Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide Slide 1 of 54

Outline 1. Definitions 2. Six Steps Things that Effect Means 1 population 1. When σ 2 is known 2. When σ 2 is not known Types of errors and correct decisions. When the alternative is true. 1. 2. Ways of increasing power 3. Choosing sample size Hypothesis Testing or How to Decide to Decide Slide 2 of 54

Hypothesis Testing Definitions A hypothesis is a statement that may or may not be true. Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions There are two kinds of hypotheses. Scientific Statistical Scientific Hypotheses is a statement about what should be observed based on some theory about a particular behavior(s).// The scientific theory provides Things that Effect Guidance about what to observe and what should happen. Explanations about why it occurs. Hypothesis Testing or How to Decide to Decide Slide 3 of 54

Scientific Hypotheses Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions Things that Effect Phenomenon: Students in academic high school programs tend to have higher achievement test scores than those in general and/or vocational/techinical programs. Scientific hypotheses (theories): Certain students are in academic programs because they are good in academic subjects, plan to go to college, etc. Students in academic programs are taught more academic subjects than those in general and/or vocational-technical programs. Hypothesis Testing or How to Decide to Decide Slide 4 of 54

Statistical Hypotheses Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions Statistical Hypotheses are statements about one or more population distribution(s) and usually about one or more population parameter. Examples: H: Reading scores of high school seniors attending academic programs are N(55, 80). Things that Effect H: Reading scores of high school seniors attending academic programs are N(55, 80) and scores for students attending general programs are N(50, 100). Hypothesis Testing or How to Decide to Decide Slide 5 of 54

Two Types of Statistical Hypotheses Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions Simple hypothesis completely specifies a population distribution the sampling distribution of any statistic is also completely known (once you know the sample size). e.g., The ones given on previous page. Composite hypotheses the population distribution is not completely specified. e.g., H: population is normal and µ = 55 Things that Effect ( exact parameter value) H: µ 50 ( range of parameter values). Hypothesis Testing or How to Decide to Decide Slide 6 of 54

Testing THE Hypothesis THE hypothesis is the null hypothesis, H o Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions Testing the null hypothesis is making a choice between two hypotheses or models. These two hypotheses are: Null hypothesis, H O : Things that Effect Assumed to be true. It determines the sampling distribution of the test statistic H o must be an exact hypothesis. The Alternative hypothesis, H a, is true if the null is false H a is a composite hypothesis. Hypothesis Testing or How to Decide to Decide Slide 7 of 54

Testing Hypotheses & Assumptions Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions The alternative hypothesis is usually what corresponds to the expected result (based on your scientific hypothesis). E.g., H o : µ = 50 and H a : µ > 50. H o only specifies a particular value for the mean. H a gives a range of possible values. Need the sampling distribution for a test statistic must make assumptions. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 8 of 54

Hypotheses versus Assumptions Hypothesis Testing Definitions Scientific Hypotheses Statistical Hypotheses Two Types of Statistical Hypotheses Testing THE Hypothesis Testing Hypotheses & Assumptions Hypotheses versus Assumptions Things that Effect The difference between assumptions and hypotheses is that assumptions are exactly the same under H o and H a but the hypotheses differ. For example, Null Alternative Assumptions Ȳ is Normal Ȳ is Normal Independence Independence σ 2 σ 2 Hypothesis H o : µ = 55 H a : µ 55 Hypothesis Testing or How to Decide to Decide Slide 9 of 54

in a Statistical Hypothesis Test For illustration, use test on mean of a single variable (i.e., HSB who attend academic programs and reading scores). in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Things that Effect 1. State the null and alternative hypotheses. H o : µ = 50 vs H a : µ 50 (or H a : µ > 50) 2. Make assumptions so that the sampling distribution of the sample statistic is completely specified. Observations are independent. Come from a population with σ 2 = 100. Sampling distribution of Ȳ is normal (note: n = 308). 3. Specify the degree of risk or α level. This equals the probability of rejecting the null hypothesis when the null hypothesis is true. α =.05 Hypothesis Testing or How to Decide to Decide Slide 10 of 54

Step 4 in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps 4. Compute the probability of obtaining the sample statistic that differs from the hypothesized valued (from H o ). This probability is called the p value. To find the p value, you need to know what is the sampling distribution of a test statistic.... Test statistic: For our example, z = ȳ 50 σ/ n = 55.89 50 10/ 308 = 10.34 Sampling distribution of z is N(0, 1) because Assumed that Ȳ is normal. A linear transformation of a normal R.V. is a normal variable. E(z) = 0 (if null is true) and var(z) = 1. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 11 of 54

Step 4 The Sampling Distribution of Ȳ assuming H o : in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Things that Effect Hypothesis Testing or How to Decide to Decide Slide 12 of 54

Step 4 The Sampling Distribution of Ȳ and test statistic in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Things that Effect Hypothesis Testing or How to Decide to Decide Slide 13 of 54

Step 4 & Step 5 in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps 4. (continued) The probability of observing a standard normal random variable with a z-score of 10.34 or larger is p =.5 10 24. 5. Make a decision. Reject H o or Retain H o. There are equivalent ways of doing this, but they all depend on Your alternative hypothesis The following probability ( Prob z α/2 Ȳ µ ) z α/2 σ Ȳ Things that Effect Hypothesis Testing or How to Decide to Decide Slide 14 of 54

Rules for Non-directional Alternatives Two-Tailed Tests H a : µ 50. in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Using p values and α, the rule is If p > α/2 then retain H o If p < α/2 then reject H o e.g., p =.5 10 24 <<.05/2 =.025 Compare the test statistic to a critical value If z α/2 z z α/2, then retain H o If z < z α/2 or z > z α/2, then reject H o e.g., z = 10.34 > z.05/2 = 1.96, so reject H o. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 15 of 54

Rejection Region in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Things that Effect Hypothesis Testing or How to Decide to Decide Slide 16 of 54

Rules for Non-directional H a Compare Ȳ to the critical mean value. Compute Ȳ lower = µ z α/2 σ Ȳ and Ȳ upper = µ + z α/w σ Ȳ. in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps The rule is If Ȳ lower Ȳ Ȳ upper, then retain H o If Ȳ < Ȳ lower or Ȳ > Ȳ upper, then reject H o e.g., Ȳlower = 50 1.96(10/ 308) = 48.88 and Ȳ upper = 50 + 1.96(10/ 308) = 51.12. The obtained sample mean Ȳ = 55.89 is not in the interval (48.88 to 51.12), reject H o. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 17 of 54

Critical Mean Values in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Things that Effect Hypothesis Testing or How to Decide to Decide Slide 18 of 54

Last Rule for Non-directional H a Ȳ fall inside the (1 α) 100% confidence interval? The (1 α) 100% confidence interval is in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Ȳ ± z α/2 σ Ȳ Rule: If the hypothesized value is in interval, retain H o, and if the hypothesized value is outside the interval, reject H o. e.g., 95% confidence interval for µ reading is 55.89 ± 1.96(10/ 308) (54.77, 57.01) The interval does not contain 50 reject H o. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 19 of 54

Directional Alternatives e.g., H a : µ reading > µ 0 or H a : µ reading < µ o in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Using p-values: If p > α, then retain H o if p < α, then reject H o Compare test statistic to critical value of the test statistic. Either If H a : µ < µ o then if z < z /alpha, reject H o or If H a : µ > µ o, then if z > z /alpha, reject H o Things that Effect Hypothesis Testing or How to Decide to Decide Slide 20 of 54

Directional Alternatives continued in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps Compare obtained sample statistic (e.g., Ȳ ) to the critical value of the statistic. If H a : µ < µ o, then compute Ȳcrit = µ o z α σ Ȳ. Reject H o if Ȳ < Ȳ crit. If H a : µ > µ o, then compute Ȳ crit = µ o + z α σ Ȳ. Reject H o if Ȳ > Ȳcrit. A one-sided confidence interval... Things that Effect Hypothesis Testing or How to Decide to Decide Slide 21 of 54

1 vs 2 Tail Tests in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps The rejection regions for 1 and 2 tailed tests with the same α level are different. The rejection region for 1 tail test is larger. You must specify H a before you look at your data; otherwise, your probability values are not valid. It s cheating to go data snooping as way to decide on H a. Statistical significance means that the difference between what you observed/obtained and what you expected assuming H O is statistically large and it was unlikely to have happened. The result is more likely under H a. It is not correct to state that H O (or H A ) is true or false. Always possible that we ve made a mistake. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 22 of 54

1 vs 2 Tail Tests The decision does not tell you that in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps The difference is important. The result is meaningful. The scientific theory (explanation) guiding your research is correct. Step 6: interpretation. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 23 of 54

Summary of the Six Steps in a Statistical Hypothesis Test & Step 5 Alternatives Rejection Region Critical Mean Values Last Rule for Non-directional Directional Alternatives Directional Alternatives continued Summary of the Six Steps 1. State the null and alternative hypotheses. 2. Make assumptions. 3. Specify the α level. 4. Compute test statistic, p-value, CI, or critical value of sample statistic. 5. Make a decision. 6. Interpret the result. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 24 of 54

When σ 2 is Unknown The process is the same, just two details change Must estimate σ 2. When σ 2 is Unknown Different Sampling Distribution Student t-distribution Standard Normal vs t-distribution Standard Normal vs t-distribution We ll use the unbiased one: s 2 = 1 n (y i ȳ) 2 n 1 i=1 and use s/ (n) as an estimate of σ Ȳ. Things that Effect The test statistic is now, t = Ȳ µ o s/ n = Ȳ µ o s Ȳ Need a different sampling distribution for our test statistic.... Hypothesis Testing or How to Decide to Decide Slide 25 of 54

Different Sampling Distribution Ȳ is the only random variable in When σ 2 is Unknown Different Sampling Distribution Student t-distribution Standard Normal vs t-distribution Standard Normal vs t-distribution Things that Effect z = Ȳ µ σ Ȳ Both Ȳ and s Ȳ are random variables in t = Ȳ µ o s/ n = Ȳ µ o s Ȳ Therefore, expect that our t statistic to be more variable than z. The sampling distribution for t must have heavier tails than the normal distribution ( leptokurtic ). Student t-distribution Hypothesis Testing or How to Decide to Decide Slide 26 of 54

Student t-distribution Symmetric around the mean µ = 0, central t-distribution When σ 2 is Unknown Different Sampling Distribution Student t-distribution Standard Normal vs t-distribution Standard Normal vs t-distribution Bell shaped. Heavier tails than normal distribution. Depends on the degrees of freedom. Notation: Greek nu, ν = degrees of freedom. Things that Effect ν is a parameter of the central t distribution. ν = n 1 Hypothesis Testing or How to Decide to Decide Slide 27 of 54

Standard Normal vs t-distribution When σ 2 is Unknown Different Sampling Distribution Student t-distribution Standard Normal vs t-distribution Standard Normal vs t-distribution The standard normal is the limiting distribution of the t-distribution: As ν, t ν N(0, 1) For very large n, the t-distribution is very close to N(0, 1). HSB reading H o : µ = 50 vs H a : µ > 50, t = p-value =.32 10 23. 55.89 50 87.14/308 = 5.89.53 = 11.098 Things that Effect Note z.025 = 1.968 compared to t.025,ν=307 = 1.969 Hypothesis Testing or How to Decide to Decide Slide 28 of 54

Standard Normal vs t-distribution When σ 2 is Unknown Different Sampling Distribution Student t-distribution Standard Normal vs t-distribution Standard Normal vs t-distribution Things that Effect Hypothesis Testing or How to Decide to Decide Slide 29 of 54

Errors and Correction Decisions Errors and Correction Decisions Errors and Correction Decisions Graphically, what s going on True State of the World H O H A H O Correct Type II Error Decision (1 α) β H A Type I Error Correct α power (1 β) 1.00 1.00 Things that Effect Hypothesis Testing or How to Decide to Decide Slide 30 of 54

Errors and Correction Decisions Errors and Correction Decisions Errors and Correction Decisions Graphically, what s going on Things that Effect Probabilities of events when the null is true: Correct P(retainH O H O is true) = 1 α Type 1 error: P(rejectH O H O is true) = α Probabilities when the alternative is true: Type 2 error: P(retainH o H A is true) = β : P(rejectH o H A is true) = 1 β These are conditional probabilities. Hypothesis Testing or How to Decide to Decide Slide 31 of 54

Generally, making a type I error is considered to be worse than making a type II error; therefore, we set the type I error rate α so that it is small. Errors and Correction Decisions Errors and Correction Decisions Errors and Correction Decisions Graphically, what s going on When we preform a test of H o, we assume that it is true. Now we ll consider what happens when H a is really true when we ve assumed or acted as if H o is true. Things that Effect Suppose that the true µ = 55.00 and σ 2 = 100 and that we test H o : µ = 50 vs H A : µ using z. > 50 Hypothesis Testing or How to Decide to Decide Slide 32 of 54

Graphically, what s going on Errors and Correction Decisions Errors and Correction Decisions Errors and Correction Decisions Graphically, what s going on Things that Effect Hypothesis Testing or How to Decide to Decide Slide 33 of 54

What s the probability of rejecting H o : µ = 50 given that the true mean is µ = 55? P(reject H o µ = 55) = POWER Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results This is the shaded area under the curve for distribution on the next page... Things that Effect Hypothesis Testing or How to Decide to Decide Slide 34 of 54

Graphically: Computing Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Things that Effect Hypothesis Testing or How to Decide to Decide Slide 35 of 54

The Algebra: Computing Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results We rejected H o : µ = 50 vs H a : µ > 50 for the reading scores of students that go to academic programs. The test statistic was z = Ȳ µ o σ/ n and z.95 = 1.645 (i.e., α =.05). Assume actual mean is µ = 55. = 55.89 50.00 10/ 308 = 10.34 We would have rejected H o for any test statistic 1.645 Things that Effect Hypothesis Testing or How to Decide to Decide Slide 36 of 54

The Algebra: Computing Would have rejected H o for any Ȳ Ȳ crit Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results After a little algebra, So for our example, z crit = 1.645 Ȳcrit µ o σ/ n Ȳ crit = z crit σ Ȳ + µ o Ȳ crit = 1.645( 100/308) + 50 = 50.937 Things that Effect Hypothesis Testing or How to Decide to Decide Slide 37 of 54

The Algebra: Computing If µ = 55, this corresponds to a z-score of z a = Ȳcrit µ a σ/ n = 50.937 55 100/308 = 7.125 Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Things that Effect Probability of Type II error β = P(retain H o µ = 55) = P(z 7.125 µ = 55) =.1 10 11 Probability of correct decision, P(reject H o µ = 55) = 1 β =.99999 Hypothesis Testing or How to Decide to Decide Slide 38 of 54

of t-test Same procedure; however, use a non-central t-distribution instead of normal. Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Example: HSB reading scores of students who attend general high school program: Hypotheses: H o : µ = 50 vs H a : µ < 50 α =.05 Summary Statistics: n = 145, Ȳ = 49.06, s 2 = 80.09, s Ȳ = 80.09/145 =.74 Things that Effect Hypothesis Testing or How to Decide to Decide Slide 39 of 54

of t-test Test statistic: t = (49.06 50)/.74 = 1.27 Using t-distribution with ν = 144: t crit = t 144,.05 = 1.6555 or p-value =.10. Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Decision: retain H o. Interpretation: The data do not provide evidence that the mean reading scores for students that attend general high school programs differs from the overall mean of 50. What about power of the test? Things that Effect Hypothesis Testing or How to Decide to Decide Slide 40 of 54

of t-test & Non-central t-distribution Need to use a Non-central t-distribution The non-centrality parameter equals Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Example... δ = µ a µ o s/ n Things that Effect Hypothesis Testing or How to Decide to Decide Slide 41 of 54

of t-test: µ = 49 Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Things that Effect Hypothesis Testing or How to Decide to Decide Slide 42 of 54

of t-test: µ = 48 Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Things that Effect Hypothesis Testing or How to Decide to Decide Slide 43 of 54

Two-Tailed Test Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Things that Effect Hypothesis Testing or How to Decide to Decide Slide 44 of 54

of t-test: SAS/ Analysis Solutions Analysis Analyst Statistics Sample Size Select the test for which you want to computer power. Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results For one sample t-test enter requested information. Note that standard deviation refers to s. Output: what you put in and the computed power. Demonstration. Things that Effect Hypothesis Testing or How to Decide to Decide Slide 45 of 54

SAS/ Analysis: Results For a one-sample t-test, 1-sided alternative with Null Mean = 50, Standard Deviation = 8.949, and Alpha = 0.05 Graphically: Computing of t-test of t-test of t-test & Non-central t-distribution of t-test: µ = 49 of t-test: µ = 48 Two-Tailed Test of t-test: SAS/ Analysis SAS/ Analysis: Results Null Mean N 49.75 145.095 49.5 145.164 49 145.379 48 145.849 47 145 >.99 Things that Effect Hypothesis Testing or How to Decide to Decide Slide 46 of 54

Things that Effect Alpha Level: as α decreases, power decreases and β increases Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase Hypothesis Testing or How to Decide to Decide Slide 47 of 54

Things that Effect µ a Alternative Mean: The greater the distance between µ o and µ a, the greater the power. Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase Hypothesis Testing or How to Decide to Decide Slide 48 of 54

Things that Effect n The variance of the sampling distribution: Sample size n Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase Hypothesis Testing or How to Decide to Decide Slide 49 of 54

Things that Effect error The variance of sampling distribution: Measurement error Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase If observed scores include (random) measurement error, which is uncorrelated with the actual score: X = True Score + e The variance of the observed score X equals σ 2 = var(x) = var(true Score) + var(e) The variance of the sample distribution of the mean equals σ 2 Ȳ = σ2 n = var(true Score) + var(e) n Hypothesis Testing or How to Decide to Decide Slide 50 of 54

Things that Effect error The variance of the sampling distribution: Error Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase Hypothesis Testing or How to Decide to Decide Slide 51 of 54

How to Increase Increase α (make it easier to reject H o ), but this should not be done. Why? True parameter deviates more from the null value. Things that Effect Things that Effect Things that Effect µa Things that Effect n Things that Effect error Things that Effect error How to Increase Decreases the variance of the sampling distribution Increase sample size Better measurement Hypothesis Testing or How to Decide to Decide Slide 52 of 54

Use concept of power to help determine sample size. You need Need an estimate of σ 2. Things that Effect Need an estimate size of effect. Sample Size and Desired level of precision. How many units from the actual mean. Demonstration of SAS/Analyst Hypothesis Testing or How to Decide to Decide Slide 53 of 54

Sample Size and From SAS/Analyst Things that Effect Sample Size and Hypothesis Testing or How to Decide to Decide Slide 54 of 54