Psych 230 Psychological Measurement and Statistics Pedro Wolf October 28, 2009

Last Time…. Hypothesis testing Statistical Errors Z-test

This Time…. T-Test Confidence intervals Practice problems

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

Steps to Hypothesis testing State the hypotheses Design the experiment Collect the data Create the statistical hypotheses Select the appropriate statistical test Decide the size of the rejection region (value of ) Calculate the obtained and critical values Make our conclusion

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

Statistical Testing Decide which test to use State the hypotheses (H0 and H1) Calculate the obtained value Calculate the critical value (size of ) Make our conclusion

One sample T-test

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

The T-test - summary Create H0 and H1 Compute tobt Compute X and s2x Compute sx Find tcrit by using the T-tables with df = N - 1 Compare tobt to tcrit

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

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

The T-distribution (Tcrit) Two different T-distributions

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

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)

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

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

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

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 What is the appropriate statistical test? Is this a one-tailed or two-tailed test? Why? What are H0 and HA? Compute Tobt With =0.05, what is Tcrit? What conclusion should we draw?

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?

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

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?

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-tailed We are interested in whether men differ from women

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?

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 enthusiastic H0 : men = 5.23 H1 : Men and women differ in enthusiasm H1 : men  5.23

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

Problem 1 5, 7, 5, 7, 8, 6, 5 3. Compute Tobt Tobt = (X - µ) / sx 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= ??

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)=6.14 s2X = [273-(1849/7)] / [7-1]=(273-264.14)/6=1.48 sX= √(1.48/7) = √(0.21) = 0.46

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= 0.46 Tobt = (6.14 - 5.23) / (0.46) Tobt = 1.97

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?

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-tailed df=N-1=7-1=6 Tcrit = 2.447

Tcrit and Tobt  Tobt=+1.97 Tcrit= -2.447 Tcrit= +2.447 a a a a

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?

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

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 What is the appropriate statistical test? Is this a one-tailed or two-tailed test? Why? What are H0 and HA? Compute the obtained value With =0.05, what is the critical value? What conclusion should we draw from this study?

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 What is the appropriate statistical test?

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 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

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?

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

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?

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 smell H0 : smokers >= 18.4 H1 : Smoking is associated with a decreased sense of smell H1 : smokers < 18.4

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

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 Compute the obtained value Tobt = (X - µ) / sx sx= √(s2x / N) =18.4; N=12 X= ?? sX= ?? s2X= ??

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 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

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 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

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?

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 With =0.05, what is the critical value? =0.05, one-tailed df=N-1=12-1=11 Tcrit = -1.796

Tcrit and Tobt  Tobt=-3.42 Tcrit=-1.796 a a a a

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?

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.

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, 23 Using =0.05, did she keep her promise?

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, 23 Using =0.05, did she keep her promise? Decide which test to use State the hypotheses (H0 and H1) Calculate the obtained value Calculate the critical value (size of ) Make our conclusion

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, 23 Using =0.05, did she keep her promise? 1. Decide which test to use

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, 23 Using =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

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, 23 Using =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.

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, 23 Using =0.05, did she keep her promise? 2. State the hypotheses (H0 and H1)

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, 23 Using =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 one H0 : new >= 30 H1 : The new response time is faster than the old one H1 : new < 30

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, 23 Using =0.05, did she keep her promise? 3. Calculate the obtained value

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, 23 Using =0.05, did she keep her promise? 3. Calculate the obtained value Tobt = (X - µ) / sx sx= √(s2x / N) =30; N=10 X= ?? sX= ?? s2X= ??

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, 23 Using =0.05, did she keep her promise? Calculate the obtained value X= (280/10) = 28 s2X= [7964-(78400/10)] / [10-1]=(7964-7840)/9=13.78 sX= √(s2x / N) = √(13.78 / 10) = 1.17

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, 23 Using =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

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, 23 Using =0.05, did she keep her promise? 4. Calculate the critical value (size of )

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, 23 Using =0.05, did she keep her promise? Calculate the critical value (size of ) =0.05, one-tailed df=N-1=10-1=9 Tcrit = -1.833

Tcrit and Tobt  Tobt=-1.71 Tcrit=-1.833 a a a a

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, 23 Using =0.05, did she keep her promise? 5. Make our conclusion

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, 23 Using =0.05, did she keep her promise? Make our conclusion As Tobs = Tcrit , we retain H0 The new sheriff has not significantly reduced response times.

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

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

Estimating m There are two ways to estimate the population mean m 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

Estimating m We perform interval estimation by creating a confidence interval The confidence interval for a particular m describes an interval containing values of m sx= √(s2x / N)

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

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

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 Compute the 95% confidence interval for the  of smokers

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 CI = [(sX)(- Tcrit) + X] <=  <= [(sX)(+ Tcrit) + X]

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 CI = [(sX)(- Tcrit) + X] <=  <= [(sX)(+ Tcrit) + X] [(0.629)(-2.201) + 16.25] <=  <= [(0.629)(+2.201) + 16.25] 14.87 <=  <= 17.63

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

Homework Exercises 14, 15, and 16 for chapter 12 Syllabus change- Read Chapter 13 for next week.