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Unformatted text preview: 4/22/11 Announcements To see your midterm, email your sec7on TA (or the sec7on TA for the sec7on you can a;end) and he or she can bring it for you to look over This Week FR OM BE FO RE Parameters and Sta7s7cs Popula7on The Distribu7on of Sample Means The Logic of Hypothesis Tes7ng Sample ? ? A ESTIM TION X s STATISTICS (ESTIMATES) PARAMETERS 1 4/22/11 O FR M FO BE RE Sampling Error The discrepancy (or amount of error) that exists between a sample sta7s7c and the corresponding popula7on parameter O FR M FO BE RE Popula7on All Individuals in the USA = 100 = 15 Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Sample Sta4s4cs X = 106 s = 15.2 FR OM BE FO RE Popula7on All Individuals in the USA = 100 = 15 FR OM BE FO RE The Distribu7on of Sample Means Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Sample Sta4s4cs X = 106 s = 15.2 2 4/22/11 O FR M FO BE RE O FR M FO BE RE The Distribu7on of Sample Means The collec7on of sample means for all the possible random samples of a par7cular size (n) Construc7ng a real Sampling Distribu7on for a small popula4on FR OM BE FO RE FR OM BE FO RE 3 4/22/11 O FR M FO BE RE O FR M FO BE RE Construc7ng a real Sampling Distribu7on 1. 2. 3. 4. 5. Choose a popula7on Take a random sample of size n Calculate the mean for that par4cular sample Record the mean to a histogram Go back to #2 and repeat 5000 7mes Construc7ng a real Sampling Distribu7on for a big popula4on FR OM BE FO RE Popula7on Each Sample _ Sampling Distribu7on of X FR OM BE FO RE = 100 = 15 4 4/22/11 O FR M FO BE RE O FR M FO BE RE = 100 = 15 = 100 = 15 Random Sample X FR OM BE FO RE FR OM BE FO RE = 100 = 15 = 100 = 15 X 5 4/22/11 O FR M FO BE RE O FR M FO BE RE = 100 = 15 = 100 = 15 Random Sample X FR OM BE FO RE FR OM BE FO RE = 100 = 15 = 100 = 15 X 6 4/22/11 O FR M FO BE RE = 100 = 15 O FR M FO BE RE n = 2 n = 4 Construc7ng a real Sampling Distribu7on for a big popula4on that is not normally distributed n = 20 n = 60 n = 120 FR OM BE FO RE FR OM BE FO RE Construc7ng a real Sampling Distribu7on 1. 2. 3. 4. 5. Choose a popula7on (beta distribu7on) Take a random sample of size n Calculate the mean for that par4cular sample Record the mean to a histogram Go back to #2 and repeat 5000 7mes Popula7on Each Sample = 0.33 = 0.30 _ Sampling Distribu7on of X 7 4/22/11 O FR M FO BE RE O FR M FO BE RE = 0.33 = 0.30 = 0.33 = 0.30 Random Sample FR OM BE FO RE FR OM BE FO RE = 0.33 = 0.30 = 0.33 = 0.30 Random Sample X X 8 4/22/11 O FR M FO BE RE O FR M FO BE RE = 0.33 = 0.30 = 0.33 = 0.30 Sampling Distribu7on of the Mean n = 2 Sampling Distribu7on of the Mean n = 5 Sampling Distribu7on of the Mean n = 20 Sampling Distribu7on of the Mean n = 60 Sampling Distribu7on of the Mean n = 120 FR OM BE FO RE Characteris7cs of the Sampling Distribu7on of X Sampling distribu7on has a mean = As sample size (n) increases, the variability of the sampling distribu7on decreases ! n Standard Devia7on of the Sampling Distribu7on of Sample Means The sampling distribu7on is normally distributed when popula4on is normal or samples are large, n > ~30 9 4/22/11 Standard (Sampling) Error Standard Error Standard Devia7on of the Sampling Distribu7on of Sample Means !X !M Standard Error = 100 = 15 n = 2 ! !2 !X = = n n n = 4 n = 20 n = 60 n = 120 10 4/22/11 = 100 = 15 Standard Error for n = 2 n = 2 n = 4 !X = ! n n = 20 n = 60 n = 120 Standard Error for n = 2 Standard Error for n = 2 !X = ! n !X = 15 2 !X = ! n !X = 15 2 ! X = 10.61 11 4/22/11 = 100 = 15 = 100 = 15 n = 2 n = 4 ! X = 10.61 n = 2 n = 4 n = 20 n = 20 n = 60 n = 120 n = 60 n = 120 = 100 = 15 ! X = 10.61 n = 2 n = 4 n = 20 n = 60 n = 120 12 4/22/11 = 100 = 15 O FR M FO BE RE ! X = 10.61 n = 2 = 0.33 = 0.30 Sampling Distribu7on of the Mean n = 2 ! X = 7.5 n = 4 Sampling Distribu7on of the Mean n = 5 ! X = 3.35 n = 20 Sampling Distribu7on of the Mean n = 20 Sampling Distribu7on of the Mean ! X = 1.94 n = 60 n = 60 n = 120 ! X = 1.37 Sampling Distribu7on of the Mean n = 120 Standard Error for n = 2 Standard Error for n = 2 !X = ! n !X = ! n !X = 0.30 2 13 4/22/11 Standard Error for n = 2 O FR M FO BE RE = 0.33 = 0.30 Sampling Distribu7on of the Mean ! !X = n 0.30 !X = 2 n = 2 Sampling Distribu7on of the Mean n = 5 Sampling Distribu7on of the Mean ! X = 0.212 n = 20 Sampling Distribu7on of the Mean n = 60 Sampling Distribu7on of the Mean n = 120 = 0.33 = 0.30 Sampling Distribu7on of the Mean = 0.33 = 0.30 Sampling Distribu7on of the Mean ! X = 0.212
Sampling Distribu7on of the Mean n = 2 ! X = 0.212
Sampling Distribu7on of the Mean n = 2 n = 5 ! X = 0.134
Sampling Distribu7on of the Mean n = 5 Sampling Distribu7on of the Mean n = 20 Sampling Distribu7on of the Mean ! X = 0.067 n = 20 n = 60 n = 60 Sampling Distribu7on of the Mean ! X = 0.039 Sampling Distribu7on of the Mean n = 120 Sampling Distribu7on of the Mean ! X = 0.027 n = 120 14 4/22/11 Popula7on All Individuals in the USA = 100 = 15 Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Sample Sta4s4cs X = 106 s = 15.2 Distribu7on of Sample Means X = !X = ! n Distribu7on of Sample Means X = !X = ! n 34% 14% 1.5 1 .5 34% 14% .5 1 1.5 Z 2.5 2 1.5 1 .5 .5 1 1.5 2 2.5 Z 2.5 2% 2 2 2% 2.5 15 4/22/11 Popula7on All Individuals in the USA = 100 = 15 Popula7on All Individuals in the USA = 100 = 15 Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... Sample Sta4s4cs X = 106 s = 15.2 Sample Sta4s4cs X = 106 s = 15.2 Distribu7on of Sample Means Popula7on All Individuals in the USA = 100 = 15 34% Treatment Sample n = 25 106, 101, 98, 102, 100, 99, 111, 92, 108, 95, 110, 104... X = 100
!X =
15 25 34% 14% Sample Sta4s4cs X = 106 s = 15.2 Z 2.5 2% 14% 1.5 1 .5 .5 1 2 1.5 2 2% 2.5 16 4/22/11 Distribu7on of Sample Means X = 100 Distribu7on of Sample Means X = 100 !X = 3
34% 14% 1.5 1 .5 .5 1 !X = 3
34% 34% 14% 1 .5 .5 1 1.5 34% 14% 1.5 Z 2.5 2% 2 2 2% 2.5 Z 2.5 2% 14% 1.5 2 2 2% 2.5 ZScore for Sample Mean ZScore for Sample Mean zX = X ! X !X zX = X! !X 17 4/22/11 = 100 x = 3 X = 106 zX = X ! "X = 100 x = 3 X = 106 zX = X ! "X zX = 106 !100 3 Distribu7on of Sample Means = 100 x = 3 X = 106 X ! zX = "X X = 100 !X = 3
34% 14% 1.5 1 .5 .5 1 zX = 106 !100 3
Z 2.5 34% 14% 1.5 2% 2 2 2% 2.5 zX = 2 ZX 18 4/22/11 Distribu7on of Sample Means Two Possible Explana6ons: 1.There is no effect of green tea on IQ scores and the 6point increase is due to sampling error 2.There is a real effect of green tea on IQ scores 14% 1.5 1 34% 34% 14% Z 2.5 2% 2 .5 .5 1 1.5 2 2% 2.5 ZX Distribu7on of Sample Means = 100 = 15 ! X = 10.61 n = 2 34% 14% 1.5 95.5 1 97 .5 98.5 100 34% 14% .5 101.5 1 103 1.5 104.5 ! X = 7.5 n = 4 Z X 2.5 92.5 2% 2 2 2% 2.5 107.5 ! X = 3.35 n = 20 94 106 X = 100 X !X = 3 ! X = 1.94 n = 60 n = 120 ! X = 1.37 19 4/22/11 = 100 = 15 = 100 = 15 ! X = 10.61 n = 2 ! X = 10.61 n = 2 ! X = 7.5 n = 4 ! X = 7.5 n = 4 ! X = 3.35 n = 20 ! X = 3.35 n = 20 ! X = 1.94 n = 60 ! X = 1.94 n = 60 n = 120 ! X = 1.37 n = 120 ! X = 1.37 Check Your Understanding For the same popula7on, the probability of obtaining a mean of 106 or greater with a sample of n = 4 scores is: A B C D More likely than with a sample of n = 25 scores Less likely than with a sample of n = 25 scores Just as likely as with a sample of n = 25 scores Cannot be determined without more informa7on Check Your Understanding For the same popula7on, the probability of obtaining
a mean of 106 or greater with a sample of n = 4 scores is: A B C D More likely than with a sample of n = 25 scores Less likely than with a sample of n = 25 scores Just as likely as with a sample of n = 25 scores Cannot be determined without more informa7on 20 4/22/11 Distribu7on of Sample Means Distribu7on of Sample Means 34% 14% 1.5 95.5 1 97 .5 98.5 100 34% 14% .5 101.5 1 103 1.5 104.5 34% 14% 1.5 95.5 88.75 1 97 92.5 .5 98.5 96.25 100 100 34% 14% .5 101.5 1 103 1.5 104.5 Z X 2.5 92.5 2% 2 2 2% 2.5 107.5 94 106 Z X X 2.5 92.5 81.25 2% 2 2 2% 2.5 107.5 118.75 94 85 106 115 103.75 107.5 111.25 X X = 100 ! X = 7.5 Distribu7on of Sample Means = 100 = 15 ! X = 10.61 n = 2 34% 14% 1.5 95.5 88.75 1 97 92.5 .5 98.5 96.25 100 100 34% 14% .5 101.5 1 103 1.5 104.5 ! X = 7.5 n = 4 Z X X 2.5 92.5 81.25 2% 2 2 2% 2.5 107.5 118.75 ! X = 3.35 n = 20 94 85 106 115 103.75 107.5 111.25 X = 100 ! X = 1.94 n = 60 X = 106 ! X = 7.5 n = 120 ! X = 1.37 21 4/22/11 Popula7on Each Sample Sampling Distribu7on of X _ Popula7on Popula7on Each Sample _ Sampling Distribu7on of X Each Sample X s _ Sampling Distribu7on of X 22 4/22/11 Popula7on Each Sample X s Sampling Distribu7on of X _ , X X This Week The Logic of Hypothesis Tes7ng The Distribu7on of Sample Means The Logic of Hypothesis Tes7ng 23 4/22/11 Hypothesis Test A hypothesis test is a sta7s7cal method that uses sample data to evaluate a hypothesis about a popula7on Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable Popula7on All Individuals in the All Individuals USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 Facebook 24 4/22/11 Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the All Individuals USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 in the USA = 15 hours = 100 = 5 hours = 15 Random Sample n = 100 Random Sample n = 100 Treatment $ Popula7on All Individuals in the All Individuals USA using Facebook Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter in the USA = 15 hours = 100 = 5 hours = 15 Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable 25 4/22/11 Sta7s7cal Hypotheses Null Hypothesis (H0) Sta7s7cal Hypotheses Alterna7ve Hypothesis (H1) Null Hypothesis (H0) In the popula7on there is no change, no difference, or no rela7onship. For an experiment, this means no effect of treatment Alterna7ve Hypothesis (H1) In the popula7on there is a change, a difference, or a rela7onship. For an experiment, this means an effect of treatment 26 4/22/11 Sta7s7cal Hypotheses Null Hypothesis (H0) the treatment does not have an effect Alterna7ve Hypothesis (H1) the treatment has an effect Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) the treatment has an effect Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) treatment without treatment Popula7on All Individuals in the All Individuals USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person 27 4/22/11 Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the All Individuals USA using Facebook in the USA = 15 hours = 100 = 5 hours = 15 in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ Hypothe7cal Hypothe7cal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours Hypothe7cal Hypothe7cal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 28 4/22/11 Popula7on All Individuals in the All Individuals USA using Facebook Popula7on Treatment $ All Individuals in the USA using Facebook Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) treatment without treatment Hypothe7cal in the USA = 15 hours = 100 = 5 hours = 15 = ? hours = 5 hours Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Sample Time on Facebook during sixth month for each person Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) treatment without treatment Hypothe7cal in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Sample Time on Facebook during sixth month for each person 29 4/22/11 Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) treatment without treatment Sta7s7cal hypotheses are always about popula7ons, not samples! Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = without treatment Alterna7ve Hypothesis (H1) treatment without treatment Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = ? hours = 5 hours in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = 15 hours = 5 hours If H0 is true Hypothe7cal Hypothe7cal Random Sample n = 100 Treatment $ Sample Time on Facebook during sixth month for each person Random Sample n = 100 Random Sample n = 100 Treatment $ Random Sample n = 100 Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person 30 4/22/11 Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = 15 Alterna7ve Hypothesis (H1) treatment 15 Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = 15 Alterna7ve Hypothesis (H1) treatment 15 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable 31 4/22/11 Distribu7on of Sample Means Sta7s7cal Hypotheses Null Hypothesis (H0) treatment = 15 Alterna7ve Hypothesis (H1) treatment 15 X = !X = ! n 2.5 2 1.5 1 .5 .5 1 1.5 2 2.5 Distribu7on of Sample Means Popula7on All Individuals in the All Individuals USA using Facebook Popula7on All Individuals in the X = 15
!X =
5 100 in the USA = 15 hours = 100 = 5 hours = 15 Treatment $ USA using Facebook = 15 hours = 5 hours If H0 is true Hypothe7cal Random Sample n = 100 Treatment $ Random Sample n = 100 2.5 2 1.5 1 .5 .5 1 1.5 2 2.5 Sample Time on Facebook during sixth month for each person Sample Time on Facebook during sixth month for each person 32 4/22/11 Distribu7on of Sample Means X = 15
! X = 0.5 Distribu7on of Sample Means X = 15
! X = 0.5 2.5 2 1.5 1 .5 .5 1 1.5 2 2.5 2.5 13.75 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5 16.25 Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable 33 4/22/11 Distribu7on of Sample Means X = 15
! X = 0.5 Distribu7on of Sample Means X = 15
! X = 0.5 2.5 13.75 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5 16.25
2.5 13.75 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5 16.25 Which samples do not support H0 Which samples do not support H0 Distribu7on of Sample Means X = 15
! X = 0.5 Low Probability Samples if H0 is True High Probability Samples if H0 is True 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 Low Probability Samples if H0 is True How unlikely is unlikely enough? 10%? 5%? 1% 0.00001%? 2.5 13.75 2 14 2 16 2.5 16.25 Which samples do not support H0 34 4/22/11 Alpha Level, The probability value that is used to define which sample outcomes are considered very unlikely if the null hypothesis is true Alpha Level, (most common) ! = 0.05 Distribu7on of Sample Means X = 15
! X = 0.5 Cri7cal Region The region of the sampling distribu7on that contains the sample outcomes that are considered very unlikely if H0 is true 2.5% High Probability Samples if H0 is True 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5% 2.5 13.75 2.5 16.25 35 4/22/11 Distribu7on of Sample Means X = 15
! X = 0.5 Unit Normal Table 2.5% High Probability Samples if H0 is True 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5% 2.5 13.75 2.5 16.25 36 4/22/11 Distribu7on of Sample Means X = 15
! X = 0.5 2.5% High Probability Samples if H0 is True 2 14 1.5 14.25 1 14.5 .5 14.75 15 .5 15.25 1 15.5 1.5 15.75 2 16 2.5% 2.5 13.75 2.5 16.25 zcri7cal = 1.96 zcri7cal = +1.96 37 4/22/11 Four Steps to Hypothesis Tes7ng 1. State a sta7s7cal hypothesis about a popula7on, usually about a popula7on parameter Check Your Understanding If H0 is true, which of the following must be true: A B C D _ X without treatment _ X = without treatment treatment without treatment treatment = without treatment 2. Use the hypothesis to predict the characteris7cs samples from that popula7on should have; pick criteria to decide when samples do not support that hypothesis 3. Obtain a random sample of size n from the popula7on and compute a test sta7s7c 4. Compare obtained sample with the predic7on made by the sta7s7cal hypothesis. If sample is consistent, conclude hypothesis is reasonable; if sample is very inconsistent, conclude hypothesis is unreasonable Check Your Understanding If H0 is true, which of the following must be true: A B C D _ X without treatment _ X = without treatment treatment without treatment treatment = without treatment Check Your Understanding What is the name of the area in the sampling distribu7on containing samples that are very unlikely if H0 is true? A B C D E Alpha Region Cri7cal Region Significance Region Null Region Alterna7ve Region 38 4/22/11 Check Your Understanding What is the name of the area in the sampling distribu7on containing samples that are very unlikely if H0 is true? A B C D E Alpha Region Cri7cal Region Significance Region Null Region Alterna7ve Region For Next Time (Re)Read Chapter 8 Do Homework 4 Review Midterm 39 ...
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