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Chapter 7 Hypothesis Testing 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) 1 7-1 Basics of Hypothesis Testing 2 Definition Hypothesis in statistics, is a statement regarding a characteristic of one or more populations 3 Steps in Hypothesis Testing 1. Statement is made about the population 2. Evidence in collected to test the statement 3. Data is analyzed to assess the plausibility of the statement 4 Components of a Formal Hypothesis Test 5 Components of a Hypothesis Test 1. Form Hypothesis 2. Calculate Test Statistic 3. Choose Significance Level 4. Find Critical Value(s) 5. Conclusion 6 Null Hypothesis: H0 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. 7 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 Reject H0 or fail to reject H0 8 Alternative Hypothesis: H1 Must be true if H0 is false , <, > ‘opposite’ of Null Ha sometimes used instead of H1 9 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. • The null hypothesis must contain the condition of equality 10 Examples Set up the null and alternative hypothesis 1. 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. 2. 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) 11 Test Statistic 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 Z* = x-µ σ n 12 • 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 13 Critical Region and Critical Value One Tailed Test Critical Region Critical Value ( z score ) 14 Critical Region and Critical Value One Tailed Test Critical Region Critical Value ( z score ) 15 Critical Region and Critical Value Two Tailed Test Critical Regions Critical Value ( z score ) Critical Value ( z score ) 16 Significance Level 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 17 Two-tailed,Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values. 18 Two-tailed Test H0: µ = 100 is divided equally between the two tails of the critical region H1: µ 100 Means less than or greater than Reject H0 Fail to reject H0 Reject H0 100 Values that differ significantly from 100 19 Right-tailed Test H0: µ 100 H1: µ > 100 Points Right Fail to reject H0 100 Reject H0 Values that differ significantly from 100 20 Left-tailed Test H0: µ 100 H1: µ < 100 Points Left Reject H0 Values that differ significantly from 100 Fail to reject H0 100 21 Conclusions in Hypothesis Testing 1. Traditional Method Reject H0 if the test statistic falls in the critical region Fail to reject H0 if the test statistic does not fall in the critical region 2. P-Value Method Reject H0 if the P-value is less than or equal Fail to reject H0 if the P-value is greater than the 22 P-Value Method of Testing Hypotheses 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 23 Finding P-values Two tailed test p(z>a) + p(z<-a) One tailed test (right) Where “a” is the value of the calculated test statistic p(z>a) One tailed test (left) p(z<-a) Used for HW # 3 – 5 – see example on next two slides 24 Determine P-value Sample data: x = 105 or z* = 2.66 Fail to Reject H0: µ = 100 Reject H0: µ = 100 * µ = 73.4 z = 1.96 or z = 0 z* = 2.66 Just find p(z > 2.66) 25 Determine P-value Sample data: x = 105 or z* = 2.66 Reject H0: µ = 100 Fail to Reject H0: µ = 100 Reject H0: µ = 100 * z = - 1.96 µ = 73.4 z = 1.96 or z = 0 z* = 2.66 Just find p(z > 2.66) + p(z < -2.66) 26 Conclusions in Hypothesis Testing Always test the null hypothesis Choose one of two possible conclusions 1. Reject the H0 2. Fail to reject the H0 27 Accept versus Fail to Reject 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) 28 Accept versus Fail to Reject Justice System - Trial Defendant Innocent Defendant Guilty Reject Presumption of Innocence (Guilty Verdict) Error Correct Fail to Reject Presumption of Innocence (Not Guilty Verdict) Correct Error 29 Conclusions in Hypothesis Testing Need to formulate correct wording of final conclusion 30 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) 31 Example State a conclusion 1. The proportion of college graduates how smoke is less than 27%. Reject Ho: 2. The mean weights of men at FLC is different from 180 lbs. Fail to Reject Ho: Used for #6 on HW 32 Type I Error 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) 33 Type II 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) 34 Type I and Type II Errors True State of Nature H0 True Reject H0 Decision Fail to Reject H0 H0 False Type I error Correct decision Correct decision Type II error 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. 35 Type I and Type II Errors = p(rejecting a true null hypothesis) = p(failing to reject a false null hypothesis) n, and are all related 36 Example 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. 37 Controlling Type I and Type II Errors For any fixed sample size n , as decreases, increases and conversely. To decrease both and , increase the sample size. 38 Definition 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. 39 7-2 Testing a claim about the mean (large samples) 40 Traditional (or Classical) Method of Testing Hypotheses Goal Identify a sample result that is significantly different from the claimed value By Comparing the test statistic to the critical value 41 Traditional (or Classical) Method of Testing Hypotheses (MAKE SURE THIS IS IN YOUR NOTES) 1. Determine H0 and H1. (and if necessary) 2. Determine the correct test statistic and calculate. 3. Determine the critical values, the critical region and sketch a graph. 4. Determine Reject H0 or Fail to reject H0 5. State your conclusion in simple non technical terms. 42 Test Statistic for Testing a Claim about a Proportion Can Use Traditional method Or P-value method 43 Three Methods Discussed 1) Traditional method 2) P-value method 3) Confidence intervals 44 Assumptions 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 s is unknown, we can use sample standard deviation s as estimate for s. 45 Test Statistic for Claims about µ when n > 30 Z* = x - µx s n 46 Decision Criterion 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 47 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 Claim: m = 69.5 years H0 : m = 69.5 H1 : m 69.5 Select if necessary level: = 0.05 48 Step 2: Identify the test statistic and calculate z* = x - µ = s n 73.4 – 69.5 8.7 35 = 2.65 49 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 z = - 1.96 0.025 1.96 Critical Values - Calculator 50 Step 4: Determine reject or fail to reject H0: Sample data: x = 73.4 or z = 2.66 Reject H0: µ = 69.5 Fail to Reject H0: µ = 69.5 Reject H0: µ = 69.5 * z = - 1.96 µ = 73.4 z = 1.96 or z = 0 P-value = P(z > 2.66) x 2 = .0078 z = 2.66 REJECT H0 51 Step 5: Restate in simple nontechnical terms Claim: m = 69.5 years REJECT H0 : m = 69.5 H1 : m 69.5 1) There is sufficient evident to conclude that the mean life span of symphony conductors is different from the general population. OR 2) There is sufficient evidence to conclude that mean life span of symphony conductors is different from 69.5 years. 52 TI-83 Calculator Hypothesis Test using z (large sample) 1. Press STAT 2. Cursor to TESTS 3. Choose ZTest 4. Choose Input: STATS 5. Enter σ and x and two tail, right tail or left tail 6. Cursor to calculate or draw *If your input is raw data, then input your raw data in L1 then use DATA 53 Testing Claims with Confidence Intervals • We reject a claim that the population parameter has a value that is not included in the confidence interval • Typically only used for two-tailed tests • For one-tailed test the degree of confidence would be 1 – 2 (don’t worry about this) 54 Testing Claims with Confidence Intervals Claim: mean age = 69.5 years, where n = 35, x = 73.4 and s = 8.7 95% confidence interval of 35 conductors (that is, 95% of samples would contain true value µ ) 70.5 < µ < 76.3 69.5 is not in this interval Therefore it is very unlikely that µ = 69.5 Thus we reject claim µ = 69.5 (same conclusion as previously stated) 55 7- 3 Testing a claim about the mean (small samples) 56 Assumptions for testing claims about population means (student t distribution) 1) The sample is a random sample. 2) The sample is small (n 30). 3) The value of the population standard deviation s is unknown. 4) population is approximately normal. 57 Test Statistic for a Student t-distribution x -µx t* = s n Critical Values Found in Table A-3 Degrees of freedom (df) = n -1 Critical t values to the left of the mean are negative 58 Choosing between the Normal and Student t-Distributions when Testing a Claim about a Population Mean µ Start Use normal distribution with Is n > 30 ? Z Yes (If s is unknown use s instead.) No Is the distribution of the population essentially normal ? (Use a histogram.) x - µx s/ n No Yes Is s known ? No Use nonparametric methods, which don’t require a normal distribution. Use normal distribution with Z x - µx s/ n (This case is rare.) Use the Student t distribution with x - µx t s/ n 59 Easier Decision Tree 1. Use z if s known or n is large 2. Use t if s is unknown and n is small and population is approximately normal MAKE SURE THIS IS IN YOUR NOTES 60 P-Value Method Table A-3 includes only selected values of Specific P-values usually cannot be found from table Use Table to identify limits that contain the P-value – very confusing Some calculators and computer programs will find exact P-values 61 TI-83 Calculator Hypothesis Test using t (small sample) 1. Press STAT 2. Cursor to TESTS 3. Choose TTest 4. Choose Input: STATS 5. Enter s and x and two tail, right tail or left tail 6. Cursor to calculate or draw *If your input is raw data, then input your raw data in L1 then use DATA 62 Example Sample statistics of GPA include n=20, x=2.35 and s=.7 1. Test the claim that the GPA is greater than 2.0 a. Use traditional method b. Use Calculator 2. Find exact p-value (see excel – TDIST function) 63 7-4 Testing a claim about a proportion 64 Assumptions for testing claims about population proportions 1) The sample observations are a random sample. 2) The conditions for a binomial experiment are satisfied 3) If np 5 and nq 5 are satisfied we • Use normal distribution to approximate binomial with µ = np and s = npq 65 Notation n = number of trials p = x/n (sample proportion) p = population proportion (used in the null hypothesis) q=1-p 66 Test Statistic for Testing a Claim about a Proportion p p * = pq n z 67 p sometimes is given directly “10% of the observed sports cars are red” is expressed as p = 0.10 p sometimes must be calculated “96 surveyed households have cable TV and 54 do not” is calculated using p x 96 =n = = 0.64 (96+54) (determining the sample proportion of households with cable TV) 68 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. 69 Test Statistic for Testing a Claim about a Proportion Z* = x-µ z = s = p-p pq n x - np npq = x n np n npq n = p-p pq n 70 TI-83 Calculator Hypothesis Test using z (proportions) 1. Press STAT 2. Cursor to TESTS 3. Choose 1-PropZTest 4. Enter x and n and two tail, right tail or left tail 5. Cursor to calculate or draw 71