM SET 11.1

1

1 A machine is set to deliver packets of a given weight 10 samples of size 5 each were recorded Data being given below:

Mean x —

Calculate the values for the central line and control limits of mean chart and the range and then comment on the state of control

Given for n=5, A2 =0.58, D3 =0, D4 =2.115

Ans

2 The data below give the number of defective bearing in samples of size 150 Construct

p-chart for these data and state your comment.

Sample no

No of defective

Compute control limits for p-chart. [Ans UCL= 0.08650, CL = 0.03905, LCL = 0] 3

3 A process produces rubber belts in lots of size 2300 Inspection of the last 20 lots reveals the following data:

.

308 342 311 285 327 230 346 221 435 230 407 221 269 131 414 198 331 285 394 456

Lot no

No of

defective belts

Compute control limits for p-chart.

[Ans UCLp = 0.1548, CLp = 0.1335, LCL p = 0.1122]

4

4 The following figure give the number of defectives in 20 samples, each sample containing 2,000 items

Calculate the control limits for fraction defective chart (p-chart) Draw the p-chart

and state the comment [Ans UCLp = 0.178, CLp = 0.154, LCL p = 0.130]

5

5 An inspection of 10 samples of size 400 each from 10 lots revealed the following no

of defective units;

17, 15, 14, 26, 9, 4, 19, 12, 9, 15

Calculate control limits for the no of defective units Plot the control limits and the observations and state whether the process is under control or not

[Ans UCLnp = 25.02679, CLnp = 14, LCL np = 2.97231]

6

6 The following data refer to visual defects found during inspection of the first 10 samples of size 100 each Use them to obtain upper and lower control limits for percentage defective in sample of 100

Sample no

No of defective

[Ans UCLnp = 16.87, CLnp = 8.5, LCLnp = 0.13]

7 The pieces of cloth out of the different rolls of equal length contained the following number of defects:

prepare a c-chart and state whether the process is in a statistical control?

[Ans UCLc = 8.38, CL c = 3.1, LCL c = 0]

8

8 The following table gives the no of defects in carpets manufactured by a company

Carpet serial no

No of defective

Determine the control line and the control limits for c-chart.

9

9 The following data relate to the number or break downs in the rubber covered wires

in 24 successive lengths of 10,000 feet each

Draw c-chart and state your comment.

[Ans UCLc = 13.0715, CLc = 5.875, LCL c = 0 (Process out of control)]

10

10 A drilling machine bores holes with a mean diameter of 0.5230 cm and a standard deviation of 0.0032 cm Calculate the 2-sigma and 3-sigma upper and lower control limits for mean of sample of 4

Ans 2–sigma

UCL 0.5262 cm LCL 0.5198 cm

CL 0.5230 cm

=





=



 =



3-sigma

UCL 0.5278 cm LCL 0.5182 cm

CL 0.5230 cm

=





=



 =



GGG

FACTORS USEFUL IN THE CONSTRUCTION OF CONTROL CHARTS

Sample Factors for Factors for Factors for control limits Factors for Factors for control limit

n A A 1 A 2 C 2 B 1 B 2 B 3 B 4 d 2 D 1 D 2 D 3 D 4

CHAPTER 12

Testing of Hypothesis

Suppose some business concern has an average sale of Rs 10000/- daily estimated over a long period A salesman claims that he will increase the average sales by Rs 700/- a day The concern

is interested in an increased sale no doubt, but how to know whether the claim of the man is justified or not? For this some such a mathematical model for the population of increased sales

is assumed which agrees to the maximum with the practical observations In the example given, let us assume that the claim of the girl about her sales is justified and that the increase in sales

is normally distributed with mean µ = 700 and variance σ2 This assumption is called statistical hypothesis Thereafter the suitability of the assumed model is examined on the basis of the sale observations made This procedure is called testing of hypothesis

A statistical hypothesis is some statement or assertion about a population or equivalently about the probability distribution characterising a population which we want to verify on the basis of information available from a sample If the statistical hypothesis specifies the population completely then it is termed as a simple statistical hypothesis, otherwise it is called a composite statistical hypothesis

Example: If X1, X2, , X n is a random sample of size n from a normal population with mean

µ and variance σ2, then the hypothesis

H0: µ=µ0, σ2 = σ02

is a simple hypothesis, whereas each of the following hypothesis is a composite hypothesis: (1) µ = µ0

(2) σ2 = σ02

(3) µ = µ0, σ2 < σ02

(4) µ < µ0, σ2 > σ02

(5) µ < µ0, σ2 = σ02

(6) µ = µ0, σ2 > σ02

(7) µ > µ0, σ2 = σ02

A hypothesis which does not specify completely ‘r’ parameters of a population is termed as

a composite hypothesis with r degrees of freedom.

492

12.2 SOME IMPORTANT DEFINITIONS

Test of a Statistical Hypothesis: A test of a statistical hypothesis is a two action decision problem after the experimental sample values have been obtained, the two actions being the acceptance

or rejection of the hypothesis under consideration

Null Hypothesis: The statistical hypothesis tested under the assumption that it is true is called null hypothesis It is tested on the basis of the sample observations and is liable to be rejected as well, depending upon the outcome of the statistical test applied There are many occasions where null hypothesis is formulated for the sole purpose of rejecting it

In other words, null hypothesis is statement of zero or no change If the original claim includes equality (< =, =, or > = ), it is the null hypothesis If the original claim does not include equality (<, not equal, >) the null hypothesis is the complement of original claim The null hypothesis always includes the equal sign The decision is based on the null hypothesis The null hypothesis

is denoted by H0

Alternative Hypothesis: Statement which is true if the null hypothesis is false is known as alternative hypothesis In other words a possible or the acceptable alternative to the null hypothesis

called alternative hypothesis, and is denoted by H1 It testing if H0 is rejected, then H1 is accepted The type of test (left, right, or two tail) is based on the alternative hypothesis

Type I Error and Type II Error: When a null hypothesis H0 is tested against an alternative

H1, then there can be either of the following two types of errors:

(a) Rejecting the null hypothesis H0 when actually it is true

(b) Failing to reject the null hypothesis when it is false

These are called errors of Type I and Type II and denoted by α and β respectively The other two possible outcomes of testing are:

(c) Rejection of H0 when it was wrong and

(d) Acceptance of H0 when it was true

H0 is true H1 is true (H0 is false)

Accept H0 Correct decision Type II error ( β )

Accept H1 (reject H0) Type I error ( α ) Correct decision

Alpha: The probability of rejecting H0, when it was true = The probability of committing type I error = The size of type I error = α

Beta: The probability of accepting H0, when it was wrong = The probability of committing type II error = The size of type II error = β

Level of Significance: Alpha, the probability of type I error is known as the level of significance of the test It is also called the size of the critical region In other words, the maximum value of type I error which we would be willing to risk is called level of significance of the test

In general, 0.05 and 0.01 are the commonly accepted values of the levels of the significance When the level of significance is 0.05, it simply means that on the average in 5 chances out of 100 we

are likely to reject a correct H0

Probability (P-Value) Value: The probability of getting the results obtained if the null hypothesis is true If this probability is too small (smaller than the level of significance), then we

reject the null hypothesis If the level of significance is the area beyond the critical values, then the probability value is the area beyond the test statistic

Test Statistic: “Sample statistic used to decide whether to reject or fail to reject the null hypothesis”

Critical Region: Set of all values which would cause us to reject H0 Suppose the sample

values x1, x2, x n determine a point E on the n-dimensional sample space S which would be the

set of the various sample points corresponding to the all possible outcomes of the experiment The testing of statistics hypothesis is made on the basis of the division of this sample space into two mutually exclusive regions:

(1) Acceptance region

(2) Rejection (critical region) region of H0

The null hypothesis H0 is rejected as soon as the sample points falls in the critical region of

the sample space S The region of rejection is denoted either by R or by C.

Critical region R ` S acceptance region A _ S

A + R = S

FIG 12.1 The null hypothesis is accepted as soon as the sample point falls in the acceptance region,

which is denoted by A The values which separates the critical region from the non-critical region

is known as critical values The critical values are determined independently of the sample statistics Decision: Decision is a statement based upon the null hypothesis It is either “reject the null hypothesis” or ”fail to reject the null hypothesis” we will never accept the null hypothesis Conclusion: Conclusion is a statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative)

Unbiased Critical Region: A critical region is said to be unbiased if the size of type II error

β comes out to be less than the size of type I error

The type of test is determined by the Alternative Hypothesis (H1) The following way explain how

to determine if the test is a left tail, right tail, or two tail test

(a) Left Tailed Test

H1: Parameter < value

Notice that the inequality points to the left

A

R

S

Decision Rule: Reject H0 if t.s.< c.v.

Critical region

Non Critical region

Critical value

α

FIG 12.2

(b) Right Tailed Test

H1: Parameter > value

Notice that the inequality points to the right

Decision Rule: Reject H0 if t.s > c.v

Critical region Non Critical region

Critical value

α

FIG 12.3

(c) Two-Tailed Test

H1: Parameter not equal value another way to write not equal is < or >

Notice that the inequality points to both sides

Decision Rule: Reject H0 if t.s < c.v (left) or t.s > c.v (right)

Critical region

Non Critical region

Critical value Critical value

FIG 12.4

Critical region