Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall MIS 214 Statistics II 2013/2014 Spring Chapter 6 7 9 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter Goals After completing this chapter, you should be able to: Describe a simple random sample and why sampling is important Explain the difference between descriptive and inferential statistics Define the concept of a sampling distribution Determine the mean and standard deviation for the sampling distribution of the sample mean, Describe the Central Limit Theorem and its importance Determine the mean and standard deviation for the sampling distribution of the sample proportion, Describe sampling distributions of sample variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 6-2

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Introduction Descriptive statistics Collecting, presenting, and describing data Inferential statistics Drawing conclusions and/or making decisions concerning a population based only on sample data Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Inferential Statistics Making statements about a population by examining sample results Sample statistics Population parameters (known) Inference (unknown, but can be estimated from sample evidence) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Inferential Statistics Drawing conclusions and/or making decisions concerning a population based on sample results. Estimation e.g., Estimate the population mean weight using the sample mean weight Hypothesis Testing e.g., Use sample evidence to test the claim that the population mean weight is 120 pounds Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling from a Population 6.1 A Population is the set of all items or individuals of interest Examples: All likely voters in the next election All parts produced today All sales receipts for November A Sample is a subset of the population Examples: 1000 voters selected at random for interview A few parts selected for destructive testing Random receipts selected for audit Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Population vs. Sample Population Sample Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Why Sample? Less time consuming than a census Less costly to administer than a census It is possible to obtain statistical results of a sufficiently high precision based on samples. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Simple Random Sample Every object in the population has the same probability of being selected Objects are selected independently Samples can be obtained from a table of random numbers or computer random number generators A simple random sample is the ideal against which other sampling methods are compared Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distributions A sampling distribution is a probability distribution of all of the possible values of a statistic for a given size sample selected from a population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Chapter Outline Sampling Distributions Sampling Distributions of Sample Means Sampling Distributions of Sample Proportions Sampling Distributions of Sample Variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distributions of Sample Means 6.2 Sampling Distributions Sampling Distributions of Sample Means Sampling Distributions of Sample Proportions Sampling Distributions of Sample Variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Sample Mean Let X1, X2, . . ., Xn represent a random sample from a population The sample mean value of these observations is defined as Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: Note that the standard error of the mean decreases as the sample size increases Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

If sample values are not independent If the sample size n is not a small fraction of the population size N, then individual sample members are not distributed independently of one another Thus, observations are not selected independently A finite population correction is made to account for this: or The term (N – n)/(N – 1) is often called a finite population correction factor Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Let X1, X2, … Xn is a random sample from a poplation, If the sample size n is a small fraction of the population size N, then individual sample members are distributed independently of one another Let X1, X2, … Xn is a random sample from a poplation, Each Xi is a random variable whose parameters are the same as populations Fixed and unknown In parametric frequencist paradigm of inference Assume a distrinutional form for population Take a random sample from the population

If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and If the sample size n is not large relative to the population size N, then Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Standard Normal Distribution for the Sample Means Z-value for the sampling distribution of : where: = sample mean = population mean = standard error of the mean Z is a standardized normal random variable with mean of 0 and a variance of 1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (both distributions have the same mean) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties (continued) Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (the distribution of has a reduced standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties (continued) As n increases, decreases Larger sample size Smaller sample size Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem (continued) Let X1, X2, . . . , Xn be a set of n independent random variables having identical distributions with mean µ, variance σ2, and X as the mean of these random variables. As n becomes large, the central limit theorem states that the distribution of approaches the standard normal distribution Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As the sample size gets large enough… n↑ Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency Sampling Distribution (becomes normal as n increases) Variation Larger sample size Smaller sample size Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

How Large is Large Enough? For most distributions, n > 25 will give a sampling distribution that is nearly normal For normal population distributions, the sampling distribution of the mean is always normally distributed Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example Suppose a large population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 25) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Example Solution (continued): (continued) Z X Population Distribution Sampling Distribution Standard Normal Distribution .1554 +.1554 ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? -0.4 0.4 7.8 8.2 Z X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Acceptance Intervals Goal: determine a range within which sample means are likely to occur, given a population mean and variance By the Central Limit Theorem, we know that the distribution of X is approximately normal if n is large enough, with mean μ and standard deviation Let zα/2 be the z-value that leaves area α/2 in the upper tail of the normal distribution (i.e., the interval - zα/2 to zα/2 encloses probability 1 – α) Then is the interval that includes X with probability 1 – α Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distributions of Sample Proportions 6.3 Sampling Distributions Sampling Distributions of Sample Means Sampling Distributions of Sample Proportions Sampling Distributions of Sample Variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distributions of Sample Proportions P = the proportion of the population having some characteristic Sample proportion ( ) provides an estimate of P: 0 ≤ ≤ 1 has a binomial distribution, but can be approximated by a normal distribution when nP(1 – P) > 5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution of p ^ Normal approximation: Properties: and Sampling Distribution .3 .2 .1 0 . 2 .4 .6 8 1 (where P = population proportion) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Z-Value for Proportions Standardize to a Z value with the formula: Where the distribution of Z is a good approximation to the standard normal distribution if nP(1−P) > 5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example If the true proportion of voters who support Proposition A is P = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45? i.e.: if P = 0.4 and n = 200, what is P(0.40 ≤ ≤ 0.45) ? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example (continued) if P = 0.4 and n = 200, what is P(0.40 ≤ ≤ 0.45) ? Find : Convert to standard normal: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Example if P = 0.4 and n = 200, what is P(0.40 ≤ ≤ 0.45) ? (continued) if P = 0.4 and n = 200, what is P(0.40 ≤ ≤ 0.45) ? Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251 Standardized Normal Distribution Sampling Distribution .4251 Standardize .40 .45 1.44 Z Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 7 Estimation: Single Population Statistics for Business and Economics 8th Edition Chapter 7 Estimation: Single Population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-38

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions Form and interpret a confidence interval estimate for a single population proportion Create confidence interval estimates for the variance of a normal population Determine the required sample size to estimate a mean or proportion within a specified margin of error Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-39

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Confidence Intervals Contents of this chapter: Confidence Intervals for the Population Mean, μ when Population Variance σ2 is Known when Population Variance σ2 is Unknown Confidence Intervals for the Population Proportion, P (large samples) Confidence interval estimates for the variance of a normal population Finite population corrections Sample-size determination Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-40

Properties of Point Estimators 7.1 An estimator of a population parameter is a random variable that depends on sample information . . . whose value provides an approximation to this unknown parameter A specific value of that random variable is called an estimate Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-41

Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Upper Confidence Limit Lower Confidence Limit Point Estimate Width of confidence interval Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-42

Point Estimates μ x P Mean Proportion We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) μ x Mean Proportion P Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-43

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Unbiasedness A point estimator is said to be an unbiased estimator of the parameter  if its expected value is equal to that parameter: Examples: The sample mean is an unbiased estimator of μ The sample variance s2 is an unbiased estimator of σ2 The sample proportion is an unbiased estimator of P Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-44

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Unbiasedness (continued) is an unbiased estimator, is biased: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-45

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Bias Let be an estimator of  The bias in is defined as the difference between its mean and  The bias of an unbiased estimator is 0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-46

Exercise In the previous exampleof student grades Population: mean 50, variance 100 or std 10 We took samples of size9 Compute sample means x_bar Replicated this 1000 times Average of these sample meands is almost 50 Standard dev. Of these x_bars is about 10/3

From N(50,10) n=9 Each raw is a sample of size 9 Then x_bar is computed in last colomn Draw histogram for x_bar s It is N(50,3.33) In reality only one sample is taken – Hypothetical – go back in time repeat the exam Due to random randomness -

Assume grades are from a population Assume a distributional form Parameters are unknown Aim of infrential statistics is Estimation of the fixed but unknown parameters Ther is no way to know the values of the population parameters

A simple high school problem Ahmets age is 2 times Ayses age Sum of their ages is 30 What are Ahmet and Ayşe s ages? Ages are fixed unknown, At the begining When the prblem is solved Determine the values

but In th frquencist parametric paradigm Population parameters are fixed bt unknown as well But ther is no way to find their values At best what can be done Estimate those paramters Point or intervfal estimates Test your clames on these parameters

Population distributons Hypothesize distributional forms Test these hypothesis Godness of fit tests chapter 14 If normality of populations is rejected For large ssmle sizes İnference about mean is based on x:bar Central limit theorem Infrence about variance or tests based on variance Use nonparametric tests

Exercise Population normal mean 50, std 10 samle size – n:3 Replicate 1000 times Generate 3 N(50,10) grades Compute x_bar Compute 3i=1(xi-x_bar)2 Compute average of these 3i=1(xi-x_bar)2 Almost 200 = (n-1)*102 E[3i=1(xi-x_bar)2] = (n-1)2 Or E[3i=1(xi-x_bar)2/ (n-1)] = 2

Exercise Population normal mean 50, std 10 sample size – n:3 Replicate 1000 times Generate 3 N(50,10) grades Compute x_bar Compute 3i=1(xi- )2 Compute average of these 3i=1(xi-x_bar)2 Almost 300 = (n-1)*102 E[3i=1(xi- )2] = (n)2 Or E[3i=1(xi-x_bar)2/ (n)] = 2

If the population of grades is uniform rather then normal The statistic 3i=1(xi-x_bar)2/ 2 is not chi-square at all

Confidence Interval Estimation How much uncertainty is associated with a point estimate of a population parameter? An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence interval estimates Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-57

Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Can never be 100% confident Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-58

Confidence Interval and Confidence Level If P(a <  < b) = 1 -  then the interval from a to b is called a 100(1 - )% confidence interval of . The quantity 100(1 - )% is called the confidence level of the interval  is between 0 and 1 In repeated samples of the population, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way. The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-59

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Estimation Process I am 95% confident that μ is between 40 & 60. Random Sample Population Mean X = 50 (mean, μ, is unknown) Sample Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-60

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Confidence Level, (1-) (continued) Suppose confidence level = 95% Also written (1 - ) = 0.95 A relative frequency interpretation: From repeated samples, 95% of all the confidence intervals that can be constructed of size n will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter No probability involved in a specific interval Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-61

Point Estimate ± Margin of Error General Formula The general form for all confidence intervals is: The value of the margin of error depends on the desired level of confidence Point Estimate ± Margin of Error Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-62

Confidence Intervals Confidence Intervals Population Mean Population Proportion Population Variance σ2 Known σ2 Unknown (From normally distributed populations) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-63

Confidence Interval Estimation for the Mean (σ2 Known) 7.2 Assumptions Population variance σ2 is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate: (where z/2 is the normal distribution value for a probability of /2 in each tail) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-64

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Confidence Limits The confidence interval is The endpoints of the interval are Upper confidence limit Lower confidence limit Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-65

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Margin of Error The confidence interval, Can also be written as where ME is called the margin of error The interval width, w, is equal to twice the margin of error Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-66

Reducing the Margin of Error The margin of error can be reduced if the population standard deviation can be reduced (σ↓) The sample size is increased (n↑) The confidence level is decreased, (1 – ) ↓ Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-67

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Finding z/2 Consider a 95% confidence interval: z = -1.96 z = 1.96 Z units: Lower Confidence Limit Upper Confidence Limit X units: Point Estimate Point Estimate Find z.025 = 1.96 from the standard normal distribution table Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-68

Chapter 9 Hypothesis Testing: Single Population Statistics for Business and Economics 8th Edition Chapter 9 Hypothesis Testing: Single Population Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-69

Concepts of Hypothesis Testing 9.1 A hypothesis is a claim (assumption) about a population parameter: population mean population proportion Example: The mean monthly cell phone bill of this city is μ = $52 Example: The proportion of adults in this city with cell phones is P = .88 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-70

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall The Null Hypothesis, H0 States the assumption (numerical) to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) Is always about a population parameter, not about a sample statistic Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-71

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall The Null Hypothesis, H0 (continued) Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo Always contains “=” , “≤” or “” sign May or may not be rejected Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-72

The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 ) Challenges the status quo Never contains the “=” , “≤” or “” sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-73

Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: Population H0: μ = 50 ) Now select a random sample x Is = 20 likely if μ = 50? Suppose the sample If not likely, REJECT mean age is 20: x = 20 Sample Null Hypothesis Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-74

Reason for Rejecting H0 Sampling Distribution of X μ = 50 20 μ = 50 If H0 is true ... then we reject the null hypothesis that μ = 50. If it is unlikely that we would get a sample mean of this value ... ... if in fact this were the population mean… Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-75

Level of Significance,  Defines the unlikely values of the sample statistic if the null hypothesis is true Defines rejection region of the sampling distribution Is designated by  , (level of significance) Typical values are 0.01, 0.05, or 0.10 Is selected by the researcher at the beginning Provides the critical value(s) of the test Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-76

Level of Significance and the Rejection Region Represents critical value a a H0: μ = 3 H1: μ ≠ 3 /2 /2 Rejection region is shaded Two-tail test H0: μ ≤ 3 H1: μ > 3 a Upper-tail test H0: μ ≥ 3 H1: μ < 3 a Lower-tail test Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-77

Errors in Making Decisions Type I Error Reject a true null hypothesis Considered a serious type of error The probability of Type I Error is  Called level of significance of the test Set by researcher in advance Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-78

Errors in Making Decisions (continued) Type II Error Fail to reject a false null hypothesis The probability of Type II Error is β Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-79

Outcomes and Probabilities Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Fail to Correct Decision (1 - ) Type II Error ( β ) Reject Key: Outcome (Probability) a H Reject Type I Error ( ) Correct Decision ( 1 - β ) H a ( 1 - β ) is called the power of the test Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-80

Consequences of Fixing the Significance Level of a Test Once the significance level α is chosen (generally less than 0.10), the probability of Type II error, β, can be found. Investigator chooses significance level (probability of Type I error) Decision rule is established Probability of Type II error follows Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-81

Type I & II Error Relationship Type I and Type II errors can not happen at the same time Type I error can only occur if H0 is true Type II error can only occur if H0 is false If Type I error probability (  ) , then Type II error probability ( β ) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-82

Factors Affecting Type II Error All else equal, β when the difference between hypothesized parameter and its true value β when  β when σ β when n Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-83

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Power of the Test The power of a test is the probability of rejecting a null hypothesis that is false i.e., Power = P(Reject H0 | H1 is true) Power of the test increases as the sample size increases Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-84

Hypothesis Tests for the Mean  Known  Unknown Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-85

Tests of the Mean of a Normal Distribution (σ Known) 9.2 Convert sample result ( ) to a z value Hypothesis Tests for  σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-86

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Rule H0: μ = μ0 H1: μ > μ0 Alternate rule: a Do not reject H0 Reject H0 Z zα μ0 Critical value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-87

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall p-Value p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H0 is true Also called observed level of significance Smallest value of  for which H0 can be rejected Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-88

p-Value Approach to Testing Convert sample result (e.g., ) to test statistic (e.g., z statistic ) Obtain the p-value For an upper tail test: Decision rule: compare the p-value to  If p-value <  , reject H0 If p-value   , do not reject H0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-89

Example: Upper-Tail Z Test for Mean ( Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume  = 10 is known) Form hypothesis test: H0: μ ≤ 52 the average is not over $52 per month H1: μ > 52 the average is greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-90

Example: Find Rejection Region (continued) Suppose that  = .10 is chosen for this test Find the rejection region: Reject H0  = .10 Do not reject H0 Reject H0 1.28 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-91

Example: Sample Results (continued) Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 64, x = 53.1 ( = 10 was assumed known) Using the sample results, Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-92

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example: Decision (continued) Reach a decision and interpret the result: Reject H0  = .10 Do not reject H0 Reject H0 1.28 z = 0.88 Do not reject H0 since z = 0.88 < 1.28 i.e.: there is not sufficient evidence that the mean bill is over $52 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-93

Example: p-Value Solution (continued) Calculate the p-value and compare to  (assuming that μ = 52.0) p-value = .1894 Reject H0  = .10 Do not reject H0 Reject H0 1.28 Z = .88 Do not reject H0 since p-value = .1894 >  = .10 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-94

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall One-Tail Tests In many cases, the alternative hypothesis focuses on one particular direction This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 H0: μ ≤ 3 H1: μ > 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H0: μ ≥ 3 H1: μ < 3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-95

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Upper-Tail Tests H0: μ ≤ 3 H1: μ > 3 There is only one critical value, since the rejection area is in only one tail a Do not reject H0 Reject H0 zα Z μ Critical value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-96

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Lower-Tail Tests H0: μ ≥ 3 H1: μ < 3 There is only one critical value, since the rejection area is in only one tail a Reject H0 Do not reject H0 -z Z μ Critical value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-97

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Two-Tail Tests In some settings, the alternative hypothesis does not specify a unique direction H0: μ = 3 H1: μ ¹ 3 /2 /2 There are two critical values, defining the two regions of rejection x 3 Reject H0 Do not reject H0 Reject H0 -z/2 +z/2 z Lower critical value Upper critical value Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-98

Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is equal to 3. (Assume σ = 0.8) State the appropriate null and alternative hypotheses H0: μ = 3 , H1: μ ≠ 3 (This is a two tailed test) Specify the desired level of significance Suppose that  = .05 is chosen for this test Choose a sample size Suppose a sample of size n = 100 is selected Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-99

Hypothesis Testing Example (continued) Determine the appropriate technique σ is known so this is a z test Set up the critical values For  = .05 the critical z values are ±1.96 Collect the data and compute the test statistic Suppose the sample results are n = 100, x = 2.84 (σ = 0.8 is assumed known) So the test statistic is: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-100

Hypothesis Testing Example (continued) Is the test statistic in the rejection region? Reject H0 if z < -1.96 or z > 1.96; otherwise do not reject H0  = .05/2  = .05/2 Reject H0 Do not reject H0 Reject H0 -z = -1.96 +z = +1.96 Here, z = -2.0 < -1.96, so the test statistic is in the rejection region Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-101

Hypothesis Testing Example (continued) Reach a decision and interpret the result  = .05/2  = .05/2 Reject H0 Do not reject H0 Reject H0 -z = -1.96 +z = +1.96 -2.0 Since z = -2.0 < -1.96, we reject the null hypothesis and conclude that there is sufficient evidence that the mean number of TVs in US homes is not equal to 3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-102

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example: p-Value Example: How likely is it to see a sample mean of 2.84 (or something further from the mean, in either direction) if the true mean is  = 3.0? x = 2.84 is translated to a z score of z = -2.0 /2 = .025 /2 = .025 .0228 .0228 p-value = .0228 + .0228 = .0456 -1.96 1.96 Z -2.0 2.0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-103

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example: p-Value (continued) Compare the p-value with  If p-value <  , reject H0 If p-value   , do not reject H0 Here: p-value = .0456  = .05 Since .0456 < .05, we reject the null hypothesis /2 = .025 /2 = .025 .0228 .0228 -1.96 1.96 Z -2.0 2.0 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-104

Tests of the Mean of a Normal Population (σ Unknown) 9.3 Convert sample result ( ) to a t test statistic Hypothesis Tests for  σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-105

Tests of the Mean of a Normal Population (σ Unknown) (continued) For a two-tailed test: Consider the test (Assume the population is normal, and the population variance is unknown) The decision rule is: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-106

Example: Two-Tail Test ( Unknown) The average cost of a hotel room in Chicago is said to be $168 per night. A random sample of 25 hotels resulted in x = $172.50 and s = $15.40. Test at the  = 0.05 level. (Assume the population distribution is normal) H0: μ = 168 H1: μ ¹ 168 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-107

Example Solution: Two-Tail Test H0: μ = 168 H1: μ ¹ 168 a/2=.025 a/2=.025 a = 0.05 n = 25  is unknown, so use a t statistic Critical Value: t24 , .025 = ± 2.064 Reject H0 Do not reject H0 Reject H0 t n-1,α/2 -t n-1,α/2 2.064 -2.064 1.46 Do not reject H0: not sufficient evidence that true mean cost is different than $168 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-108

Tests of the Population Proportion 9.4 Involves categorical variables Two possible outcomes “Success” (a certain characteristic is present) “Failure” (the characteristic is not present) Fraction or proportion of the population in the “success” category is denoted by P Assume sample size is large Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-109

Tests of the Population Proportion (continued) The sample proportion in the success category is denoted by When nP(1 – P) > 5, can be approximated by a normal distribution with mean and standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-110

Hypothesis Tests for Proportions The sampling distribution of is approximately normal, so the test statistic is a z value: Hypothesis Tests for P nP(1 – P) > 5 nP(1 – P) < 5 Not discussed in this chapter Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-111

Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the  = .05 significance level. Check: Our approximation for P is = 25/500 = .05 nP(1 - P) = (500)(.05)(.95) = 23.75 > 5  Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-112

Z Test for Proportion: Solution Test Statistic: H0: P = .08 H1: P ¹ .08 a = .05 n = 500, = .05 Decision: Critical Values: ± 1.96 Reject H0 at  = .05 Reject Reject Conclusion: .025 .025 There is sufficient evidence to reject the company’s claim of 8% response rate. z -1.96 1.96 -2.47 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-113

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall p-Value Solution (continued) Calculate the p-value and compare to  (For a two sided test the p-value is always two sided) Do not reject H0 Reject H0 Reject H0 p-value = .0136: /2 = .025 /2 = .025 .0068 .0068 -1.96 1.96 Z = -2.47 Z = 2.47 Reject H0 since p-value = .0136 <  = .05 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9-114