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How to Write a Hypothesis

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Chapter 7 Hypothesis Testing

Aug 14, 2012

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7-1 Basics of Hypothesis Testing 7-2 Testing a Claim about a Mean: Large Samples 7-3 Testing a Claim about a Mean: Small Samples 7-4 Testing a Claim about a Proportion 7- 5 Testing a Claim about a Standard     Deviation (will cover with chap 8). Chapter 7 Hypothesis Testing. 7-1.

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7-1 Basics of Hypothesis Testing 7-2 Testing a Claim about a Mean: Large Samples 7-3 Testing a Claim about a Mean: Small Samples 7-4 Testing a Claim about a Proportion 7- 5 Testing a Claim about a Standard     Deviation (will cover with chap 8) Chapter 7Hypothesis Testing

7-1 Basics of Hypothesis Testing

Hypothesis in statistics, is a statement regarding a characteristic of one or more populations Definition

Statement is made about the population Evidence in collected to test the statement Data is analyzed to assess the plausibility of the statement Steps in Hypothesis Testing

Components of aFormal Hypothesis Test

Form Hypothesis Calculate Test Statistic Choose Significance Level Find Critical Value(s) Conclusion Components of a Hypothesis Test

A hypothesis set up to be nullified or refuted in order to support an alternate hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. Null Hypothesis: H0

Statement about value of population parameter like m, p or s Must contain condition of equality =, , or Test the Null Hypothesis directly RejectH0 or fail to rejectH0 Null Hypothesis: H0

Must be true if H0 is false , <, > ‘opposite’ of Null sometimes used instead of Alternative Hypothesis: H1 H1 Ha

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. The null hypothesis must contain the condition of equality Note about Forming Your Own Claims (Hypotheses)

Set up the null and alternative hypothesis The packaging on a lightbulb states that the bulb will last 500 hours. A consumer advocate would like to know if the mean lifetime of a bulb is different than 500 hours. A drug to lower blood pressure advertises that it drops blood pressure by 20%. A doctor that prescribes this medication believes that it is less. Set up the null and alternative hypothesis. (see hw # 1) Examples

a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis Testing claims about the population proportion Test Statistic x - µ σ Z*= n

Critical Region - Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Value - Value or values that separate the critical region from the values of the test statistics that do not lead to a rejection of the null hypothesis

One Tailed Test Critical Region and Critical Value Critical Region Critical Value ( z score )

Two Tailed Test Critical Region and Critical Value Critical Regions Critical Value ( z score ) Critical Value ( z score )

Denoted by  The probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common choices are 0.05, 0.01, and 0.10 Significance Level

Two-tailed,Right-tailed,Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

H0: µ = 100 H1: µ  100 Two-tailed Test  is divided equally between the two tails of the critical region Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100

H0: µ  100 H1: µ > 100 Fail to reject H0 Reject H0 Right-tailed Test Points Right Values that differ significantly from 100 100

H0: µ  100 H1: µ < 100 Left-tailed Test Points Left Reject H0 Fail to reject H0 Values that differ significantly from 100 100

Traditional Method Reject H0if the test statistic falls in the critical region Fail to reject H0if the test statistic does not fall in the critical region P-Value Method Reject H0if the P-value is less than or equal  Fail to reject H0if the P-value is greater than the  Conclusions in Hypothesis Testing

Finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low Uses test statistic to find the probability. Method used by most computer programs and calculators. Will prefer that you use the traditional method on HW and Tests P-Value Methodof Testing Hypotheses

Two tailed test p(z>a) + p(z<-a) One tailed test (right) p(z>a) One tailed test (left) p(z<-a) Finding P-values Where “a” is the value of the calculated test statistic Used for HW # 3 – 5 – see example on next two slides

Determine P-value Sample data: x = 105 or z* = 2.66 Reject H0: µ = 100 Fail to Reject H0: µ = 100 * µ = 73.4 or z = 0 z = 1.96 z* = 2.66 Just find p(z > 2.66)

Determine P-value Sample data: x = 105 or z* = 2.66 Reject H0: µ = 100 Reject H0: µ = 100 Fail to Reject H0: µ = 100 * z = - 1.96 µ = 73.4 or z = 0 z = 1.96 z* = 2.66 Just find p(z > 2.66) + p(z < -2.66)

Always test the null hypothesis Choose one of two possible conclusions 1. Reject the H0 2. Fail to reject the H0 Conclusions in Hypothesis Testing

Never “accept the null hypothesis, we will fail to reject it. Will discuss this in more detail in a moment We are not proving the null hypothesis Sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect – guilty vs. not guilty) Accept versus Fail to Reject

Accept versus Fail to Reject

Need to formulate correct wording of finalconclusion Conclusions in Hypothesis Testing

Wording of final conclusion 1. Reject the H0 Conclusion: There is sufficient evidence to conclude………………………(what ever H1 says) 2. Fail to reject the H0 Conclusion: There is not sufficient evidence to conclude ……………………(what ever H1 says) Conclusions in Hypothesis Testing

State a conclusion The proportion of college graduates how smoke is less than 27%. Reject Ho: The mean weights of men at FLC is different from 180 lbs. Fail to Reject Ho: Example Used for #6 on HW

The mistake of rejecting the null hypothesis when it is true. (alpha) is used to represent the probability of a type I error Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 (test question) Type I Error

the mistake of failing to reject the null hypothesis when it is false. ß (beta) is used to represent the probability of a type II error Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6 (test question) Type II Error

Type I and Type II Errors True State of Nature H0 True H0 False Reject H0 Correct decision Type I error  Decision Fail to Reject H0 Type II error  Correct decision In this class we will focus on controlling a Type I error. However, you will have one question on the exam asking you to differentiate between the two.

a = p(rejecting a true null hypothesis) b = p(failing to reject a false null hypothesis) n, a and b are all related Type I and Type II Errors

Identify the type I and type II error. The mean IQ of statistics teachers is greater than 120. Type I: We reject the mean IQ of statistics teachers is 120 when it really is 120. Type II: We fail to reject the mean IQ of statistics teachers is 120 when it really isn’t 120. Example

For any fixed sample size n, as  decreases,  increases and conversely. To decrease both  and , increase the sample size. Controlling Type I and Type II Errors

Power of a Hypothesis Test is the probability (1 - ) of rejecting a false null hypothesis. Note: No exam questions on this. Usually covered in a more advanced class in statistics. Definition

7-2 Testing a claim about the mean (large samples)

Goal Identify a sample result that is significantly different from the claimed value By Comparing the test statistic to the critical value Traditional (or Classical) Method of Testing Hypotheses

Determine H0 and H1. (and if necessary) Determine the correct test statistic and calculate. Determine the critical values, the critical region and sketch a graph. Determine Reject H0 or Fail to reject H0 State your conclusion in simple non technical terms. Traditional (or Classical) Method of Testing Hypotheses (MAKE SURE THIS IS IN YOUR NOTES)

Test Statistic for Testing a Claim about a Proportion Can Use Traditional method Or P-value method

1) Traditional method 2) P-value method 3) Confidence intervals Three Methods Discussed

for testing claims about population means 1) The sample is a random sample. 2) The sample is large (n > 30). a) Central limit theorem applies b) Can use normal distribution 3) If  is unknown, we can use sample standard deviation s as estimate for . Assumptions

Test Statistic for Claims about µ when n > 30 x - µx Z*=  n

Reject the null hypothesis if the test statistic is in the critical region Fail to reject the null hypothesis if the test statistic is not in the critical region Decision Criterion

Claim:  = 69.5 years H0 :  = 69.5 H1 :  69.5 Example:A newspaper article noted that the mean life span for 35 male symphony conductors was 73.4 years, in contrast to the mean of 69.5 years for males in the general population. Test the claim that there is a difference. Assume a standard deviation of 8.7 years. Choose your own significance level. Step 1: Set up Claim, H0, H1 Select if necessary level:  = 0.05

Step 2: Identify the test statistic and calculate x - µ 73.4 – 69.5 z*=== 2.65  8.7 n 35

Step 3: Determine critical region(s) and critical value(s) & Sketch = 0.05 /2= 0.025 (two tailed test) 0.4750 0.4750 0.025 0.025 z = - 1.96 1.96 Critical Values - Calculator

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