Reasoning in Psychology Using Statistics 2017

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Testing Hypotheses Hypothesis testing: a five step program Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data from your sample Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Testing Hypotheses

Testing Hypotheses Hypothesis testing: a five step program Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data from your sample Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Testing Hypotheses

Performing your statistical test What are we doing when we test the hypotheses? Consider a variation of our memory experiment example Population of memory patients MemoryTest μ & σ known Memory treatment patients Test X Compare these two means We test this one Conclusions: H0: The memory treatment sample are the same as those in the population of memory patients. HA: They are not the same as those in the population of memory patients Performing your statistical test

Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H0: is true (no treatment effect) H0: is false (is a treatment effect) One population XA H0: The memory treatment sample are the same as those in the population of memory patients. HA: They are not the same as those in the population of memory patients Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Performing your statistical test

Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H0: is true (no treatment effect) H0: is false (is a treatment effect) One population Two populations XA XA H0: The memory treatment sample are the same as those in the population of memory patients. HA: They are not the same as those in the population of memory patients Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Performing your statistical test

Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) We test this one H0: is true (no treatment effect) H0: is false (is a treatment effect) One population Two populations XA XA H0: The memory treatment sample are the same as those in the population of memory patients. HA: They are not the same as those in the population of memory patients Real world (‘truth’) H0 is correct H0 is wrong Experimenter’s conclusions Reject H0 Fail to Reject H0 Type I error Type II error Performing your statistical test

“Generic” statistical test The generic test statistic distribution (a transformation of the distribution of sample means) To reject the H0, you want a computed test statistics that is large The probability of having a sample with that mean is very low What is large enough? The alpha level gives us the decision criterion Distribution of the test statistic α-level determines where these boundaries go “Generic” statistical test

“Generic” statistical test The generic test statistic distribution (a transformation of the distribution of sample means) To reject the H0, you want a computed test statistics that is large The probability of having a sample with that mean is very low What is large enough? The alpha level gives us the decision criterion Distribution of the test statistic If test statistic is here Reject H0 If test statistic is here Fail to reject H0 “Generic” statistical test

“Generic” statistical test The alpha level gives us the decision criterion Two -tailed z .00 .01 … .06 : 1.0 1.9 3.0 .5000 .1587 .0287 .0013 .4960 .1562 .0281 .0250 α = 0.05 0.025 split up into the two tails Reject H0 Fail to reject H0 Go to the table (unit normal table for z-test) and find the z that has 0.050 in the tails. Zcritical = ±1.96 “Generic” statistical test

“Generic” statistical test The alpha level gives us the decision criterion Two -tailed One -tailed Reject H0 Fail to reject H0 α = 0.05 0.05 all of it in one tail Reject H0 Reject H0 Fail to reject H0 Fail to reject H0 Go to the table (unit normal table for z-test) and find the z that has 0.050 in the tail. Zcritical = +1.645 z .00 … .05 : 1.6 0.5000 0.0548 …. .4801 .0495 “Generic” statistical test

“Generic” statistical test The alpha level gives us the decision criterion Two -tailed One -tailed Reject H0 Fail to reject H0 α = 0.05 all of it in one tail 0.05 Reject H0 Reject H0 Fail to reject H0 Fail to reject H0 Go to the table (unit normal table for z-test) and find the z that has 0.050 in the tail. Zcritical = -1.645 z .00 … .05 : 1.6 0.5000 0.0548 …. .4801 .0495 “Generic” statistical test

1-sample z-test Population of memory patients Memory Test σ is known Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. Population of memory patients Memory Test σ is known μ is known Memory treatment patients Test X Compare these two means 1-sample z-test

1-sample z-test 1 sample Population of memory patients Memory Test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. 1 sample Population of memory patients Memory Test σ is known μ is known Memory treatment patients Test X Compare these two means 1-sample z-test

1-sample z-test 1 sample 1 score per subject Population of Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. 1 sample 1 score per subject Population of memory patients Memory Test σ is known μ is known Memory treatment patients Test X Compare these two means 1-sample z-test

1-sample z-test 1-sample z-test 1 sample 1 score per subject Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. 1 sample 1 score per subject Population mean (μ) and standard deviation (σ) are known (assume Normal dist) Population of memory patients Memory Test σ is known μ is known Memory treatment patients Test X Compare these two means 1-sample z-test 1-sample z-test

Performing your statistical test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. One -tailed Step 1: State your hypotheses μTreatment > μpop = 60 Step 2: Set your decision criteria H0: the memory treatment sample are the same as those in the population of memory patients (or even worse). Step 3: Collect your data Step 4: Compute your test statistics HA: the memory treatment sample perform better (fewer errors) than those in the population of memory patients μTreatment < μpop = 60 Step 5: Make a decision about your null hypothesis Performing your statistical test

Performing your statistical test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 One -tailed α = 0.05 Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Performing your statistical test

Performing your statistical test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 n = 16, X = 55 One -tailed α = 0.05 Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Performing your statistical test

Performing your statistical test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 One -tailed α = 0.05 n = 16, X = 55 Step 1: State your hypotheses = -2.5 Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Performing your statistical test

Performing your statistical test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05. H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60 One -tailed α = 0.05 n = 16, X = 55 Step 1: State your hypotheses = -2.5 Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics 5% Step 5: Make a decision about your null hypothesis Reject H0 - Support for our HA, the evidence suggests that the treatment decreases the number of memory errors Performing your statistical test

1-sample t-test The 1-sample t-test Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). The 1-sample t-test Population standard deviation (s) is NOT known 1-sample t-test

One sample t-test The 1-sample t-test Hypotheses: H0: HA: Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). The 1-sample t-test Population of memory patients MemoryTest σ is NOT known μ is known Memory treatment patients Test X Compare these two means Hypotheses: The memory treatment sample are the same as those in the population of memory patients (or even worse), that is, μA ≥ μ0. The memory treatment sample perform better (fewer errors) than those in the population of memory patients, that is, μA < μ0. H0: HA: One sample t-test

Testing Hypotheses Hypothesis testing: a five step program The 1-sample t-test Hypothesis testing: a five step program Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis Same Different Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom Testing Hypotheses

Based on standard error or an estimate of the standard error What are we doing when we test the hypotheses? Computing a test statistic: Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error Hypothesis testing

Based on standard error or an estimate of the standard error What are we doing when we test the hypotheses? Computing a test statistic: 1-sample t-test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error 1-sample t-test

Based onandard error or estimate of standard error What are we doing when we test the hypotheses? Computing a test statistic: 1-sample t-test Could be difference between a sample and a population, or between different samples Based onandard error or estimate of standard error 1-sample t-test

1-sample z 1-sample t identical Test statistic 1-sample t-test

Use our best guess, sample standard deviation (s) 1-sample z 1-sample t Test statistic different Diff. Expected by chance Standard error If don’t know this, so need to estimate it Use our best guess, sample standard deviation (s) XA s 1-sample t-test

Use our best guess, sample standard deviation (s) 1-sample z 1-sample t Test statistic different Diff. Expected by chance Standard error If don’t know this, so need to estimate it Estimated standard error Degrees of freedom Use our best guess, sample standard deviation (s) 1-sample t-test

The t-statistic distribution (a transformation of the distribution of sample means) To reject the H0, you want a computed test statistics that is large The alpha level gives us the decision criterion New table: the t-table; set up to find critical t-value for given p-value Distribution of the t-statistic If test statistic is beyond here Reject H0 If test statistic is here, fail to reject H0 Note: Bottom row: same as z. The t-distribution becomes the Normal distribution when df = ∞ The t-distribution

Each row corresponds to a different curve The t distributions are like the z distribution (μ = 0; σ =1). α levels New table: the t-table 1-tailed - or - 2-tailed Degrees of freedom df Critical values of t tcrit Each row corresponds to a different curve The t-distribution

The t-distribution New table: the t-table As df gets smaller, need larger tcrit, shown here for α = .05, 2-tailed. Curve flattens (becomes platykurdic). df= df = df = The t-distribution

What is the tcrit for a 2-tailed hypothesis test with a sample size of n = 6 and an α level of 0.05? α = 0.05 2-tailed df = n - 1 = 5 tcrit = +2.571 The t-distribution

What is the tcrit for a 1-tailed hypothesis test with a sample size of n = 6 and an α level of 0.05? α = 0.05 n = 6 1-tailed df = n - 1 = 5 tcrit = +2.015 1-tailed, larger critical region in 1-tail, so smaller critical value needed. tcrit = 2.015 vs. ±2.571 The t-distribution

1-sample t-test An example: 1-sample t-test H0: HA: Memory treatment sample same as or make more errors than population of memory patients. Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed H0: μA > μ0 = 60 HA: Sample make fewer errors than population of memory patients HA: μA < μ0 = 60 Step 1: Hypotheses 1-sample t-test

1-sample t-test An example: 1-sample t-test H0: μA > μ0 = 60 HA: μA < μ0 = 60 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed α = 0.05 Step 1: Hypotheses Step 2: Criterion for decision 1-sample t-test

1-sample t-test An example: 1-sample t-test = 55, s = 8 H0: μA > μ0 = 60 HA: μA < μ0 = 60 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed α = 0.05 = 55, s = 8 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics 1-sample t-test

1-sample t-test An example: 1-sample t-test = 55, s = 8 H0: μA > μ0 = 60 HA: μA < μ0 = 60 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed α = 0.05 = 55, s = 8 Step 1: Hypotheses = -2.5 Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic 1-sample t-test

1-sample t-test An example: 1-sample t-test = 55, s = 8 tcrit = -1.753 H0: μA > μ0 = 60 HA: μA < μ0 = 60 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed α = 0.05 = 55, s = 8 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic Step 5: Compare observed to critical value & make a decision about your null hypothesis tcrit = -1.753 1-sample t-test

1-sample t-test An example: 1-sample t-test = 55, s = 8 tobs=-2.5 H0: μA > μ0 = 60 HA: μA < μ0 = 60 Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05. 1-tailed α = 0.05 = 55, s = 8 tobs=-2.5 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic Step 5: Compare observed to critical value & make a decision about your null hypothesis -1.753 = tcrit “Evidence supports the hypothesis that the memory treatment improved performance” 1-sample t-test

SPSS output for 1-sample t-test = 8 – 10 = -5 .4 Cover later: Estimate difference between -3 and -1 In SPSS, compare observed & critical p-values. 1-tail p = .0005 Less than α = .05? SPSS output for 1-sample t-test

In lab: Practice using 1-sample t-tests and the hypothesis testing framework Questions? Wrap up