Chapter 9

Chapter 9

Hypothesis Testing

Understanding Basic Statistics Fifth Edition

By Brase and Brase Prepared by Jon Booze

9 | 2Copyright © Cengage Learning. All rights reserved.

Methods for Drawing Inferences

• We can draw inferences on a population parameter in two ways:

1) Estimation (Chapter 8)

2) Hypothesis Testing (Chapter 9)


Hypothesis Testing

• Hypothesis testing is the process of making decisions about the value of a population parameter.


Establishing the Hypotheses

• Null Hypothesis: A hypothesis about a parameter that often denotes a theoretical value, a historical value, or a production specification.– Denoted as H0

• Alternate Hypothesis: A hypothesis that differs from the null hypothesis, such that if we reject the null hypothesis, we will accept the alternate hypothesis.– Denoted as H1 (in other sources HA).


Hypotheses Restated


Statistical Hypotheses

• The null hypothesis is always a statement of equality.– H0: μ = k, where k is a specified value

• The alternate hypothesis states that the parameter (μ or p) is less than, greater than, or not equal to a specified value.



Which of the following is an acceptable null hypothesis?

a). H0: 1.2 b). H0: > 1.2

c). H0: = 1.2 d). H0: 1.2



Which of the following is an acceptable null hypothesis?

a). H0: 1.2 b). H0: > 1.2

c). H0: = 1.2 d). H0: 1.2


Types of Tests

• Left-Tailed Tests: H1: μ < k

H1: p < k

• Right-Tailed Tests: H1: μ > k

H1: p > k

• Two-Tailed Tests: H1: μ ≠ k

H1: p ≠ k


Types of Tests

A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim?

a). H0: 150; H1: > 150

b). H0: > 150; H1: = 150

c). H0: = 150; H1: 150

d). H0: = 150; H1: > 150


Types of Tests

A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim?

a). H0: 150; H1: > 150

b). H0: > 150; H1: = 150

c). H0: = 150; H1: 150

d). H0: = 150; H1: > 150


Hypothesis Testing Procedure

1) Select appropriate hypotheses.2) Draw a random sample.3) Calculate the test statistic.4) Assess the compatibility of the test statistic

with H0.5) Make a conclusion in the context of the

problem.


Hypothesis Test of μx is Normal, σ is Known


P-Value

P-values are sometimes called the probability of chance.

Low P-values are a good indication that your test results are not due to chance.


P-Value for Left-Tailed Test


P-Value for Right-Tailed Test


P-Value for Two-Tailed Test


Types of Errors in Statistical Testing

• Since we are making decisions with incomplete information (sample data), we can make the wrong conclusion.

– Type I Error: Rejecting the null hypothesis when the null hypothesis is true.

– Type II Error: Accepting the null hypothesis when the null hypothesis is false.


Type I and Type II Errors


Errors in Statistical Testing

• Unfortunately, we usually will not know when we have made an error.

• We can only talk about the probability of making an error.

• Decreasing the probability of making a type I error will increase the probability of making a type II error (and vice versa).

• We can only decrease the probability of both types of errors by increasing the sample size (obtaining more information), but this may not be feasible in practice.


Level of Significance

• Good practice requires us to specify in advance the risk level of type I error we are willing to accept.

• The probability of type I error is the level of significance for the test, denoted by α (alpha).


Type II Error

• The probability of making a type II error is denoted by β (Beta).

• 1 – β is called the power of the test.

– 1 – β is the probability of rejecting H0 when H0 is false (a correct decision).


Type II Error

• The probability of making a type II error is denoted by β (Beta).

• 1 – β is called the power of the test.

– 1 – β is the probability of rejecting H0 when H0 is false (a correct decision).


The ProbabilitiesAssociated with Testing


Concluding a Statistical Test

For our purposes, significant is defined as follows:

At our predetermined level of risk α, the evidence against H0 is sufficient to reject H0. Thus we adopt H1.



For a particular experiment, P = 0.17 and = 0.05. What is the appropriate conclusion?

a). Reject the null hypothesis.b). Do not reject the null hypothesis.c). Reject both the null hypothesis and the

alternative hypothesis.d). Accept both the null hypothesis and the

alternative hypothesis.



For a particular experiment, P = 0.17 and = 0.05. What is the appropriate conclusion?

a). Reject the null hypothesis.b). Do not reject the null hypothesis.c). Reject both the null hypothesis and the

alternative hypothesis.d). Accept both the null hypothesis and the

alternative hypothesis.


Statistical Testing Comments

• Frequently, the significance is set at α = 0.05 or α = 0.01.

• When we “accept” the null hypothesis, we are not proving the null hypothesis to be true. We are only saying that the sample evidence is not strong enough to justify the rejection of H0.– Some statisticians prefer to say “fail to reject

H0 ” rather than “accept H0 .”


Interpretation of Testing Terms


Testing µ When σ is Known

1) State the null hypothesis, alternate hypothesis, and level of significance.

2) If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required.

Calculate:



3) Use the standard normal table and the type of test (one or two-tailed) to determine the P-value.

4) Make a statistical conclusion:

If P-value ≤ α, reject H0.

If P-value > α, do not reject H0.5) Make a context-specific conclusion.



Suppose that the test statistic z = 1.85 for a right-tailed test. Use Table 3 in the Appendix to find the corresponding P-value.

a). 0.2514b). 0.0322c). 0.9678d). 0.0161



Suppose that the test statistic z = 1.85 for a right-tailed test. Use Table 3 in the Appendix to find the corresponding P-value.

a). 0.2514b). 0.0322c). 0.9678d). 0.0161


Testing µ When σ is Unknown


2) If x is normally distributed (or mound-shaped), any sample size will suffice. If not, n ≥ 30 is required. Calculate:


Testing µ When σ is Unknown

3) Use the Student’s t table and the type of test (one or two-tailed) to determine (or estimate) the P-value.





Using Table 4 to Estimate P-values

0.025 < P-value < 0.050

Suppose we calculate t = 2.22 for a one-tailed test from a sample size of 6. df = n – 1 = 5.


Testing µ Using theCritical Value Method

• The values of that will result in the rejection of the null hypothesis are called the critical region of the distribution.

• When we use a predetermined significance level α, the Critical Value Method and the P-Value Method are logically equivalent.

x

x


Critical Regions for H0: µ = k






Testing µ When σ is Known (Critical Region Method)


2) If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required.

Calculate:


Testing µ When σ is Known (Critical Region Method)

3) Show the critical region and critical value(s) on a graph (determined by the alternate hypothesis and α).

4) Conclude in favor of the alternate hypothesis if z is in the critical region.

5) State a conclusion within the context of the problem.


Left-Tailed Tests


Right-Tailed Tests


Two-Tailed Tests


Testing a Proportion p

Binomial Experiments:r (# of successes) is a binomial variablen is the number of independent trialsp is the probability of success on each trial

Test Assumption: np > 5 and n(1 – p) > 5


Testing a Proportion pTest Assumption: np > 5 and n(1 – p) > 5

The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution?

a). n = 48, p = 0.39b). n = 843, p = 0.09c). n = 52, p = 0.93d). n = 12, p = 0.51


Testing a Proportion pTest Assumption: np > 5 and n(1 – p) > 5

The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution?

a). n = 48, p = 0.39b). n = 843, p = 0.09c). n = 52, p = 0.93d). n = 12, p = 0.51


Types of Proportion Tests


The Distributionof the Sample Proportion

ˆRecall the distribution of

is approximately normal with:

(1 ) and

rp

n

p pp

n


Converting the Sample Proportion to z


Testing p


2) Check np > 5 and nq > 5 (recall q = 1 – p). Compute:

p = the specified value in H0


Testing p

3) Use the standard normal table and the type of test (one or two-tailed) to determine the P-value.





Using the Critical Value Method for p

• As when testing for means, we can use the critical value method when testing for p.

• Use the critical value graphs exactly as when testing µ.


Critical Thinking: Issues Related to Hypothesis Testing

• Central question – Is the value of test statistic too different from zero for the difference to be due to chance alone?

• The P-value gives the probability that the test statistic’s value is due to chance alone.


Critical Thinking: Issues Related to Hypothesis Testing

• If the P-value is close to α, then we might attempt to clarify the results by

- increasing the sample size - controlling the experiment to reduce the

standard deviation• How reliable is the study and the

measurements in the sample? – Consider the source of the data and the reliability of the organization doing the study.

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Chapter 9