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

2. Last Time….
Hypothesis testing
Statistical Errors
Z-test

3. This Time….
T-Test
Confidence intervals
Practice problems

4. 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

5. 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

6. 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

7. 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

8. One sample T-test

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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)

16. 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  =  = 0.01

17. 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

18. 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
N=15;  = 0.01; One-tailed

19. 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 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?

20. 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 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?

21. 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 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

22. 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 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?

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 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

24. 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 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?

25. 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 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

26. 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 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

27. 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 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= ??

28. 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 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]=( )/6=1.48
sX= √(1.48/7) = √(0.21) = 0.46

29. 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 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 = ( ) / (0.46)
Tobt = 1.97

30. 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 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?

31. 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 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

32. Tcrit and Tobt 
Tobt=+1.97
Tcrit=
Tcrit= a a a a

33. 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 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?

34. 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 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

35. 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 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?

36. 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 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?

37. 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 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

38. 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 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?

39. 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 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

40. 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 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?

41. 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 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

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 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

43. 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 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= ??

44. 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 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]=( )/11=4.75
sX= √(4.75/12) = √(0.396) = 0.629

45. 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 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 = ( ) / (0.629)
Tobt = -3.42

46. 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 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?

47. 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 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 =

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

49. 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 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?

50. 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 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.

51. 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?

52. 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

53. 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

54. 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

55. 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.

56. 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)

57. 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

58. 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

59. 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= ??

60. 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]=( )/9=13.78
sX= √(s2x / N) = √(13.78 / 10) = 1.17

61. 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 = ( ) / (1.17)
Tobt = -1.71

62. 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 )

63. 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 =

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

65. 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

66. 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.

67. 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

68. 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

69. 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

70. 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)

71. 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

72. 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

73. 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

74. 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]

75. 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) ] <=  <= [(0.629)(+2.201) ]
14.87 <=  <= 17.63

76. 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

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