Section 8.4-1Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Lecture Slides

Elementary Statistics Twelfth Edition

and the Triola Statistics Series

by Mario F. Triola

Section 8.4-2Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Chapter 8Hypothesis Testing

8-1 Review and Preview

8-2 Basics of Hypothesis Testing

8-3 Testing a Claim about a Proportion

8-4 Testing a Claim About a Mean

8-5 Testing a Claim About a Standard Deviation or Variance

Section 8.4-3Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Key Concept

This section presents methods for testing a claim about a population mean.

Part 1 deals with the very realistic and commonly used case in which the population standard deviation σ is not known.

Part 2 discusses the procedure when σ is known, which is very rare.

Section 8.4-4Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Part 1

When σ is not known, we use a “t test” that incorporates the Student t distribution.

Section 8.4-5Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Notation

n = sample size

= sample mean

= population mean

x

x

Section 8.4-6Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Requirements

1) The sample is a simple random sample.

2) Either or both of these conditions is satisfied: The population is normally distributed or n > 30.

Section 8.4-7Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Test Statistic

xxt

s

n

Section 8.4-8Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Running the Test

P-values: Use technology or use the Student t distribution in Table A-3 with degrees of freedom df = n – 1.

Critical values: Use the Student t distribution with degrees of freedom df = n – 1.

Section 8.4-9Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Important Properties of the Student t Distribution

1.The Student t distribution is different for different sample sizes (see Figure 7-5 in Section 7-3).

2.The Student t distribution has the same general bell shape as the normal distribution; its wider shape reflects the greater variability that is expected when s is used to estimate σ.

3.The Student t distribution has a mean of t = 0.

4.The standard deviation of the Student t distribution varies with the sample size and is greater than 1.

5.As the sample size n gets larger, the Student t distribution gets closer to the standard normal distribution.

Section 8.4-10Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

Listed below are the measured radiation emissions (in W/kg) corresponding to a sample of cell phones.

Use a 0.05 level of significance to test the claim that cell phones have a mean radiation level that is less than 1.00 W/kg.

The summary statistics are: .

0.38 0.55 1.54 1.55 0.50 0.60 0.92 0.96 1.00 0.86 1.46

0.938 and 0.423x s

Section 8.4-11Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Requirement Check:

1.We assume the sample is a simple random sample.

2.The sample size is n = 11, which is not greater than 30, so we must check a normal quantile plot for normality.

Section 8.4-12Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

The points are reasonably close to a straight line and there is no other pattern, so we conclude the data appear to be from a normally distributed population.

Section 8.4-13Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Step 1: The claim that cell phones have a mean radiation level less than 1.00 W/kg is expressed as μ < 1.00 W/kg.

Step 2: The alternative to the original claim is μ ≥ 1.00 W/kg.

Step 3: The hypotheses are written as:

0

1

: 1.00 W/kg

: 1.00 W/kg

H

H

Section 8.4-14Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Step 4: The stated level of significance is α = 0.05.

Step 5: Because the claim is about a population mean μ, the statistic most relevant to this test is the sample mean: .x

Section 8.4-15Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Step 6: Calculate the test statistic and then find the P-value or the critical value from Table A-3:

0.938 1.000.486

0.423

11

xxt

s

n

Section 8.4-16Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Step 7: Critical Value Method: Because the test statistic of t = –0.486 does not fall in the critical region bounded by the critical value of t = –1.812, fail to reject the null hypothesis.

Section 8.4-17Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

Step 7: P-value method: Technology, such as a TI-83/84 Plus calculator can output the P-value of 0.3191. Since the P-value exceeds α = 0.05, we fail to reject the null hypothesis.

Section 8.4-18Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

Step 8: Because we fail to reject the null hypothesis, we conclude that there is not sufficient evidence to support the claim that cell phones have a mean radiation level that is less than 1.00 W/kg.

Section 8.4-19Copyright © 2014, 2012, 2010 Pearson Education, Inc.

a) In a left-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –2.007.

b) In a right-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = 1.222.

c) In a two-tailed hypothesis test, the sample size is n = 12, and the test statistic is t = –3.456.

Assuming that neither software nor a TI-83 Plus calculator is available, use Table A-3 to find a range of values for the P-value corresponding to the given results.

Finding P-Values

Section 8.4-20Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example – Confidence Interval Method

We can use a confidence interval for testing a claim about μ.

For a two-tailed test with a 0.05 significance level, we construct a 95% confidence interval.

For a one-tailed test with a 0.05 significance level, we construct a 90% confidence interval.

Section 8.4-21Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example – Confidence Interval Method

Using the cell phone example, construct a confidence interval that can be used to test the claim that μ < 1.00 W/kg, assuming a 0.05 significance level.

Note that a left-tailed hypothesis test with α = 0.05 corresponds to a 90% confidence interval.

Using methods described in Section 7.3, we find:

0.707 W/kg < μ < 1.169 W/kg

Section 8.4-22Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example – Confidence Interval Method

Because the value of μ = 1.00 W/kg is contained in the interval, we cannot reject the null hypothesis that μ = 1.00 W/kg .

Based on the sample of 11 values, we do not have sufficient evidence to support the claim that the mean radiation level is less than 1.00 W/kg.

Section 8.4-23Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Part 2

When σ is known, we use test that involves the standard normal distribution.

In reality, it is very rare to test a claim about an unknown population mean while the population standard deviation is somehow known.

The procedure is essentially the same as a t test, with the following exception:

Section 8.4-24Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Test Statistic for Testing a Claim About a Mean (with σ Known)

The test statistic is:

The P-value can be provided by technology or the standard normal distribution (Table A-2).

The critical values can be found using the standard normal distribution (Table A-2).

xxz

n

Section 8.4-25Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

If we repeat the cell phone radiation example, with the assumption that σ = 0.480 W/kg, the test statistic is:

The example refers to a left-tailed test, so the P-value is the area to the left of z = –0.43, which is 0.3336 (found in Table A-2).

Since the P-value is large, we fail to reject the null and reach the same conclusion as before. 0.938 1.000.43

0.480

11

xxz

n