Reasoning in Psychology Using Statistics

Reasoning in Psychology Using Statistics
2015

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Hypothesis testing Design Test statistic (Estimated) Standard error
One sample, σ known One sample, σ unknown df Two related samples, σ unknown Hypothesis testing

1-sample t-test An example: 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). Population standard deviation (σ) is NOT known 1-sample t-test

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-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 = tcrit Reject H0: “Evidence supports the hypothesis that the memory treatment improved performance” 1-sample t-test

Related-samples t-test
The related samples t-test can be used with 2 different designs Note: Different names Repeated-measures t-test Within-persons t-test Matched-pairs t-test Related-samples t-test Related-samples t-test

Related-samples t-test: Within Persons
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment. Memory patients treatment Test scores Pre-test Post-test Compare pair-wise differences Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment. 1 sample Memory Test scores Pre-test Memory Test scores Post-test Memory patients Memory treatment Compare pair-wise differences Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment. 1 sample 2 scores per subject Memory Test scores Pre-test Memory Test scores Post-test Memory patients Memory treatment Compare pair-wise differences Hypotheses: Related-samples t-test: Within Persons

The related-samples t-test can be used when: 1 sample 2 scores per subject Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment. Average the pair-wise differences should be 0 if no effect Memory Test scores Pre-test Memory Test scores Post-test Memory patients Memory treatment Compare pair-wise differences Hypotheses: Memory performance at the post-test is equal to memory performance at the pre-test, that is, H0: HA: Memory performance at the post-test is NOT equal to memory performance at the pre-test, that is, Related-samples t-test: Within Persons

Related-samples t-test: Matched Pairs
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it, he recruits 50 patients and matches them into 2 related samples. He then gives one sample the new treatment;, while the other sample comprises the no-treatment control group.) Following the treatment period, all participants take a memory test. Treatment participants averaged 5 fewer errors than their matched partners. No Memory treatment Memory patients Memory treatment Related-samples t-test: Matched Pairs

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it, he recruits 50 patients and matches them into 2 related samples. He then gives one sample the new treatment;, while the other sample comprises the no-treatment control group.) Following the treatment period, all participants take a memory test. Treatment participants averaged 5 fewer errors than their matched partners. 2 samples, matched related On a pair-by-pair basis every person in the No Treatment group is matched (related) to a person in the Memory Treatment group & the average difference is calculated No Memory treatment Memory patients Memory treatment Not Independent groups participants matched Pair 1, condition A = male, 26, Age = 21 Pair 1, condition B= male, 24, Age = 21 pre existing relationship twins - one in each group couples - one in each group matched Red Short 21yrs Blue tall 23yrs Green average 22yrs Brown Related-samples t-test: Matched Pairs

The related samples t-test can be used when: 2 samples matched Related-samples t-test: Matched Pairs

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it, he recruits 50 patients and matches them into 2 related samples. He then gives one sample the new treatment;, while the other sample comprises the no-treatment control group.) Following the treatment period, all participants take a memory test. Treatment participants averaged 5 fewer errors than their matched partners. Memory Test scores No Memory treatment Average the pair-wise differences; should be 0 if no effect Memory patients related Memory treatment Hypotheses: H0: Memory performance by the treatment group is equal to memory performance by the matched no-treatment group Memory performance by the treatment group is NOT equal to memory performance by the matched no-treatment group HA: Related-samples t-test: Matched Pairs

Testing Hypotheses 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 Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom Step 5: Make a decision about your null hypothesis One-sample t Related-samples t These are computed differently than last time Testing Hypotheses

Performing your statistical test
What are all of these “D’s” referring to? Mean of the differences Test statistic Diff. Expected by chance Estimated standard error of the differences Number of difference scores Performing your statistical test

(Pre-test) - (Post-test) What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2 6 All positive Ds since lower on post-test 5 9 If a 1-tailed test, be careful about the order in forming Ds. Pre-Post = positive if Pre score higher, so positive if lower score is improvement (fewer memory errors, as above) Post-Pre = positive if Post score higher, so positive if higher score is improvement (more items correct) Note: SPSS always calculates D = Gr1 – Gr2 Related-samples t-test: Within Persons

You can think of this as doing a 1-sample t-test, where your sample is the difference scores, and your null hypothesis is that the average of the differences is equal to 0 Once you do the subtraction, you can ignore these numbers Difference scores 1-sample t Related-samples t Person Pre-test Post-test 45 55 40 60 43 49 35 51 1 2 2 6 3 5 4 9 Is this a likely sample if μD = 0? Related-samples t-test: Within Persons

(Pre-test) - (Post-test) What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 = 5.5 Related-samples t-test: Within Persons

Related-samples t-test Within Persons
What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 45 43 2 2 55 49 6 3 40 35 5 4 60 51 9 22 D = 5.5 Related-samples t-test Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 - 5.5 = -3.5 12.25 2 6 0.5 - 5.5 = 0.25 3 5 -0.5 - 5.5 = 0.25 4 9 3.5 - 5.5 = 12.25 22 25 = SSD D = 5.5 Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 ? 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 Think back to null hypothesis 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 H0: No difference in memory performance. 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit α = 0.05 2-tailed 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD Note how large tcrit is when df = 3. Related-samples t-test: Within Persons

What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit α = 0.05 2-tailed 4 9 3.5 12.25 +3.18 = tcrit 22 25 = SSD tobs=3.8 D = 5.5 2.9 = sD - Reject H0 1.45 = sD Why is tobs in upper tail (critical region)? Because positive difference (improvement) Related-samples t-test: Within Persons

Related-samples t-test
What are all of these “D’s” referring to? Mean of differences Estimated standard deviation of differences Estimated standard error of differences Degrees of freedom of difference scores Related-samples t-test

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment, sD = 2. Test this with α = 0.05. Memory Test scores Pre-test Memory Test scores Post-test Memory patients Memory treatment Compare pair-wise differences Hypotheses: Memory performance at post-test is equal to memory performance at the pre-test, that is, H0: HA: Memory performance at the post-test is NOT equal to memory performance at the pre-test, that is, Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment, sD = 4. Test this with α = 0.05. H0: HA: 2-tailed α = 0.05 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic = - 5, sD = 4 df = n-1 = 24 Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment, sD = 4. Test this with α = 0.05. H0: HA: 2-tailed, α = 0.05 t = df = 24 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic Step 5: Compare observed to critical value = 5, sD = 4 -6.25 > |2.064| Related-samples t-test: Within Persons

Dr. Mnemonic develops a new treatment for patients with a memory disorder. He isn’t certain what impact, if any, it will have. To test it he collects a sample of 25 patients and gives them his new treatment. Before the treatment he gives them a pre-treatment memory test and after the treatment a post-treatment memory test. His sample averaged 5 fewer errors after the treatment, sD = 4. Test this with α = 0.05. H0: HA: 2-tailed, α = 0.05 t = df = 24 Step 1: Hypotheses Step 2: Criterion for decision Step 3: Sample statistics Step 4: Test statistic Step 5: Compare observed to critical value Decide about hypotheses Conclude about relationship tobs = -6.25 = 5, sD = 4 tcrit = ± 2.064 “Reject H0: Evidence supports the hypothesis that the memory treatment improved performance.” Related-samples t-test: Within Persons

In lab: Practice using related sample t-tests, by hand and using SPSS
Questions? Wrap up

Reasoning in Psychology Using Statistics

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