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Sta+s+cs
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Unformatted text preview: MGCR 271 Business
Sta+s+cs
 Make
Inference
 Inference on Means Ramnath
Vaidyanathan
 Procedure for Hypothesis Testing Fail to Reject H0 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! Fail to RejectFail to Reject H0 H0 7
 0.3 Reach a H Conclusion0 population 0.2 : µ1 − µ2 = d0 1
 Ha : Formulate µ1 − µ2 =ypothesis H d0 0.1 x1 − x2 ¯ ¯ p 2 H0 : µ = µ0 2
 Choosegroup group a b Significance bc H c: Level a ! ! ! H0 4
 µ1 − µ2 = d0 : µ1 − µ2 = d0 Compute p-Value α 2 Ha : µ = µ0 Identify Test 0 Statistic x α 2 x1 6
 x2 ¯ −¯ p 2 5
 Identify !3 Critical Region 0.0 z∗ −z !2 ∗ !1 1 LC µ0 L∗ C zz −z ∗ U 2 3 3
 C x ¯ Compute Test Score 1
 Formulate Hypothesis A hypothesis is an assumption or a theory about the characteristics of one or more variables in one or more populations. Parameter Null Value of Parameter H0 : µ = µ0 Ha : µ = µ0 2
 Identify Test Statistic… A test statistic is a measure of how far the sample statistic is from the population parameter. Test Statistic Statistic Parameter Z= ¯ X −µ √ σ/ n Std. Error 2
 …and its Distribution !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! N (0, 1) 0.3 population group ! 0.2 a b c ! ! 0.1 !3 !2 Z= x !1 0 ¯ X −µ √ σ/ n 1 2 3 3
 Compute Test Score A test score is the the test statistic calculated value of for an observed sample under the assumption of the null hypothesis. Statistic (Observed Value) Test Score Parameter (Null Value) z= x−µ0 ¯√ σ/ n Std. Error 4
 Choose Significance Level Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! Fail to RejectFail to Reject H0 H0 4
 population 1
 0.2 Formulate Hypothesis H0 : µ = µ0 2
 Choose Significance Level ! ! ! group group a b b c c 0.1 α 2 Ha : µ = µ0 Identify Test 0 Statistic x α 2 0.0 z∗ !3 −z ∗ !2 !1 1 µ0 z∗ z −z ∗ 2 3 3
 x ¯ Compute Test Score 5
 Identify Critical Region Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! Fail to RejectFail to Reject H0 H0 4
 population 1
 0.2 Formulate Hypothesis H0 : µ = µ0 2
 Choose Significance Level ! ! ! group group a b b c c 0.1 α 2 Ha : µ = µ0 Identify Test 0 Statistic x α 2 5
 Identify !3 Critical Region 0.0 z∗ −z !2 ∗ !1 1 LC µ0 L∗ C zz −z ∗ U 2 3 3
 C x ¯ Compute Test Score 6
 Compute p-Value The p-value is the probability, if H0
is
true,
 of
 drawing
 a
 sample
 like
 the
 one
 obtained
 or
 more
extreme,
in
the
direc6on
of

Ha. One‐sided

 (one‐tailed)

 Two‐sided

 (two‐tailed)

 hypothesis is true there is only a 7.71% chance of seeing a sample as extreme or more extreme = = ∼ N (0, 1) ˆ qP −p p(1−p) n than what we observed. Since p ≤ α, we conclude that there is sufficient evidence to reject the null hypothesis and the inventor’s claim that the mean run time is 300 minutes. kjhjk 10. Based on these results, should the principal accept or reject her original hypothesis? Assume a 1. gfhgfgh significance level of 0.01. ˆ q p−p0 of students p0 (1−p0 ) n 3. Bon Air Elementary School has 300 students. The principal of the school thinks that the average IQ at Bon Air is at least 110. To prove her point, she administers an IQ test to 20 randomly selected students. Among the sampled students, the average IQ is 108 with a standard deviation of 4. One of the best indicators of the health of a baby is his/her weight at birth. Birthweight is an outcome that is sensitive to and alternative hypotheses. experienced pregnancy, particularly to issues of (a) State the null the conditions in which mothers Use only letters and mathematical symbols to write the hypotheses. Do not use words. deprivation and poor diet, which are tied to lower birthweight. It is also an excellent predictor of some diffiIdentify the entity, property, population, sample, of life. The National Center for Health (b) culties that infants may experience in their first weeks parameter and statisStatistics reports that although infants weighing(a). pounds (88 ounces) or less account for only 7% tic for the hypothesis being tested in 5 1/2 (c) Identify the Test Statistic and its Distribution of births, they account for nearly 2/3 of infant deaths. (d) Compute the Critical Test Statistic(s) birthweight for babies born in the U.S. is approxithose who do not live in poverty. While the average (e) Compute the the average birthweightp-Value living in poverty is 2800 grams. Eliminating the mately 3300 grams, Test Score and the for women linkage between label and low birthweight status has been prominent dimension (f) Draw and povertydiagrams of the distribution of athe test statistic. of health policy for the past decade. Recently, a local hospital introduced an innovative new prenatal care program to Clearly indicate the critical region(s), the critical test statistic(s), reduce the number of Shade the areababies born in the to the p-value.first year, 25 mothers, all of the test score. low birthweight corresponding hospital. In the In the United States, mothers who live in poverty generally have babies with lower birthweight than whom live in poverty, participated in this program. Data drawn null hypothesis) (g) State your conclusion (whether or not you reject thefrom hospital records reveals that the babies born to these women had a birthweight of 3075 grams. The standard deviation of birth weights and state why. is known to be 500 grams. The question posed to you, the researcher, is whether this program has been effective at improving the birthweights of babies born to poor women. (a) Formulate Hypothesis Clearly indicate the test score. Sh Formulate Hypothesis Null Value of Parameter (g) State your conclu and state why. Parameter H0 : µ ≤ 110 Ha : µ > 110 Identify Test Statistic… Statistic Test Statistic Parameter T= ¯ X −µ √ s/ n Std. Error t-Test …and its Distribution t-Test (Pooled) 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! Two Sample µ1 − µ2 t(df = n − 1) population group ! 0.2 a b c ! ! 0.1 !3 T= !2 !1 0 x ¯ X −µ √ s/ n 1 2 3 Compute Test Score Test Score Statistic (Observed Value) Parameter (Null Value) Std. Error 1 Two Sample One-Sample t-Test Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t-Test (Pooled) One Sample µ1 − µ2 t-Test Two Sample z-Test d0 t(df = n − 1) µ Fail to RejectFail to Reject H0 H0 population H0 : µ1 − µ2 = d0 0.2 Ha : µ1 − µ2 = d0 p 2 0.1 x1 − x2 ¯ ¯ α 2∗ !3 H0 : µ = µ0 Two Sample µ H0 : µ1 − µ2 = d0 t-Test Ha : µ1 − µ2 = d0 group group a b b c c ! ! ! Ha : µ = µ0 t∗ t Two− x2 x1 Sample ¯ ¯ α 2t-Test (Pooled) p 2 3 µ 0.0 −t −t∗ !2 !1 0 1 2 x t∗ t = ∗ −tx ¯ x−µ0 ¯√ s/ n µ0 Robustness The t procedures are exactly correct when the population is distributed exactly normally. However, most real data are not exactly normal. The t procedures are robust to small deviations from normality – the results will not be affected too much. Factors that strongly matter: –  Random sampling. The sample must be an SRS from the population. –  Outliers and skewness. They strongly influence the mean and therefore the t procedures. However, their impact diminishes as the sample size gets larger because of the Central Limit Theorem. Specifically:
       When
n
<
15,
the
data
must
be
close
to
normal
and
without
outliers.
 When
15
>
n
>
40,
mild
skewness
is
acceptable
but
not
outliers.
 When
n
>
40,
the
t‐sta+s+c
will
be
valid
even
with
strong
skewness.

 Problem 1
 
An exam consists of two parts. Part I is a multiple choice section and the time to complete it is a normally distributed random variable with a mean of 55 minutes and a standard deviation of 6 minutes. Part II is in essay format and the time to complete it is normally distributed with a mean of 110 minutes and a standard deviation of 9 minutes. Assume the time to complete each part is independent. What percentage of students will finish the entire exam in less than three hours? Problem
2
 
 The academic motivation and study habits of female students as a group are better than those of males. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures these factors. The distribution of SSHA scores among the women at a college has mean 120 and standard deviation 28, and the distribution of scores among the men has mean 105 and standard deviation 35. A single male student and a single female student are selected at random and given the SSHA test. Assume the scores of the two students are independent: (a) What are the mean and standard deviation of the difference (female minus male) between their scores? (b) What’s the probability that the woman chosen scores higher than the man? Assume the distributions are normal. Degree of Reading Power (DRP): Can directed reading activities in the classroom help in improve these reading ability? A class of 15 thirdgraders participates activities for 8 weeks while a control class of 15 third-graders follows the same curriculum without the activities. After 8 weeks, all take a DRP test . Treatment
 24
 43
 58
 71
 43
 49
 61
 44
 67
 49
 53
 56
 59
 52
 62
 Control
 42
 43
 55
 26
 62
 37
 33
 41
 19
 54
 20
 85
 46
 10
 17
 Formulate Hypothesis Parameter Null Value of Parameter H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 > 0 Identify Test Statistic… µ1 − µ2 µ µ− 1 ¯ X −X ¯¯ X1 −1X2 ¯ 2 Test Statistic n1 2 S s2 s 2 2 2 tatistic s2 s1 1 2 n n + 1 + T= T (X −X )−(µ 2−µ ) 2 1 2 2 r 2 s1 s1 n n2 s2 1 s2 1+ 2 n1 n2 ¯ ¯ (X1 −X2 )−(µ1 −µ2 ) r = ¯ ¯ + n2 2 x1 − x2 ¯ ¯ Parameter s2 1 n1 µ1 − µ2 + s2 2 n2 s2 1 n1 s2 T= µ1 − µ2 s2 1 n1 x1 − Error ¯ Std. x2 ¯ s2 2 n2 Group Shoes On Shoes Off (¯1 −x2 )−(µ1 −µ2 ) x ¯ r + n2 2 n 15 15 + x1 − x2 = 1.57 ¯ ¯ x ¯− r T = (¯1 −x2 )2 (µ1 −µ2 ) d0 = 0 s s2 1 + 2 z= p= ˆ …and its Distribution !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! n1 p1 +n2 p2 ˆ ˆ n1 +n2 q p(1−p) ˆ ˆ p(1−p) ˆ ˆ +n n1 2 (p1 −p2 )−d0 ˆˆ t(df = min(n1 , n2 ) − 1) t(df = n1 + n2 − 2) 0.3 population group ! 0.2 µ1 − µ2 s2 1 n1 !2 a b c ! ! 0.1 ¯ ¯ X1 − X2 + s2 2 n2 !3 T= ¯ ¯ (X1 −X2 )−(µ1 −µ2 ) x r !1 0 1 2 3 s2 1 n1 s2 + n2 2 s2 1 1 + s2 2 n2 = 5.93 = 0.82 1.57−0 1.91 Compute Test Score Group Treatment Control n 15 15 x ¯ 52.73 39.33 s 11.62 19.82 Parameter (Null Value) Test Score Statistic (Observed Value) Std. Error z= 3 2-Sample Independent t-Test (Unpooled) p= ˆ n1 p1 +n2 p2 ˆ ˆ n1 +n2 q p(1−p) ˆ ˆ p(1−p) ˆ ˆ +n n1 2 (p1 −p2 )−d0 ˆˆ Fail to Reject H0 0.3 population H0 : µ1 − µ2 = d0 0.2 Ha : µ1 − µ2 = d0 p 2 0.1 x1 − x2 ¯ ¯ α 2∗ !3 0.0 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t(df = min(n1 , n2 ) − 1) t(df = n1 + n2 − 2) Fail to RejectFail to Reject H0 H0 H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 ! ! group group a b b c c ! H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 tt ∗ H0 : 3 α 2 x1 − x2 ¯ ¯ p µ2 1 − µ2 = d0 s1 n1 −t −t∗ !2 !1 0 1 2 x H0 : µ1 − µ2 = d0 ¯ : (¯ −x2 2 = r t∗Ha= µx1−2 µ)−2d0 d0 1 t ¯1 ¯ −x∗ − x2 t + n2 s 2 A company was wondering which style of pepperoni pizza was most popular. It set up an experiment where ten people were each given two types of pizza to eat, Type A and Type B. Each pizza was carefully weighed at exactly 16 oz. After fifteen minutes, the remainders of the pizza were weighed, and the amount of each type pizza remaining per person was calculated. It is assumed that the subject would eat more of the type of pizza he or she preferred. ID
 Pizza
A
 Pizza
B
 1 2 3 4 5 6 7 8 9 10 12.9 5.7 16.0 14.3 2.4 1.6 14.6 10.2 4.3 6.6 16.0 7.5 16.0 15.7 13.2 5.4 15.5 11.3 15.4 10.6 Formulate Hypothesis Parameter Null Value of Parameter H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 > 0 Identify Test Statistic… µ1 − µ2 µ µ− 1 ¯ X −X ¯¯ X1 −1X2 ¯ 2 Test Statistic n1 2 S s2 s 2 2 2 tatistic s2 s1 1 2 n n + 1 + T= T (X −X )−(µ 2−µ ) 2 1 2 2 r 2 s1 s1 n n2 s2 1 s2 1+ 2 n1 n2 ¯ ¯ (X1 −X2 )−(µ1 −µ2 ) r = ¯ ¯ + n2 2 x1 − x2 ¯ ¯ Parameter s2 1 n1 µ1 − µ2 + s2 2 n2 s2 1 n1 s2 T= µ1 − µ2 s2 1 n1 x1 − Error ¯ Std. x2 ¯ s2 2 n2 Group Shoes On Shoes Off (¯1 −x2 )−(µ1 −µ2 ) x ¯ r + n2 2 n 15 15 + x1 − x2 = 1.57 ¯ ¯ x ¯− r T = (¯1 −x2 )2 (µ1 −µ2 ) d0 = 0 s s2 1 + 2 z= p= ˆ …and its Distribution !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! n1 p1 +n2 p2 ˆ ˆ n1 +n2 q p(1−p) ˆ ˆ p(1−p) ˆ ˆ +n n1 2 (p1 −p2 )−d0 ˆˆ t(df = min(n1 , n2 ) − 1) t(df = n1 + n2 − 2) 0.3 population group ! 0.2 µ1 − µ2 s2 1 n1 !2 a b c ! ! 0.1 ¯ ¯ X1 − X2 + s2 2 n2 !3 T= ¯ ¯ (X1 −X2 )−(µ1 −µ2 ) x r !1 0 1 2 3 s2 1 n1 s2 + n2 2 Compute Test Score Test Score Statistic (Observed Value) Parameter (Null Value) Std. Error Your manager comes along and tells you: “ This does not make sense. Is it not clear that Pizza A is more preferred as all customers in the sample left more of Pizza B than Pizza A. You with your fancy statistical knowledge are telling me that I cannot conclude that way, but is the result not intuitive” What would you tell your manager? ID
 Pizza
A
 Pizza
B
 1 2 3 4 5 6 7 8 9 10 12.9 5.7 16.0 14.3 2.4 1.6 14.6 10.2 4.3 6.6 16.0 7.5 16.0 15.7 13.2 5.4 15.5 11.3 15.4 10.6 Formulate Hypothesis H : µ ≤ 110 0 Ha P:arameter 110 µ> H0 : µd = 0 Ha : µd > 0 Null Value of Parameter IdentifyH:a d:>µ > 110 Test Statistic… Ha µ 0 H:0µ > 110 ≤ 110 :µ H a H0 : µd = 0 T= D −µd √ sd / n H0¯ : µd = 0 ¯ Test Statistic Ha : µd > 0 Statistic D = X1 − X2 Parameter T= D −µd √ sd / n Std. Error 1 t-Test and state why µ1 − µ2 …and its Distribution t-Test (Pooled) 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! Two Sample H0 : µ ≤df1101) t( = n − Ha : µ > 110 H0 : µd = 0 T= !2 !1 population group ! 0.2 a b c ! ! 0.1 Ha : µd > 0 x !3 D −µd √ sd / n 0 1 2 3 Compute Test Score Test Score Statistic (Observed Value) Parameter (Null Value) Std. Error 2 Two Sample Matched Pair t-Test Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t-Test (Pooled) µ1 − µ2 d0 t(df = n − 1) Fail to RejectFail to Reject H0 H0 z= q p1 (1−p1 ) ˆ ˆ p (1−p ) ˆ ˆ + 2n 2 n1 2 (p1 −p2 )−d0 ˆˆ population H0 : µ1 − µ2 p1 − p2 =0 ˆ dˆ 0.2 Ha : µ1 − µ2 N (0d1) = ,0 p 2 H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 ! ! group group a b b c c ! 0.1 x1 − x2 ¯ ¯ α 2∗ !3 H0 : µd = d0 Ha : µd = d0 tt ∗ H0 : 3 α 2 x1 − x2 ¯ ¯ p µ2 1 0.0 − µ2 = d0 d −t −t∗ !2 !1 0 1 2 x −¯ ) µ t∗Ha= µx1− x√n= 0d0 t : (¯ s /22 −d 1 H0 : µ1 − µ2 = d0 ¯1 ¯ −x∗ − x2 t Inference on Means Summary Response Variable Explanatory Variable 1 sample None 2 samples Independent >2 samples Pooled 2-sample t-test, Pooled ANOVA 4 Matched Unpooled 1-sample t-test Matched Pair t-test, 1 2 3 2-sample t-test, Unpooled Quant. Categ. 9 Quant. Multiple Regression 10
 1 Two Sample One-Sample t-Test Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t-Test (Pooled) One Sample µ1 − µ2 t-Test Two Sample z-Test d0 t(df = n − 1) µ Fail to RejectFail to Reject H0 H0 population H0 : µ1 − µ2 = d0 0.2 Ha : µ1 − µ2 = d0 p 2 0.1 x1 − x2 ¯ ¯ α 2∗ !3 H0 : µ = µ0 Two Sample µ H0 : µ1 − µ2 = d0 t-Test Ha : µ1 − µ2 = d0 group group a b b c c ! ! ! Ha : µ = µ0 t∗ t Two− x2 x1 Sample ¯ ¯ α 2t-Test (Pooled) p 2 3 µ 0.0 −t −t∗ !2 !1 0 1 2 x t∗ t = ∗ −tx ¯ x−µ0 ¯√ s/ n µ0 Response Variable Explanatory Variable 1 sample None 2 samples Independent >2 samples Pooled 2-sample t-test, Pooled ANOVA 4 Matched Unpooled 1-sample t-test Matched Pair t-test, 1 2 3 2-sample t-test, Unpooled Quant. Categ. 9 Quant. Multiple Regression 10
 z= 3 2-Sample Independent t-Test (Unpooled) p= ˆ n1 p1 +n2 p2 ˆ ˆ n1 +n2 q p(1−p) ˆ ˆ p(1−p) ˆ ˆ +n n1 2 (p1 −p2 )−d0 ˆˆ Fail to Reject H0 0.3 population H0 : µ1 − µ2 = d0 0.2 Ha : µ1 − µ2 = d0 p 2 0.1 x1 − x2 ¯ ¯ α 2∗ !3 0.0 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t(df = min(n1 , n2 ) − 1) t(df = n1 + n2 − 2) Fail to RejectFail to Reject H0 H0 H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 ! ! group group a b b c c ! H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 tt ∗ H0 : 3 α 2 x1 − x2 ¯ ¯ p µ2 1 − µ2 = d0 s1 n1 −t −t∗ !2 !1 0 1 2 x H0 : µ1 − µ2 = d0 ¯ : (¯ −x2 2 = r t∗Ha= µx1−2 µ)−2d0 d0 1 t ¯1 ¯ −x∗ − x2 t + n2 s 2 Response Variable Explanatory Variable 1 sample None 2 samples Independent >2 samples Pooled 2-sample t-test, Pooled ANOVA 4 Matched Unpooled 1-sample t-test Matched Pair t-test, 1 2 3 2-sample t-test, Unpooled Quant. Categ. 9 Quant. Multiple Regression 10
 2 Two Sample Matched Pair t-Test Fail to Reject H0 0.3 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! t-Test (Pooled) µ1 − µ2 d0 t(df = n − 1) Fail to RejectFail to Reject H0 H0 z= q p1 (1−p1 ) ˆ ˆ p (1−p ) ˆ ˆ + 2n 2 n1 2 (p1 −p2 )−d0 ˆˆ population H0 : µ1 − µ2 p1 − p2 =0 ˆ dˆ 0.2 Ha : µ1 − µ2 N (0d1) = ,0 p 2 H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 ! ! group group a b b c c ! 0.1 x1 − x2 ¯ ¯ α 2∗ !3 H0 : µd = d0 Ha : µd = d0 tt ∗ H0 : 3 α 2 x1 − x2 ¯ ¯ p µ2 1 0.0 − µ2 = d0 d −t −t∗ !2 !1 0 1 2 x −¯ ) µ t∗Ha= µx1− x√n= 0d0 t : (¯ s /22 −d 1 H0 : µ1 − µ2 = d0 ¯1 ¯ −x∗ − x2 t Response Variable Explanatory Variable 1 sample None 2 samples Independent >2 samples Pooled 2-sample t-test, Pooled ANOVA 4 Matched Unpooled 1-sample t-test Matched Pair t-test, 1 2 3 2-sample t-test, Unpooled Quant. Categ. 9 Quant. Multiple Regression 10
 z= 4 p= ˆ t(df = min( , n ) − 1) 2-Sample Independent nt-Test (Pooled) 1 2 !! !! ! !!! !! !!! !! !! !!! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! !! ! !! ! !! !! ! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! !! ! !! ! !! !! !! !! !!! !!! !!! !!! !!! !!! !!! !!! !!! !!!! !!!! !!!! !!! !!! !!! !!! !!!!! !! !!! !!!!!!! !!! !!!!!!! !!! !!!!!!! !!!!!! !! !!!!!!! ! !!!!!!! ! !!!!!!!! !!!!!!!! n1 p1 +n2 p2 ˆ ˆ n1 +n2 q p(1−p) ˆ ˆ p(1−p) ˆ ˆ +n n1 2 (p1 −p2 )−d0 ˆˆ t(df = n1 + n2 − 1) Fail to Reject H0 0.3 Fail to RejectFail to Reject H0 H0 z= population H0 : µ1 − µ2 = d0 0.2 Ha : µ1 − µ2 = d0 p 2 0.1 x1 − x2 ¯ ¯ α 2∗ !3 H0 : µ1 − µ2 = d0 Ha : µ1 − µ2 = d0 H0 : µ1 − µ2 = d0 p1 − p 2 ˆ ˆ Ha : µ1 − µ2 = d0 N (0, 1) x1 − x2 ¯ ¯ H0 : µd = d0 α group group a b b c c ! ! ! q p1 (1−p1 ) ˆ ˆ p (1−p ) ˆ ˆ + 2n 2 n1 2 (p1 −p2 )−d0 ˆˆ tt ∗ 0.0 p H0 a: : µd− µd0= 1 H µ2 = 2 3 2 d0 −t −t∗ !2 !1 0 1 2 x H0 : µ1 − µ2 = d0 x r t∗Ha= µx1 −2¯2 )2 2d0 d0 t : (¯ − µ − = 1 sp ¯1 −x∗ t − x2 ¯ n1 +n sp 2 Response Variable Explanatory Variable 1 sample None 2 samples Independent >2 samples Pooled 2-sample t-test, Pooled ANOVA 4 Matched Unpooled 1-sample t-test Matched Pair t-test, 1 2 3 2-sample t-test, Unpooled Quant. Categ. 9 Quant. Multiple Regression 10
 What Type of Test? •  Comparing
 vitamin
 content
 of
 bread
 immediately
 aVer
 baking
 vs.
 3
 days
 later
(the
same
loaves
are
used
on
day
 one
and
3
days
later).
 •  Comparing
 vitamin
 content
 of
 bread
 immediately
 aVer
 baking
 vs.
 3
 days
 later
 (tests
 made
 on
 independent
 loaves).
 •  Average
 fuel
 efficiency
 for
 2005
 vehicles
 is
 21
 miles
 per
 gallon.
 Is
 average
 fuel
 efficiency
 higher
 in
 the
 new
genera+on
“green
vehicles”?

 •  Is
blood
pressure
altered
by
use
of
an
 oral
 contracep+ve?
 Comparing
 a
 group
 of
 women
 not
 using
 an
 oral
 contracep+ve
with
a
group
taking
it.
 •  Review
 insurance
 records
 for
 dollar
 amount
 paid
 aVer
 fire
 damage
 in
 houses
 equipped
 with
 a
 fire
 ex+nguisher
 vs.
 houses
 without
 one.
 Was
there
a
difference
in
the
average
 dollar
amount
paid?
 ...
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This note was uploaded on 08/31/2010 for the course MANAGEMENT MGCR 271 taught by Professor Vaidyanathan during the Summer '10 term at McGill.

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