Created by Erin Hodgess, Houston, Texas Section 7-1 & 7-2 Overview and Basics of Hypothesis Testing

Chapter 7 Hypothesis Testing 7-1 Overview 7-2 Basics of Hypothesis Testing 7-4 Testing a Claim About a Mean:  Known 7-5Testing a Claim About a Mean:  Not Known 7-3 Testing a Claim About a Proportion

7.2 Fundamentals of Hypothesis Testing Recall: Inferential Statistics – draw conclusions about a population based on sample data:  Confidence Intervals: Estimate the value of a population parameter (using sample statistics).  Hypothesis Tests:Tests a claim (hypothesis) about a population parameter (using sample statistics).

 State the hypothesis.  Gather sample data (“evidence”).  Make a decision about your hypothesis, based on sample data. Basic Idea for Hypothesis Tests

Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct.

A.The Null and Alternative Hypotheses B.The test statistic and making decisions C.Types of errors in hypothesis tests D.Writing conclusions Components of a Hypothesis Test

A. The Hypotheses  Null Hypothesis, H 0 :  statement about the parameter (μ, p, σ)  assumed true until proven otherwise  must contain equality sign:  Alternative Hypothesis, H 1 or H a :  statement about the parameter that must be true if H 0 is false.  must contain:

The Hypotheses Symbolic Form of the Hypotheses Note: The original claim* may be in the null or the alternative hypothesis.

B. Making Decisions 1.State Hypotheses. **Assume H 0 is true. 2.Collect sample data (test statistic) 3.**Make a decision about H 0: a)Reject H 0, or b)Fail to Reject the H 0 Note: a) Reject H a → Accept H a b) Fail to Reject the H 0 → Can’t Accept H a

C. Errors in Decision Making  Type I error is the mistake of rejecting the null when it is true: Reject Ho | Ho is true  A Type II error is the mistake of failing to reject the null when it is false: FTR Ho | Ho is false  The symbol  (alpha) is used to represent the probability of a type I error. P(Type I error) =   The symbol  (beta) is used to represent the probability of a type II error. P(Type II error) = 

D. Writing Conclusions All conclusions are written in terms of the original claim. How the conclusion is worded depends on: 1.Whether the original claim was in the null or the alternative, and 2.Whether the decision was Reject Ho (which implies you accept the Ha) or FTR Ho (which implies you cannot accept the Ha). A.Ho contains the claim, and you: 1.Reject Ho: “There’s enough evidence to reject the claim that ….. 2.FTR Ho: “There is not enough evidence to reject the claim that…” B.Ha contains the claim, and you: 1.Reject Ho: “There is enough evidence to support the claim that….” 2.FTR Ho: “There is not enough evidence to support the claim that…”

Wording of Final Conclusion Figure 7-7

Note about Forming Your Own Claims (Hypotheses) If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.

How to Make a Decision about the Ho We start with tests about the population mean, μ. 1.State Ho, Ha. **Assume H 0 is true: μ = μ 0 2.Collect sample data. Compute the test statistic. (For tests about μ, use ). 3.Recall: For x~N or n>30 and σ known:

How to Make a Decision about the Ho 4.Determine the Tails of the Test: a)Left-tailed Test b)Right-Tailed Test c)Two-Tailed Test Basic Idea: Assuming the Ho is true, (i.e, μ = μ0 ) where would the sample mean have to fall to convince you to reject the Ho and accept the Ha?

Left-tailed Test H 0 : μ ≥ μ 0 H a : μ < μ 0 Points Left

Right-tailed Test H 0 : μ ≤ μ 0 H a : μ > μ 0 Points Right

Two-tailed Test H 0 : μ = μ 0 H a : μ ≠ μ 0 Means less than or greater than

How to Make a Decision about the Ho 5.Once you have determined the tails of the test, use your sample mean to make a decision about the Ho using either: a)The Traditional Method b)*The P-Value Method Main Idea**: How unusual would your sample mean have to be for you to reject the Ho and accept the Ha??

The Traditional Method  Critical Region: contains values of the test statistic that would be considered unusual, assuming the Ho is true.  If the test statistic falls in the critical region, that would be considered unusual, assuming the Ho was true. DECISION: REJECT HO.  If the test statistic falls in the non-critical regions, then that would not be considered unusual. DECISION: FTR Ho.

Finding the Critical Region  Critical Region: contains values of the test statistic that would be considered unusual, assuming the Ho is true.  Area of the Critical Region = α (α is also known as the significance level of the test). It is also the probability of a Type I error.  Critical Values: values that mark the boundaries of the critical region, z α

Created by Erin Hodgess, Houston, Texas Section 7-4 Testing a Claim About a Mean:  Known

Assumptions for Testing Claims About Population Means 1) The sample is a simple random sample. 2) The value of the population standard deviation  is known. 3) Either or both of these conditions is satisfied: The population is normally distributed or n > 30.

H.T about a Mean (with  known) 1.Use the claim to write Ho, Ha. **Assume the Ho is true. 2.Determine whether you’re doing a left- tailed, right-tailed, or 2-tailed test. 3.Note α, the significance level of the test. 4.Get sample mean, x. Sketch the sampling distribution.

4.Convert the sample mean to a z-score: 5.Make a decision about Ho using either: a.Traditional Method b.P-value Method 6.Write a conclusion in terms of the original claim. H.T about a Mean (with  Known)

Traditional Method 1.Shade the critical region 2.Find the critical values 3.Make a decision: a.If z is in the critical region, reject Ho. b.If z is not in the critical region, FTR the Ho.

P -Value Method for Making Decisions Instead of determining whether your sample mean is unusual just by whether or not it falls in a “critical region”, be more precise: “ Find the “P-value”

P -Value Method for Making Decisions P-value = probability of getting a test statistic as extreme or more as the one from the sample data (assuming the Ho is true to begin with). If P-value ≤ α (unusual), Reject Ho. If Pvalue > α (not unusual), FTR HO.

Example: Finding P-values. Figure 7-6

Created by Erin Hodgess, Houston, Texas Section 7-5 Testing a Claim About a Mean:  Not Known

1.Follow the same procedure as For population ≈ normal or n > 30 and σ unknown, the sampling distribution of x has a t-distribution with n-1 degrees of freedom. 3.Convert sample mean x to a standard t-score: H.T about Mean (with  unknown)

Choosing between the Normal and Student t Distributions when Testing a Claim about a Population Mean µ Use the Student t distribution when  is not known and either or both of these conditions is satisfied:The population is normally distributed or n > 30.

Created by Erin Hodgess, Houston, Texas Section 7-3 Testing a Claim About a Proportion

Hypotheses About p Note: The original claim* may be in the null or the alternative hypothesis.

Assumptions about a Hypotheses Test for a Proportion 1.To conduct a hypothesis test about a population proportion p, use a sample proportion, p. 2.We assume the sample was a simple random sample. 3.If np ≥ 5, nq ≥ 5, the sampling distribution of p is: 4.Follow the procedures of 7.3:

Hypothesis Test about a Population Proportion 1.Use the claim to write Ho, Ha. **Assume the Ho is true. 2.Determine whether you’re doing a left- tailed, right-tailed, or 2-tailed test. 3.Note α, the significance level of the test. 4.Get sample proportion, p. Sketch the sampling distribution.

4.Convert the sample proportion to a z- score: 5.Make a decision about Ho using either: a.Traditional Method b.P-value Method 6.Write a conclusion in terms of the original claim. Hypothesis Test about a Proportion

CAUTION  When the calculation of p results in a decimal with many places, store the number on your calculator and use all the decimals when evaluating the z test statistic.  Large errors can result from rounding p too much.  

Created by Erin Hodgess, Houston, Texas Section 7-6 Testing a Claim About a Standard Deviation or Variance

Assumptions for Testing Claims About  or  2 1. The sample is a simple random sample. 2) The population has values that are normally distributed (a strict requirement).

Chi-Square Distribution Test Statistic  2 = ( n – 1) s 2  2

n = sample size s 2 = sample variance  2 = population variance (given in null hypothesis)  2 = ( n – 1) s 2  2 Chi-Square Distribution Test Statistic

P-values and Critical Values for Chi-Square Distribution  Use Table A-4.  The degrees of freedom = n –1.

Properties of Chi-Square Distribution  All values of  2 are nonnegative, and the distribution is not symmetric (see Figure 7-12).  There is a different distribution for each number of degrees of freedom (see Figure 7-13).  The critical values are found in Table A-4 using n – 1 degrees of freedom.

Properties of Chi-Square Distribution Figure 7-12 Properties of the Chi-Square Distribution There is a different distribution for each number of degrees of freedom. Chi-Square Distribution for 10 and 20 Degrees of Freedom Figure 7-13

Example: For a simple random sample of adults, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. A simple random sample of 13 statistics professors yields a standard deviation of s = 7.2. Assume that IQ scores of statistics professors are normally distributed and use a 0.05 significance level to test the claim that  = 15. H 0 :  = 15 H 1 :   15  = 0.05 n = 13 s = 7.2 =  2 = ( n – 1) s 2  2 (13 – 1)(7.2) =

Example: For a simple random sample of adults, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. A simple random sample of 13 statistics professors yields a standard deviation of s = 7.2. Assume that IQ scores of statistics professors are normally distributed and use a 0.05 significance level to test the claim that  = 15. H 0 :  = 15 H 1 :   15  = 0.05 n = 13 s = 7.2  2 = 2.765

Example: For a simple random sample of adults, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. A simple random sample of 13 statistics professors yields a standard deviation of s = 7.2. Assume that IQ scores of statistics professors are normally distributed and use a 0.05 significance level to test the claim that  = 15. H 0 :  = 15 H 1 :   15  = 0.05 n = 13 s = 7.2  2 = The critical values of and are found in Table A-4, in the 12th row (degrees of freedom = n – 1) in the column corresponding to and

Example: For a simple random sample of adults, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. A simple random sample of 13 statistics professors yields a standard deviation of s = 7.2. Assume that IQ scores of statistics professors are normally distributed and use a 0.05 significance level to test the claim that  = 15. H 0 :  = 15 H 1 :   15  = 0.05 n = 13 s = 7.2  2 = Because the test statistic is in the critical region, we reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that the standard deviation is equal to 15.