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Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.
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Transcript of Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.
![Page 1: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/1.jpg)
Psych 230
Psychological Measurement and Statistics
Pedro Wolf
October 28, 2009
![Page 2: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/2.jpg)
Last Time….
• Hypothesis testing
• Statistical Errors
• Z-test
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This Time….
• T-Test
• Confidence intervals
• Practice problems
![Page 4: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/4.jpg)
Hypothesis Testing
• Experimental hypotheses describe the predicted outcome we may or may not find in an experiment– As scientists, we try to be conservative– we assume no relationship
• The Null Hypothesis (H0)– there is no relationship between the variables
• The Alternative Hypothesis (H1)– there is a real relationship between the variables
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Steps to Hypothesis testing
1. State the hypotheses2. Design the experiment 3. Collect the data4. Create the statistical hypotheses5. Select the appropriate statistical test6. Decide the size of the rejection region
(value of )7. Calculate the obtained and critical values8. Make our conclusion
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Statistical tests (so far)
• The statistical tests we have used so are concentrate on finding whether a sample is representative of a known population
• Two characteristics of these tests:– one sample is drawn– we know the population mean
• Z-test– we also know the population variance
• T-test (one sample)– we do not know the population variance
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Statistical Testing
1. Decide which test to use 2. State the hypotheses (H0 and H1)
3. Calculate the obtained value4. Calculate the critical value (size of )5. Make our conclusion
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One sample T-test
![Page 9: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/9.jpg)
One-sample T-test
• We use the one sample T-test when we do not know the population variance
• Only differences from before:
• Tobt uses a slightly different formula
• Tcrit comes from a different distribution (the T-distribution), and so we need different tables to get this value
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The T-test - summary
1. Create H0 and H1
2. Compute tobt
1. Compute X and s2x
2. Compute sx
3. Compute tobt
3. Find tcrit by using the T-tables with df = N - 1
4. Compare tobt to tcrit
![Page 11: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/11.jpg)
The T-value of our sample (Tobt)
• Calculating Tobt
Tobt = X - µ sx
sx= √(s2x / N) : estimated standard error of the mean
In General:Test statistic = Observed - Expected
Standard Error
![Page 12: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/12.jpg)
The T-distribution (Tcrit)
• When using z-scores, we always looked at the same distribution (the Z-distribution)
• The T-distribution is actually a family of curves, all which look slightly different depending on how many samples were used to create them– Therefore, as N changes, the exact curve we will use
will change
• For small samples (a small N) the curve is only roughly similar to the standard normal curve
• Large samples (a big N) look very close to the standard normal curve
![Page 13: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/13.jpg)
The T-distribution (Tcrit)
• Two different T-distributions
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The T-distribution (Tcrit)
• We choose the curve to sample from based on not N exactly, but rather the quantity N-1
• This is termed the degrees of freedom (df)– Degrees of freedom: the number of observations
in a set of data that are variable
• The larger the df, the closer the t-distribution resembles a standard normal curve– When df > 120, the t-distribution is virtually
identical to the standard curve, and in fact tcrit = zcrit
![Page 15: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/15.jpg)
The T-distribution (Tcrit)
• To decide whether the observed value (Tobs) is in the region of rejection, we need to know Tcrit
• Tcrit is defined as the value that marks the most extreme 5% (usually) of the distribution– 5% when = 0.05
• Different distributions are different shapes and so will have different critical values for the extreme 5% of scores
• So, when performing a t-test, we use one specific curve (and one set of critical values) depending on the value of df (or, N-1)
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The T-distribution (Tcrit)
• Example: Assume the experiment had N=22 and = 0.05, and we want a two-tailed test
• df = N-1 = 22-1 = 21• Look up t-tables (page 551 of book)• df = 0.05 = 0.01• 1 12.706 63.657• 2 4.303 9.925• 3 3.182 5.841• 21 2.080 2.831
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The T-distribution (Tcrit)
• Practice: What are the Tcrit values for each of the following scenarios
• N=16; = 0.05; Two-tailed• N=31; = 0.05; Two-tailed• N=28; = 0.01; Two-tailed• N=9; = 0.05; Two-tailed• N=25; = 0.05; One-tailed• N=15; = 0.01; One-tailed
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The T-distribution (Tcrit)
• Practice: What are the Tcrit values for each of the following scenarios
• N=16; = 0.05; Two-tailed ±2.131• N=31; = 0.05; Two-tailed ±2.042• N=28; = 0.01; Two-tailed ±2.771• N=9; = 0.05; Two-tailed ±2.306• N=25; = 0.05; One-tailed 1.711• N=15; = 0.01; One-tailed 2.624
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Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
1. What is the appropriate statistical test?2. Is this a one-tailed or two-tailed test? Why?3. What are H0 and HA?4. Compute Tobt
5. With =0.05, what is Tcrit?6. What conclusion should we draw?
![Page 20: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/20.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
1. What is the appropriate statistical test?
![Page 21: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/21.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
1. What is the appropriate statistical test?
We are comparing a sample of scores to a population mean, therefore we will use a one-sample test.
As we do not know the population variance, we must estimate it and use a one-sample T-test
![Page 22: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/22.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
1. Is this a one-tailed or two-tailed test? Why?
![Page 23: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/23.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
1. Is this a one-tailed or two-tailed test? Why?
Two-tailedWe are interested in whether men differ from women
![Page 24: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/24.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
2. What are H0 and H1?
![Page 25: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/25.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
2. What are H0 and H1?H0 : Men and women are equally enthusiasticH0 : men = 5.23H1 : Men and women differ in enthusiasmH1 : men 5.23
![Page 26: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/26.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
3. Compute Tobt
![Page 27: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/27.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
3. Compute Tobt Tobt = (X - µ) / sx
sx= √(s2x / N)
=5.23; N=7
X= ??sX= ?? s2X= ??
![Page 28: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/28.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
sx= √(s2x / N)
3. Compute Tobt
X=(43/7)=6.14s2
X = [273-(1849/7)] / [7-1]=(273-264.14)/6=1.48sX= √(1.48/7) = √(0.21) = 0.46
1
)( 22
2
NNX
XsX
![Page 29: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/29.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
3. Compute Tobt Tobt = (X - µ) / sx
=5.23; N=7; X=6.14; sX= 0.46Tobt = (6.14 - 5.23) / (0.46) Tobt = 1.97
![Page 30: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/30.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
4. With =0.05, what is Tcrit?
![Page 31: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/31.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
4. With =0.05, what is Tcrit?
=0.05, two-taileddf=N-1=7-1=6
Tcrit = 2.447
![Page 32: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/32.jpg)
Tcrit and Tobt
a a a
a
Tcrit= -2.447
Tobt=+1.97
Tcrit= +2.447
![Page 33: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/33.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
5. What conclusion should we draw?
![Page 34: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/34.jpg)
Problem 1
Your instructor thinks that men and women have different levels of enthusiasm about statistics classes. When asked for their ratings of how much they were looking forward to a stats class, the for women is 5.23. A sample of 7 male students gave the following scores for how excited they were about this class:5, 7, 5, 7, 8, 6, 5
5. What conclusion should we draw?As Tobs = Tcrit , we retain H0
Men do not differ significantly from women on how enthusiastic they are about this statistics class
![Page 35: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/35.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. What is the appropriate statistical test? Is this a one-tailed or two-tailed test? Why?
2. What are H0 and HA?3. Compute the obtained value4. With =0.05, what is the critical value?5. What conclusion should we draw from this study?
![Page 36: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/36.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. What is the appropriate statistical test?
![Page 37: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/37.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. What is the appropriate statistical test?
We are comparing a sample of scores to a population mean, therefore we will use a one-sample test.
As we do not know the population variance, we must estimate it and use a one-sample T-test
![Page 38: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/38.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. Is this a one-tailed or two-tailed test? Why?
![Page 39: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/39.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. Is this a one-tailed or two-tailed test? Why?
It will be a one-tailed test, as we are predicting the direction that the scores will change. That is, we are specifically asking whether smoking leads to a decreased sense of smell
![Page 40: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/40.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
2. What are H0 and H1?
![Page 41: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/41.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
2. What are H0 and H1?H0 : Smoking is not associated with decreased sense of smellH0 : smokers >= 18.4H1 : Smoking is associated with a decreased sense of smellH1 : smokers < 18.4
![Page 42: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/42.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
3. Compute the obtained value
![Page 43: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/43.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
3. Compute the obtained value Tobt = (X - µ) / sx
sx= √(s2x / N)
=18.4; N=12
X= ??sX= ??s2
X= ??
1
)( 22
2
NNX
XsX
![Page 44: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/44.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
3. Compute the obtained value
X=(195/12)=16.25
s2X = [3221-(38025/12)] /
[12-1]=(3221-3168.75)/11=4.75
sX= √(4.75/12) = √(0.396) = 0.629
1
)( 22
2
NNX
XsX
![Page 45: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/45.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
3. Compute the obtained value Tobt = (X - µ) / sx
=18.4; N=12; X=16.25; sX= 0.629
Tobt = (16.25 - 18.4) / (0.629)
Tobt = -3.42
![Page 46: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/46.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
4. With =0.05, what is the critical value?
![Page 47: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/47.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
4. With =0.05, what is the critical value?
=0.05, one-taileddf=N-1=12-1=11
Tcrit = -1.796
![Page 48: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/48.jpg)
Tcrit and Tobt
a a a
a
Tcrit=-1.796
Tobt=-3.42
![Page 49: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/49.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
5. What conclusion should we draw from this study?
![Page 50: Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009.](https://reader036.fdocuments.us/reader036/viewer/2022062407/56649f445503460f94c650a8/html5/thumbnails/50.jpg)
Problem 2
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. On a test of olfactory sensitivity, the for nonsmokers is 18.4. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
5. What conclusion should we draw from this study?
As Tobs < Tcrit , we reject H0 and accept H1.
People who smoke have a significantly decreased sense of smell.
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
1. Decide which test to use 2. State the hypotheses (H0 and H1)3. Calculate the obtained value4. Calculate the critical value (size of )5. Make our conclusion
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
1. Decide which test to use
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
1. Decide which test to use We are comparing a sample of scores to a population mean, therefore we will use a one-sample test.As we do not know the population variance, we must estimate it and use a one-sample T-test
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
1. One or Two-tailed? It will be a one-tailed test, as we are predicting the direction that the scores will change. That is, we are specifically asking whether the new sheriff has decreased response times.
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
2. State the hypotheses (H0 and H1)
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
2. State the hypotheses (H0 and H1)H0 : The new response time is not faster then the old oneH0 : new >= 30H1 : The new response time is faster than the old one H1 : new < 30
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
3. Calculate the obtained value
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
3. Calculate the obtained value Tobt = (X - µ) / sx
sx= √(s2x / N)
=30; N=10X= ??sX= ?? s2
X= ??
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
3. Calculate the obtained value X= (280/10) = 28s2
X= [7964-(78400/10)] / [10-1]=(7964-7840)/9=13.78sX= √(s2
x / N) = √(13.78 / 10) = 1.17
1
)( 22
2
NNX
XsX
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
3. Calculate the obtained value Tobt = (X - µ) / sx
=30; N=10; X=28; sX= 1.17
Tobt = (28 - 30) / (1.17) Tobt = -1.71
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
4. Calculate the critical value (size of )
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
4. Calculate the critical value (size of )=0.05, one-taileddf=N-1=10-1=9 Tcrit = -1.833
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Tcrit and Tobt
a a a
a
Tcrit=-1.833
Tobt=-1.71
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
5. Make our conclusion
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Problem 3
A candidate running for local sheriff claims that she will reduce the average speed of emergency response to less than the current average, which is 30 minutes. Thanks to this campaign, she is elected to office and records are kept. The response times (in minutes) for the first month are:26, 30, 28, 29, 25, 28, 32, 35, 24, 23Using =0.05, did she keep her promise?
5. Make our conclusionAs Tobs = Tcrit , we retain H0
The new sheriff has not significantly reduced response times.
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Reporting t
• In scientific papers, we report the result of our statistical test like this:
• t(df) = tobs, p < 0.05 [if significant]; p > 0.05 [if not significant]
• For stats enthusiasm study (problem 1)
• t(6) = 1.97, p > 0.05
• For smoking study (problem 2)
• t(11) = -3.42, p < 0.05
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Reporting t
• For break-up study
• t(9) = 4.09, p < 0.05
• But, the tcrit at a 0.01 level is 3.25
• So, the result we found is actually significant at a higher level
• t(9) = 4.09, p < 0.01
• Statistical programs will give the exact level of significance - the exact probability that the sample mean would occur if H0 is true
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Estimating
• There are two ways to estimate the population mean
• Point estimation in which we describe a point on the variable at which the population mean is expected to fall
• Interval estimation in which we specify an interval (or range of values) within which we expect the population mean to fall
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Estimating
• We perform interval estimation by creating a confidence interval
• The confidence interval for a particular describes an interval containing values of
sx= √(s2x / N)
XtsXts XX ))(())(( critcrit
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Confidence Intervals
• I pick 5 exam scores at random. How can I estimate the average score of the class on the this exam? The scores are: 75, 63, 68, 86, 73
• I can use the sample mean to estimate the population mean
• X = (365/5) = 73
• For more precision, I can also construct a confidence interval for the population mean
• A 95% confidence interval means I am 95% sure that the true population mean lies within the range of values I find
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Confidence Intervals
• I pick 5 exam scores at random. How can I estimate the average score of the class on the this exam? The scores are: 75, 63, 68, 86, 73
(3.86)(-2.776) + 73 <= <= (3.86)(+2.776) + 73
62.29 <= <= 83.71
XtsXts XX ))(())(( critcrit
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Confidence Intervals
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. Compute the 95% confidence interval for the of smokers
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Confidence Intervals
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. Compute the 95% confidence interval for the of smokersCI = [(sX)(- Tcrit) + X] <= <= [(sX)(+ Tcrit) + X]
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Confidence Intervals
A researcher predicts that smoking cigarettes decreases a person’s sense of smell. People who smoke a pack a day produced the following scores:16, 14, 19, 17, 16, 18, 17, 15, 18, 19, 12, 14
1. Compute the 95% confidence interval for the of smokersCI = [(sX)(- Tcrit) + X] <= <= [(sX)(+ Tcrit) + X]
[(0.629)(-2.201) + 16.25] <= <= [(0.629)(+2.201) + 16.25]
14.87 <= <= 17.63
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Next Time….
• What do we do when we don’t know the population mean?
• Variants of the T-Test– Independent samples T-test– Related samples T-test
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Homework
• Exercises 14, 15, and 16 for chapter 12
• Syllabus change- Read Chapter 13 for next week.