Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-1 σ σ

Single Population/Sample Hypothesis Testing: What are you trying to do? –Have parameters (mean, standard deviation) for the population (“truth”) µ, σ –Have sampled data and associated summary statistics X_bar, s –Want to know if the sample is statistically significantly different from the population Can’t simply compare the means Need to allow for that the sample’s mean can vary from 1 sample to the next, i.e., allow for variability Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-2

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-3 Hypothesis Testing for Single Populations - How do we allow for variability? - Depends on what we know about the data Population Mean σ Unknown Hypothesis Testing Population Proportion σ Known Section 9.2 Section 9.4 Section 9.3

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-4 A hypothesis is an assumption about a population parameter such as a mean or a proportion Example: population mean –The mean data use for smartphone users is μ = 1.8 gigabytes per month Example: population proportion –The proportion of cell phone users with 4G contracts is π = An Introduction to Hypothesis Testing

Stating the Hypothesis Every hypothesis test has both a null hypothesis and an alternative hypothesis The null hypothesis (H 0 ) represents the status quo –States a belief that the population parameter is ≤, =, or ≥ a specific value –The null hypothesis is believed to be true unless there is overwhelming evidence to the contrary The alternative hypothesis (H 1 ) represents the opposite of the null hypothesis –Is believed to be true if the null hypothesis is found to be false –The alternative hypothesis always states that the population parameter is >, ≠, or < a specific value Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-5

Stating the Hypothesis Stating the null and alternative hypotheses depends on the nature of the test and the motivation of the person conducting it Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-6 H 0 : μ ≥ 1.8 H 1 : μ < 1.8 H 0 : μ ≤ 1.8 H 1 : μ > 1.8 H 0 : μ = 1.8 H 1 : μ ≠ 1.8 This test would be used by someone who thinks that data use has gone up, and wants to support that the average data use is now more than 1.8 gigabytes per month This would be used by someone who wants to test an assumption that data use has gone down (rejecting the null hypothesis would support the alternative that the average data use is less than 1.8 gigabytes per month) This test would be used by someone who has no specific expectation, but wants to test the assumption that the average data use is 1.8 gigabytes per month

Two-Tail Hypothesis Tests A two-tail hypothesis test is used whenever the alternative hypothesis is expressed as ≠ Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-7 H 0 : μ = 1.8 H 1 : μ ≠ 1.8 We assume that μ = 1.8 unless the sample mean is much higher or much lower than 1.8 Reject H 0 Do not reject H 0 Reject H 0 Do not reject H 0 1.8

One-Tail Hypothesis Tests A one-tail hypothesis test is used when the alternative hypothesis is stated as Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-8 Upper tail test: We assume that μ = 1.8 unless the sample mean is much higher than 1.8 Reject H 0 Do not reject H 0 Reject H 0 Do not reject H H 0 : μ ≥ 1.8 H 1 : μ < 1.8 H 0 : μ ≤ 1.8 H 1 : μ > 1.8 Lower tail test: We assume that μ = 1.8 unless the sample mean is much lower than 1.8 Do not reject H 0 1.8

The Logic of Hypothesis Testing The null hypothesis can never be accepted The only two options available are to (1) reject the null hypothesis, or (2) fail to reject the null hypothesis The null hypothesis is tested using sample data –The sample result provides enough evidence to reject the null or does not provide enough evidence to reject Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-9

The Difference Between Type I and Type II Errors Sample evidence is not perfect due to sampling error, so a conclusion about the population can be wrong A Type I error occurs when the null hypothesis is rejected when it is true –The probability of making a Type I error is known as α, the level of significance A Type II error occurs when we fail to reject the null hypothesis when it is not true –The probability of making a Type II error is known as β Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-10

Calculating the Probability of Type II Errors Occurring Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-11 Reject H 0 :   50 Do not reject H 0 :   α = 0.05 Example: Suppose we fail to reject H 0 : μ ≥ 50 when in fact the true mean is μ = 52 This is the actual distribution of if  = 52 This is the range of where H 0 is not rejected

Decision Rules for the Two Types of Hypothesis Errors Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-12 Actual State of H 0 Decision Reject H 0 Type I Error P(Type I Error) = Type II Error P(Type II Error) = β Do Not Reject H 0 Possible Hypothesis Test Outcomes H 0 is False H 0 is True Correct Outcome The Difference Between Type I and Type II Errors

When doing a hypothesis test, decide on a value for α before selecting the sample Once α has been set, the value of β can be calculated For a given sample size, reducing the value of α will result in an increase in the value of (or the opposite, α ↑ → β ↓) The only way to reduce both α and β simultaneously is to increase the sample size Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-13 The Difference Between Type I and Type II Errors

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Hypothesis Testing for the Population Mean when σ is Known Population Mean σ Unknown Hypothesis Testing Population Proportion σ Known Section 9.2

Hypothesis testing when σ is known: 1.If the sample size is small (n < 30) the population must follow the normal distribution 2. If the sample size is large (n ≥ 30) the Central Limit Theorem states that the sampling distribution follows the normal distribution, so there is no restriction on the population distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-15 Hypothesis Testing for the Population Mean when σ is Known

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: A new CFL bulb is claimed to have an average life that exceeds 8,000 hours –Suppose the average life of a random sample of 36 of the new bulbs is 8,120 hours –Assume that the population standard deviation for the life of CFL bulbs is 500 hours The following slides show six steps to complete this hypothesis test Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-16

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 1: Identify the null and alternative hypotheses H 0 : μ ≤ 8,000 hours (status quo: average life is not greater than 8,000 hours) H 1 : μ > 8,000 hours (the new bulb does last longer than 8,000 hours) Step 2: Set a value for the significance level, α –The level of significance represents the probability of making a Type I error –Suppose that α = 0.05 is chosen Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-17

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 3: Determine the appropriate critical value –σ is known so use a z-score; the critical z-score identifies the rejection region for this one-tail test –Since this is a one-tail test the entire area for α = 0.05 is placed on the right side (upper tail) of the sampling distribution: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-18 Reject H 0 α = 0.05 Do not reject H 0 8,

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 4: Calculate the appropriate test statistic Formula for the z-test statistic for a hypothesis test for the population mean (when σ is known): Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-19 where: = The z-test statistic = The sample mean = The mean of the sampling distribution, which is assumed to be true for the null hypothesis σ = The standard deviation of the population n = The sample size

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 4: Calculate the appropriate test statistic Step 5: Compare the z-test statistic with the critical z-score –For a one-tail upper tail test, reject the null hypothesis if > –Here, 1.44 is not greater than 1.645, so do not reject H 0 –(Illustrated on the next slide) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-20

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-21 Do not reject H 0 8,000 Reject H 0 α = 0.05 is not greater than = 1.645, so do not reject H 0 An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Step 5: (continued)

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-22 An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Rule for this example

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 5a: (Optional) Compare the sample mean with the critical sample mean The critical sample mean,, is the sample mean that marks the boundary of the rejection region Formula for the critical sample mean for a hypothesis test for the population mean (when σ is known) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-23

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-24 Do not reject H 0 8,000 Reject H 0 α = 0.05 An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Step 5a: (continued) The sample result of 8,120 is not greater than 8,137.1, so do not reject H 0

An Example of a One-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 6: (Final Step) State the conclusion According to our sample of 36 new CFL bulbs, there is not enough evidence to support Edalight’s claim that the average life of these bulbs exceeds 8,000 hours. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-25

The p-value Approach to Hypothesis Testing: One- Tail Tests The p-value is the probability of observing a sample mean at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true The p-value is sometimes referred to as the observed level of significance Provides a third approach to deciding whether or not to reject the null hypothesis Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-26

Use the CFL bulb example to illustrate: –Claim: μ ≥ 8,000 –Sample result: n = 36; = 8,120 –σ = 500 was assumed known The p-value represents the probability of obtaining a sample mean of 8,120 hours or greater if the true population mean is 8,000 hours Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-27 The p-value Approach to Hypothesis Testing: One- Tail Tests

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall , ,000 α = 0.05 p-value = The p-value Approach to Hypothesis Testing: One- Tail Tests Compare the p-value with α: Since > 0.05, we do not reject H 0

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: The mean data use for smartphone users is claimed to be μ = 1.8 gigabytes per month –Suppose data use is recorded for 49 randomly selected smartphone users and the average use is found to be 1.86 gigabytes per month –Assume that the population standard deviation is 0.20 gigabytes per month The following slides show six steps to complete this hypothesis test Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-29

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 1: Identify the null and alternative hypotheses H 0 : μ = 1.8 (status quo: average use is 1.8 gigabytes per month) H 1 : μ ≠ 1.8 (average use is not equal to 1.8 gigabytes per month) Step 2: Set a value for the significance level, α –The level of significance represents the probability of making a Type I error, chosen before conducting the hypothesis test –Suppose that α = 0.05 is chosen Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-30

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 3: Determine the appropriate critical values –σ is known so use a z-score; the critical z-score identifies the rejection region for this one-tail test –Since this is a two-tail test, α = 0.05 is split evenly into two tails: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-31 Reject H 0 α/2 = Do not reject H Reject H 0 α/2 = Reject H 0

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 4: Calculate the appropriate test statistic Step 5: Compare the z-test statistic with the critical z-score –For a two-tail test, reject the null hypothesis if –Here, 2.10 is greater than 1.96, so reject H 0 –(Illustrated on the next slide) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-32

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-33 Do not reject H 0 Reject H 0 α/2 = is greater than = 1.96, so reject H 0 An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Step 5: (continued) Reject H 0 α/2 =

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 5a: (Optional) Compare the sample mean with the critical sample means A two-tail hypothesis test has two critical means, one on either side of the distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-34

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-35 An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Step 5a: (continued) is not within the interval (1.744, 1.856), so the null hypothesis is rejected The sample result of 1.86 is greater than 1.856, so reject H 0 Do not reject H 0 Reject H 0 α/2 = Reject H 0 α/2 =

An Example of a Two-Tail Hypothesis Test for the Population Mean (When σ Is Known) Example: (continued) Step 6: (Final Step) State the conclusion Our sample of 49 smartphone users provides sufficient evidence to reject the null hypothesis, so we support the alternative hypothesis that the average data use is not equal to 1.8 gigabytes per month. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-36

For a two-tail hypothesis test, the p-value is a sum of two tail areas Illustrate using the smartphone example: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-37 α/2 = Area = α/2 = The p-value Approach to Hypothesis Testing: One- Tail Tests

The Role α Plays in Hypothesis Testing Changing α changes the critical z-score in the hypothesis test, which, in turn, changes the rejection region in the sampling distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 9-38