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### Inference about means and mean differences

• 1. © aSup-2007 Inference about Means and Mean Differences    1 PART III Inference about Means and Mean Differences
• 2. © aSup-2007 Inference about Means and Mean Differences    2 Chapter 8 INTRODUCTION TO HYPOTHESIS TESTING
• 3. © aSup-2007 Inference about Means and Mean Differences    3 The Logic of Hypothesis Testing  It usually is impossible or impractical for a researcher to observe every individual in a population  Therefore, researchers usually collect data from a sample and then use the sample data to answer question about the population  Hypothesis testing is statistical method that uses sample data to evaluate a hypothesis about the population
• 4. © aSup-2007 Inference about Means and Mean Differences    4 The Hypothesis Testing Procedure 1. State a hypothesis about population, usually the hypothesis concerns the value of a population parameter 2. Before we select a sample, we use hypothesis to predict the characteristics that the sample have. The sample should be similar to the population 3. We obtain a sample from the population (sampling) 4. We compare the obtain sample data with the prediction that was made from the hypothesis
• 5. © aSup-2007 Inference about Means and Mean Differences    5 PROCESS OF HYPOTHESIS TESTING  It assumed that the parameter μ is known for the population before treatment  The purpose of the experiment is to determine whether or not the treatment has an effect on the population mean Known population before treatment μ = 30 TREATMENT Unknown population after treatment μ = ?
• 6. © aSup-2007 Inference about Means and Mean Differences    6 EXAMPLE  It is known from national health statistics that the mean weight for 2-year-old children is μ = 26 pounds and σ = 4 pounds  The researcher’s plan is to obtain a sample of n = 16 newborn infants and give their parents detailed instruction for giving their children increased handling and stimulation  NOTICE that the population after treatment is unknown
• 7. © aSup-2007 Inference about Means and Mean Differences    7 STEP-1: State the Hypothesis  H0 : μ = 26 (even with extra handling, the mean at 2 years is still 26 pounds)  H1 : μ ≠ 26 (with extra handling, the mean at 2 years will be different from 26 pounds)  Example we use α = .05 two tail
• 8. © aSup-2007 Inference about Means and Mean Differences    8 STEP-2: Set the Criteria for a Decision  Sample means that are likely to be obtained if H0 is true; that is, sample means that are close to the null hypothesis  Sample means that are very unlikely to be obtained if H0 is false; that is, sample means that are very different from the null hypothesis  The alpha level or the significant level is a probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true
• 9. © aSup-2007 Inference about Means and Mean Differences    9 The location of the critical region boundaries for three different los -1.96 1.96 -2.58 2.58 -3.30 3.30 α = .05 α = .01 α = .001
• 10. © aSup-2007 Inference about Means and Mean Differences    10 STEP-3: Collect Data and Compute Sample Statistics  After obtain the sample data, summarize the appropriate statistic σM = σ √n z = M - μ σM NOTICE  That the top of the z-scores formula measures how much difference there is between the data and the hypothesis  The bottom of the formula measures standard distances that ought to exist between the sample mean and the population mean
• 11. © aSup-2007 Inference about Means and Mean Differences    11 STEP-4: Make a Decision  Whenever the sample data fall in the critical region then reject the null hypothesis  It’s indicate there is a big discrepancy between the sample and the null hypothesis (the sample is in the extreme tail of the distribution)
• 12. © aSup-2007 Inference about Means and Mean Differences    12 HYPOTHESIS TEST WITH z  A standardized test that are normally distributed with μ = 65 and σ = 15. The researcher suspect that special training in reading skills will produce a change in scores for individuals in the population. A sample of n = 25 individual is selected, the average for this sample is M = 70.  Is there evidence that the training has an effect on test score? LEARNING CHECK
• 13. © aSup-2007 Inference about Means and Mean Differences    13 FACTORS THAT INFLUENCE A HYPOTHESIS TEST  The size of difference between the sample mean and the original population mean  The variability of the scores, which is measured by either the standard deviation or the variance  The number of score in the sample σM = σ √n z = M - μ σM
• 14. © aSup-2007 Inference about Means and Mean Differences    14 DIRECTIONAL (ONE-TAILED) HYPOTHESIS TESTS  Usually a researcher begin an experiment with a specific prediction about the direction of the treatment effect  For example, a special training program is expected to increase student performance  In this situation, it possible to state the statistical hypothesis in a manner that incorporates the directional prediction into the statement of H0 and H1
• 15. © aSup-2007 Inference about Means and Mean Differences    15 A psychologist has developed a standardized test for measuring the vocabulary skills of 4- year-old children. The score on the test form a normal distribution with μ = 60 and σ = 10. A researcher would like to use this test to investigate the hypothesis that children who grow up as an only child develop vocabulary skills at a different rate than children in large family. A sample of n = 25 only children is obtained, and the mean test score for this sample is M = 63. LEARNING CHECK
• 16. © aSup-2007 Inference about Means and Mean Differences    UNCERTAINTY AND ERRORS IN HYPOTHESIS TESTING  Hypothesis testing is a inferential process, which means that it uses limited information as the basis for reaching a general conclusion  Although sample data usually representative of the population, there is always a chance that the sample is misleading and will cause a researcher to make the wrong decision about the research results 16
• 17. © aSup-2007 Inference about Means and Mean Differences    Type I Error  … occurs when a researcher rejects a null hypothesis that is actually true.  In a typical research situation, a Type I Error means the researcher conclude that a treatment does have an effect when in fact it has no effect. 17
• 18. © aSup-2007 Inference about Means and Mean Differences    Type II Error  … occurs when a researcher fail to reject a null hypothesis that is really false.  In a typical research situation, a Type II Error means that the hypothesis test has failed to detect real treatment effect 18
• 19. © aSup-2007 Inference about Means and Mean Differences    Actual Situation No Effect, H0 False Effect Exist, H0 False Reject H0 Type I Error Decision Correct Retain H0 Decision Correct Type II Error 19
• 20. © aSup-2007 Inference about Means and Mean Differences    20 Chapter 9 INTRODUCTION TO t STATISTIC
• 21. © aSup-2007 Inference about Means and Mean Differences    21 THE t STATISTIC: AN ALTERNATIVE TO z  In the previous chapter, we presented the statistical procedure that permit researcher to use sample mean to test hypothesis about an unknown population  Remember that the expected value of the distribution of sample means is μ, the population mean
• 22. © aSup-2007 Inference about Means and Mean Differences    22 The statistical procedure were based on a few basic concepts: 1. A sample mean (M) is expected more or less to approximate its population mean (μ). This permits us to use sample mean to test a hypothesis about the population mean. 2. The standard error provide a measure of how well a sample mean approximates the population mean. Specially, the standard error determines how much difference between M and μ is reasonable to expect just by chance.
• 23. © aSup-2007 Inference about Means and Mean Differences    23 The statistical procedure were based on a few basic concepts: 3. To quantify our inferences about the population, we compare the obtained sample mean (M) with the hypothesized population mean (μ) by computing a z- score test statistic
• 24. © aSup-2007 Inference about Means and Mean Differences    24 THE t STATISTIC: AN ALTERNATIVE TO z The goal of the hypothesis test is to determine whether or not the obtained result is significantly greater than would be expected by chance.
• 25. © aSup-2007 Inference about Means and Mean Differences    25 THE PROBLEM WITH z-SCORE  A z-score requires that we know the value of the population standard deviation (or variance), which is needed to compute the standard error  In most situation, however, the standard deviation for the population is not known  In this case, we cannot compute the standard error and z-score for hypothesis test. We use t statistic for hypothesis testing when the population standard deviation is unknown
• 26. © aSup-2007 Inference about Means and Mean Differences    26 Introducing t Statistic σM = σ √n Now we will estimates the standard error by simply substituting the sample variance or standard deviation in place of the unknown population value SM = s √n Notice that the symbol for estimated standard error of M is SM instead of σM , indicating that the estimated value is computed from sample data rather than from the actual population parameter
• 27. © aSup-2007 Inference about Means and Mean Differences    27 z-score and t statistic σM = σ √n z = M - μ σM SM = s √n t = M - μ SM
• 28. © aSup-2007 Inference about Means and Mean Differences    28 The t Distribution  Every sample from a population can be used to compute a z-score or a statistic  If you select all possible samples of a particular size (n), then the entire set of resulting z-scores will form a z-score distribution  In the same way, the set of all possible t statistic will form a t distribution
• 29. © aSup-2007 Inference about Means and Mean Differences    29 The Shape of the t Distribution  The exact shape of a t distribution changes with degree of freedom  There is a different sampling distribution of t (a distribution of all possible sample t values) for each possible number of degrees of freedom  As df gets very large, then t distribution gets closer in shape to a normal z-score distribution
• 30. © aSup-2007 Inference about Means and Mean Differences    30 HYPOTHESIS TESTS WITH t STATISTIC  The goal is to use a sample from the treated population (a treated sample) as the determining whether or not the treatment has any effect Known population before treatment Unknown population after treatment μ = 30 μ = ? TREATMENT
• 31. © aSup-2007 Inference about Means and Mean Differences    31 HYPOTHESIS TESTS WITH t STATISTIC  As always, the null hypothesis states that the treatment has no effect; specifically H0 states that the population mean is unchanged  The sample data provides a specific value for the sample mean; the variance and estimated standard error are computed t = sample mean (from data) Estimated standard error (computed from the sample data) population mean (hypothesized from H0)-
• 32. © aSup-2007 Inference about Means and Mean Differences    32 A psychologist has prepared an “Optimism Test” that is administered yearly to graduating college seniors. The test measures how each graduating class feels about it future. The higher the score, the more optimistic the class. Last year’s class had a mean score of μ = 19. A sample of n = 9 seniors from this years class was selected and tested. The scores for these seniors are as follow: 19 24 23 27 19 20 27 21 18 On the basis of this sample, can the psychologist conclude that this year’s class has a different level of optimism than last year’s class? LEARNING CHECK
• 33. © aSup-2007 Inference about Means and Mean Differences    33 STEP-1: State the Hypothesis, and select an alpha level  H0 : μ = 19 (there is no change)  H1 : μ ≠ 19 (this year’s mean is different)  Example we use α = .05 two tail
• 34. © aSup-2007 Inference about Means and Mean Differences    34 STEP-2: Locate the critical region  Remember that for hypothesis test with t statistic, we must consult the t distribution table to find the critical t value. With a sample of n = 9 students, the t statistic will have degrees of freedom equal to df = n – 1 = 9 – 1 = 8  For a two tailed test with α = .05 and df = 8, the critical values are t = ± 2.306. The obtained t value must be more extreme than either of these critical values to reject H0
• 35. © aSup-2007 Inference about Means and Mean Differences    35 STEP-3: Obtain the sample data, and compute the test statistic  Find the sample mean  Find the sample variances  Find the estimated standard error SM  Find the t statistic SM = s √n t = M - μ SM
• 36. © aSup-2007 Inference about Means and Mean Differences    36 STEP-4: Make a decision about H0, and state conclusion  The obtained t statistic (t = -4.39) is in the critical region. Thus our sample data are unusual enough to reject the null hypothesis at the .05 level of significance.  We can conclude that there is a significant difference in level of optimism between this year’s and last year’s graduating classes t(8) = -4.39, p<.05, two tailed
• 37. © aSup-2007 Inference about Means and Mean Differences    37 The critical region in the t distribution for α = .05 and df = 8 Reject H0 Reject H0 Fail to reject H0 -2.306 2.306
• 38. © aSup-2007 Inference about Means and Mean Differences    38 DIRECTIONAL HYPOTHESES AND ONE-TAILED TEST  The non directional (two-tailed) test is more commonly used than the directional (one- tailed) alternative  On other hand, a directional test may be used in some research situations, such as exploratory investigation or pilot studies or when there is a priori justification (for example, a theory previous findings)
• 39. © aSup-2007 Inference about Means and Mean Differences    39 A fund raiser for a charitable organization has set a goal of averaging at least \$ 25 per donation. To see if the goal is being met, a random sample of recent donation is selected. The data for this sample are as follows: 20 50 30 25 15 20 40 50 10 20 LEARNING CHECK
• 40. © aSup-2007 Inference about Means and Mean Differences    40 The critical region in the t distribution for α = .05 and df = 9 Reject H0 Fail to reject H0 1.883
• 41. © aSup-2007 Inference about Means and Mean Differences    41 Chapter 10 THE t TEST FOR TWO INDEPENDENT SAMPLES
• 42. © aSup-2007 Inference about Means and Mean Differences    42 OVERVIEW  Single sample techniques are used occasionally in real research, most research studies require the comparison of two (or more) sets of data  There are two general research strategies that can be used to obtain of the two sets of data to be compared: ○ The two sets of data come from the two completely separate samples (independent-measures or between-subjects design) ○ The two sets of data could both come from the same sample (repeated-measures or within subject design)
• 43. © aSup-2007 Inference about Means and Mean Differences    43 Do the achievement scores for students taught by method A differ from the scores for students taught by method B? In statistical terms, are the two population means the same or different? Unknown µ =? Sample A Unknown µ =? Sample B Taught by Method A Taught by Method B
• 44. © aSup-2007 Inference about Means and Mean Differences    44 THE HYPOTHESES FOR AN INDEPENDENT-MEASURES TEST  The goal of an independent-measures research study is to evaluate the mean difference between two population (or between two treatment conditions) H0: µ1 - µ2 = 0 (No difference between the population means) H1: µ1 - µ2 ≠ 0 (There is a mean difference)
• 45. © aSup-2007 Inference about Means and Mean Differences    45 THE FORMULA FOR AN INDEPENDENT- MEASURES HYPOTHESIS TEST  In this formula, the value of M1 – M2is obtained from the sample data and the value for µ1 - µ2 comes from the null hypothesis  The null hypothesis sets the population mean different equal to zero, so the independent- measures t formula can be simplifier further t = sample mean difference estimated standard error population mean difference- = M1 – M2 S (M1 – M2)
• 46. © aSup-2007 Inference about Means and Mean Differences    46 THE STANDARD ERROR To develop the formula for S(M1–M2) we will consider the following points:  Each of the two sample means represent its own population mean, but in each case there is some error SM = s2 n√ SM1-M2 = s1 2 n1√ s2 2 n2 +
• 47. © aSup-2007 Inference about Means and Mean Differences    47 POOLED VARIANCE  The standard error is limited to situation in which the two samples are exactly the same size (that is n1 – n2)  In situations in which the two sample size are different, the formula is biased and, therefore, inappropriate  The bias come from the fact that the formula treats the two sample variance
• 48. © aSup-2007 Inference about Means and Mean Differences    48 POOLED VARIANCE  for the independent-measure t statistic, there are two SS values and two df values) SP 2 = SS n SM1-M2 = s1 2 n1√ s2 2 n2 +
• 49. © aSup-2007 Inference about Means and Mean Differences    49 HYPOTHESIS TEST WITH THE INDEPENDENT-MEASURES t STATISTIC In a study of jury behavior, two samples of participants were provided details about a trial in which the defendant was obviously guilty. Although Group-2 received the same details as Group-1, the second group was also told that some evidence had been withheld from the jury by the judge. Later participants were asked to recommend a jail sentence. The length of term suggested by each participant is presented. Is there a significant difference between the two groups in their responses?
• 50. © aSup-2007 Inference about Means and Mean Differences    50 THE LENGTH OF TERM SUGGESTED BY EACH PARTICIPANT Group-1 scores: 4 4 3 2 5 1 1 4 Group-2 scores: 3 7 8 5 4 7 6 8 There are two separate samples in this study. Therefore the analysis will use the independent-measure t test
• 51. © aSup-2007 Inference about Means and Mean Differences    51 STEP-1: State the Hypothesis, and select an alpha level  H0 : μ1 - μ2 = 0 (for the population, knowing evidence has been withheld has no effect on the suggested sentence)  H1 : μ1 - μ2 ≠ 0 (for the population, knowledge of withheld evidence has an effect on the jury’s response)  We will set α = .05 two tail
• 52. © aSup-2007 Inference about Means and Mean Differences    52 STEP-2: Identify the critical region  For the independent-measure t statistic, degrees of freedom are determined by df = n1 + n2 – 2 = 8 + 8 – 2 = 14  The t distribution table is consulted, for a two tailed test with α = .05 and df = 14, the critical values are t = ± 2.145.  The obtained t value must be more extreme than either of these critical values to reject H0
• 53. © aSup-2007 Inference about Means and Mean Differences    53 STEP-3: Compute the test statistic  Find the sample mean for each group M1= 3 and M2 = 6  Find the SS for each group SS1 = 16 and SS2 = 24  Find the pooled variance, and SP 2 = 2.86  Find estimated standard error S(M1-M2) = 0.85
• 54. © aSup-2007 Inference about Means and Mean Differences    54 STEP-3: Compute the t statistic t = M1 – M2 S (M1 – M2) = -3 0.85 = -3.53
• 55. © aSup-2007 Inference about Means and Mean Differences    55 STEP-4: Make a decision about H0, and state conclusion  The obtained t statistic (t = -3.53) is in the critical region on the left tail (critical t = ± 2.145). Therefore, the null hypothesis is rejected.  The participants that were informed about the withheld evidence gave significantly longer sentences, t(14) = -3.53, p<.05, two tails
• 56. © aSup-2007 Inference about Means and Mean Differences    56 The critical region in the t distribution for α = .05 and df = 14 Reject H0 Reject H0 Fail to reject H0 -2.145 2.145
• 57. © aSup-2007 Inference about Means and Mean Differences    57 LEARNING CHECK The following data are from two separate independent-measures experiments. Without doing any calculation, which experiment is more likely to demonstrate a significant difference between treatment A and B? Explain your answer. EXPERIMENT A EXPERIMENT B Treatment A Treatment B Treatment A Treatment B n = 10 n = 10 n = 10 n = 10 M = 42 M = 52 M = 61 M = 71 SS = 180 SS = 120 SS = 986 SS = 1042
• 58. © aSup-2007 Inference about Means and Mean Differences    58 A psychologist studying human memory, would like to examine the process of forgetting. One group of participants is required to memorize a list of words in the evening just before going to bed. Their recall is tested 10 hours latter in the morning. Participants in the second group memorized the same list of words in he morning, and then their memories tested 10 hours later after being awake all day. LEARNING CHECK
• 59. © aSup-2007 Inference about Means and Mean Differences    59 LEARNING CHECK The psychologist hypothesizes that there will be less forgetting during less forgetting during sleep than a busy day. The recall scores for two samples of college students are follows: Asleep Scores Awake Scores 15 13 14 14 15 13 14 12 16 15 16 15 14 13 11 12 16 15 17 14 13 13 12 14
• 60. © aSup-2007 Inference about Means and Mean Differences    60  Sketch a frequency distribution for the ‘asleep’ group. On the same graph (in different color), sketch the distribution for the ‘awake’ group. Just by looking at these two distributions, would you predict a significant differences between two treatment conditions?  Use the independent-measures t statistic to determines whether there is a significant difference between the treatments. Conduct the test with α = .05 LEARNING CHECK
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