Dr. Allen Back. Oct. 28, 2016

Transcription

1 Dr. Allen Back Oct. 28, 2016

2 A coffee vending machine dispenses coffee into a paper cup. You re supposed to get 10 ounces of coffee., but the amount varies slightly from cup to cup. The amounts measured in a random sample of 20 cups have summary statistics as given below. Is there evidence that the machine is shortchanging customers? x = s =.1986.

3 Notation: Let µ denote the mean amount of coffee in a dispensed cup. Hypotheses: H 0 : µ = 10 (or µ 10) H a : µ < 10

4 As usual with, we are interested in whether the observed statistic of x = is reasonably consistent with the sampling distribution assuming H 0 is true.

5 If s = σ, we d look at a Z-statistic Z = x µ 0 σ n = where we ve written H 0 more abstractly as µ = µ 0, µ 0 being the hypothesized value, 10 in this case.

6 Because s will not exactly match σ, we actually get a bit of extra error here. This is compensated for by viewing as a t-statistic. t = x µ 0 s = n = =

7 Since the error in approximating σ by s varies with the sample size, there is a different t-distribution for each sample size. These are labeled by the degrees of freedom which for a 1-sample t-test is: df = n 1.

8 These are all critical values t. For example P(t > 1.328) =.10 for the t distribution with 19 df.

9 Our t-statistic of is more extreme than any on the df=19 row of the table. The picture shows what the critical value t = for a tail prob. of.005 means.

10 So by symmetry P(T < 2.861) =.005 as well.

11 So our tail probability and p-value are both less than.005 and we reject the null. The machine does appear to be shortchanging.

12 Sps. our t-statistic had been with the same 1-sided hypotheses H 0 : µ = 10 (or µ 10) H a : µ < 10 What P-value would we report?

13 Sps. our t-statistic had been with the same 1-sided hypotheses H 0 : µ = 10 (or µ 10) H a : µ < 10 What P-value would we report? Answer: A tail probability and P-value of between.025 and.05.

14 Sps. instead our t-statistic had been 2.00 with 2-sided hypotheses H 0 : µ = 10 H a : µ 10 What P-value would we report?

15 Sps. instead our t-statistic had been 2.00 with 2-sided hypotheses H 0 : µ = 10 H a : µ 10 What P-value would we report? Answer: Our tail probability is still between.025 and.05 but our P-value is now between.05 and.10.

16 Sps. instead our t-statistic had been 2.00 with 2-sided hypotheses H 0 : µ = 10 H a : µ 10 What P-value would we report? In using the table, always be conservative. If you want df=138 and there is a df=120 as well as a df=140 row, the principle of being conservative means to use the df=120 row. When you say I am 95% confident... or I reject H 0, you ll be delivering what you promised; with better tables you d just be able to report your level of confidence as somewhat higher.

17 By Calculator TI 83/84: 2nd distr tcdf(-100,-2,19) to find P(T < 2) for a t-distribution with df = 19.

18 By Calculator TI 83/84: 2nd distr tcdf(-100,-2,19) to find P(T < 2) for a t-distribution with df = 19. TI 89: catalog F3 2nd-alpha t... tcdf(-100,-2,19) to find P(T < 2) for a t-distribution with df = 19.

19 A t-hypothesis test with is resolved using a t-statistic of H 0 : µ = µ 0 t = x µ 0 s. n

20 A t-hypothesis test with is resolved using a t-statistic of H a can be any of the three 1 µ µ 0 2 µ > µ 0 3 µ < µ 0 H 0 : µ = µ 0 t = x µ 0 s. n

21 A t confidence interval for µ is x ± t s n for the same reason as the corresponding formula in the proportion case:

22 The two sampling distributions at the border of the CI. The blue lines delimit central regions on the sampling distributions. The CI is between the green lines on the horizontal axis.

23 In the coffee machine example, this gives a 95% CI for µ of ± 2.093(.0444) = ±.093 = (9.752, 9.938) (since n = 20 and SE( x) = s =.1986 =.0444 here.) n 20 You did expect it to not include 10, didn t you?

24 t-distributions Let X 0, X 1,... X d be d + 1 independent standard normal random variables.

25 t-distributions Let X 0, X 1,... X d be d + 1 independent standard normal random variables. The t-distribution with d degrees of freedom is defined to be the random variable X 0. X1 2 + X X d 2

26 t-distributions The t-distribution with d degrees of freedom is defined to be the random variable X 0 X X X 2 d Think of this as keeping track of the t-statistic x µ 0 s = n x µ 0 σ n s σ with the top X 0 keeping track of the numerator and the denominator X1 2 + X X d 2 keeping track of the ratio of s to σ..

27 t-distributions Just as the normal distribution has the density formula f (x) = e (x µ) 2 2σ 2 2πσ

28 t-distributions Methods of calculus show that the density formula for the t-distribution with d degrees of freedom is f (x) = Γ( d+1 2 ) Γ( d 2 ) 1 dπ 1 (1 + x2 d ) d+1 2 where d is the number of degrees of freedom and Γ(n) = 0 t n 1 e t dt is the gamma function, a generalized factorial.

29 t-distributions It will be important when we think about 2-sample tests to realize that the degrees of freedom is just a parameter in the above formula.

30 t-distributions A little more probability in the tails

31 t-distributions Cumulative Dist. Fcn (Graphical Form of Table Z)

32 t-distributions This picture shows why t can be very different from z.

33 t-distributions

34 Conditions for t Tests random sampling 10% condition nearly normal condition n < 15 unimodal and symmetric or only slight skew 15 <= n < 30 avoid strong (maybe even moderate) skewness and outliers (unimodal and sym best) 30 n < 60 moderate skewness ok n 60 pretty much (not really always) ok even with strong skew

35 Conditions for t Tests random sampling 10% condition nearly normal condition n < 15 unimodal and symmetric or only slight skew 15 <= n < 30 avoid strong (maybe even moderate) skewness and outliers (unimodal and sym best) 30 n < 60 moderate skewness ok n 60 pretty much (not really always) ok even with strong skew What Happens if Not Satisfied: random sampling - could be critical; might be ok if representative representative hard/impossible to define

36 Conditions for t Tests random sampling 10% condition nearly normal condition n < 15 unimodal and symmetric or only slight skew 15 <= n < 30 avoid strong (maybe even moderate) skewness and outliers (unimodal and sym best) 30 n < 60 moderate skewness ok n 60 pretty much (not really always) ok even with strong skew What Happens if Not Satisfied: 10% condition - results in overestimation of samp. dist. st dev gradual breakdown in formulas, not method

37 Conditions for t Tests random sampling 10% condition nearly normal condition n < 15 unimodal and symmetric or only slight skew 15 <= n < 30 avoid strong (maybe even moderate) skewness and outliers (unimodal and sym best) 30 n < 60 moderate skewness ok n 60 pretty much (not really always) ok even with strong skew What Happens if Not Satisfied: nearly normal - no guarantee progressive reduction of accuracy

38 Conditions for t Tests random sampling 10% condition nearly normal condition n < 15 unimodal and symmetric or only slight skew 15 <= n < 30 avoid strong (maybe even moderate) skewness and outliers (unimodal and sym best) 30 n < 60 moderate skewness ok n 60 pretty much (not really always) ok even with strong skew What Happens if Not Satisfied:

39 Conditions for t Tests For 2-sample inference, we add the independence groups assumption. The chance of an individual in one of the groups assuming a certain value should be independent of the values assumed by any of the individuals in the other group.

40 Formulas for t-inference and regression inference are based on assumptions of normality of the data. Yet most distributions are not normal. (Although the CLT makes averages of lots normal.) So it is perhaps remarkable that t-inference methods for moderate size data sets without outliers are typically pretty good. Why is this?

41 Formulas for t-inference and regression inference are based on assumptions of normality of the data. Yet most distributions are not normal. (Although the CLT makes averages of lots normal.) So it is perhaps remarkable that t-inference methods for moderate size data sets without outliers are typically pretty good. Why is this? One answer is robustness against non-normality.

42 Actual confidence level vs kurtosis for some symmetric n = 25 test distributions in a 1975 Biometrika paper of Pearson and Please. (kurtosis=3 in the normal case.)

43 Actual confidence level vs kurtosis for some skewed n = 25 test distributions in a 1975 Biometrika paper of Pearson and Please.

44 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? haiku: A Japanese poem composed of three unrhymed lines of five, seven, and five syllables. Haiku often reflect on some aspect of nature.

45 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this?

46 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Randomized Experiment on Creativity in poetry: Grades on creativity recorded. (Higher is better.) Both groups filled out questionnaires before writing poetry. Intrinsic group first did questionnaire emphasizing intrinsic motivations. Extrinsic group first did questionnaire emphasizing extrinsic motivations.

47 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? People then wrote Haiku poems which were graded by experienced poets.

48 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? The Results Group n x s Intrinsic Extrinsic

49 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Notation: Let µ i and µ e denote the respective mean haiku scores for the intrinsically and the extrinsically motivated groups.

50 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Notation: Let µ i and µ e denote the respective mean haiku scores for the intrinsically and the extrinsically motivated groups. Hypotheses: H 0 : µ i = µ e H a : µ i µ e

51 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e Under H 0, the sampling distribution of x i x e has mean 0 and approx. std deviation si 2 SE( x i x e ) = + s2 e n i n e

52 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e Under H 0, the sampling distribution of x i x e has mean 0 and approx. std deviation si 2 SE( x i x e ) = + s2 e n i n e SE( x i x e ) = = 1.42.

53 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e SE( x i x e ) = = Our t-statistic is t = = 2.92.

54 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e Our t-statistic is SE( x i x e ) = t = = = For homework and hand computation on exams, we suggest you use df the smaller of 24-1 and 23-1 to interpret the t-statistic. To be explained later.

55 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e With df = 22, 2.92 is more extreme than the biggest t in table T (which is 2.819) so our tail probability is less than.005 and our P-value is less than.01.

56 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e With df = 22, 2.92 is more extreme than the biggest t in table T (which is 2.819) so our tail probability is less than.005 and our P-value is less than.01. We reject H 0. Intrinsic vs. extrinsic motivation does seem to make a difference.

57 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e We reject H 0. Intrinsic vs. extrinsic motivation does seem to make a difference. A 95% CI for µ i µ e would be ± = 4.15 ± 2.95 = (1.20, 7.10).

58 Haiku Poetry: Intrinsic or Extrinsic Motivation Different in Effectiveness? How can you study this? Hypotheses: H 0 : µ i = µ e H a : µ i µ e Scope of the Inference: Causal relationship seems to be established. Doesn t say much about a population since not an SRS.

59 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y.

60 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. But the sampling distribution of x ȳ can be well approximated by a t-distribution with the following degrees of freedom:

61 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2

62 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 Let min(n x 1, n y 1) denote the smallest of n x 1 and n y 1. This df is always between min(n x 1, n y 1) and n x + n y 2.

63 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 Special Case s x = s y = s, n x = n y = n. df = ( s 2 y n y ) 2 ( ) 2s 2 2 n = 2(n 1). 1 2s 4 n 1 n 2

64 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. Special Case s2 x n x df = << s2 y n y df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 ) 2 y n y 1 n y 1 ( s 2 y n y ) 2 ( s 2 y n y ) 2 = n y 1.

65 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 Calculators and computers will use this df formula. For hand computation on HW and exams, we suggest you never use this formula. Instead use the conservative min(n x 1, n y 1).

66 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 Calculators and computers will use this df formula. For hand computation on HW and exams, we suggest you never use this formula. Instead use the conservative min(n x 1, n y 1). Your t will be a little too big so your CI will be a little too wide and your HT a little harder to show statistical significance. But this is not so bad.

67 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 In your published papers and lab reports, use a computer which will likely automatically use this formula.

68 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 derived this formula using the fact that the exact mean and variance of the sampling distribution of x ȳ is easy to calculate. He asked, What df in the t-distribution would give the right µ and σ. The answer is the above.

69 In the 2-independent sample case, even if the individual data is normal, the difference x ȳ does not follow a t-distribution unless σ x = σ y. df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2 f (x) = Γ( d+1 2 ) Γ( d 2 ) 1 dπ 1 (1 + x2 d ) d+1 2

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82 In the 2-independent sample case, x ȳ can usually be well approximated by a t-distribution with the following degrees of freedom: df = ( s 2 x n x + s2 y n y ) 2 ( ) 1 s 2 2 x n x 1 n x + 1 n y 1 ( s 2 y n y ) 2

83 The graphs below, from the java applet at illustrate the fact that this df is always between min(n x 1, n y 1) and n x + n y 2.

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86 Pulse Rate Does regular exercise reduce resting pulse rates? 10 volunteers do 20 minutes of exercise 3 times a week for 6 weeks. Their resting pulse rates before and after were measured. (Beats/min.)

87 Pulse Rate Subject Before After Allen Brandon Carlos David Edwin Franco Graeme Hans Ivan Jorge 79 76

88 Pulse Rate Subject Before After Allen Brandon Carlos David Edwin Franco Graeme Hans Ivan Jorge Notice that 8 out of 10 subjects had reduced RPR s. There is a non-parametric test called the sign test that could be used to conclude that RPR s tend to decrease as hypothesized. Can you see how binomial distribution ideas could be used for this?

89 Pulse Rate Notation: Let µ b be the mean resting pulse rate before the exercise program and µ a the mean resting pulse rate after.

90 Pulse Rate Notation: Let µ b be the mean resting pulse rate before the exercise program and µ a the mean resting pulse rate after. Hypotheses: H 0 : µ b = µ a (or µ b µ a 0 or... ) H a : µ b > µ a

91 Pulse Rate Since individuals have important characteristics besides their exercise programs which determine their RPR s, statistical tests based on Var( x b x a ) = Var( x b ) + Var( x a ) would not be appropriate here.

92 Pulse Rate In MP, we look at the differences and apply 1-sample ideas to the differences.

93 Pulse Rate Differences d i Subject Before - After Allen 0 Brandon 4 Carlos 4 David 1 Edwin 4 Franco 8 Graeme 3 Hans 2 Ivan -1 Jorge 3

94 Pulse Rate The mean d = 2.8 with a standard deviation s of Note you need to directly calculate s for the differences; it is not determined by s a and s b.

95 Pulse Rate The mean d = 2.8 with a standard deviation s of t = = 3.50 with df=9. Our P-value is <.005 and we reject H 0. Exercise does seem to reduce RPR s.

96 vs. Indep. Group1 Group 2 x 1 y 1 x 2 y x n... y m x ȳ mean s x s y std. dev

97 vs. Indep. Group1 Group 2 x 1 y 1 x 2 y x n... y m x ȳ mean s x s y std. dev Independent Each x i and each y j are (except to the extent that they come from one of the groups) independent of each other. The sample sizes m and n can be different.

98 vs. Indep. Group1 Group 2 x 1 y 1 x 2 y x n... y m x ȳ mean s x s y std. dev The i th observation x i in group 1 is more closely related (in a relevant way) to the i th observation y i in group 2 than a general x i is to a general y j. The sample sizes m and n must be the same.

99 or Indep? Chapter 25 # 24: In a test of breaking performance, a tire manufacturer measured the stopping distance for one of its tire models. On a test track, a car made repeated stops from 60 miles per hour. The test was run on both dry and wet pavement, with results as shown in the table. Are wet stopping distances greater?

100 or Indep? Stopping Distance (ft) Dry Pavement Wet Pavement

101 or Indep? Chapter 25 # 24: In a test of breaking performance, a tire manufacturer measured the stopping distance for one of its tire models. On a test track, a car made repeated stops from 60 miles per hour. The test was run on both dry and wet pavement, with results as shown in the table. Are wet stopping distances greater? The wet pavement figure on e.g. the fifth line bears no closer relationship to the fifth line in the dry column than it does to any other dry measurement. There is no matching. This is independent 2-sample.

102 or Indep? Had the problem referred to 10 different cars or 10 different drivers each trying both wet and dry, it would have been matched pairs.

103 or Indep? Summary of the Data Mean Std. Dev. dry wet

104 or Indep? Notation: Let µ d be the mean dry stopping distance. and µ w be the mean wet stopping distance.

105 or Indep? Notation: Let µ d be the mean dry stopping distance. and µ w be the mean wet stopping distance. Hypotheses: H 0 : µ d = µ w (or µ w µ d 0 or... ) H a : µ w > µ d

106 or Indep? Hypotheses: H 0 : µ d = µ w (or µ w µ d 0 or... ) H a : µ w > µ d n w = n d = 10, so we use df = 9 for hand calculation.

107 or Indep? Hypotheses: H 0 : µ d = µ w (or µ w µ d 0 or... ) H a : µ w > µ d n w = n d = 10, so we use df = 9 for hand calculation SE = =

108 or Indep? Hypotheses: H 0 : µ d = µ w (or µ w µ d 0 or... ) H a : µ w > µ d n w = n d = 10, so we use df = 9 for hand calculation SE = = t = =

109 or Indep? Hypotheses: H 0 : µ d = µ w (or µ w µ d 0 or... ) H a : µ w > µ d n w = n d = 10, so we use df = 9 for hand calculation SE = = t = = Our P-value is very tiny (<.005 from the tables) so we reject the null; Wet stopping distances appear to be longer.

110 or Indep? Chapter 25 # 19: A company institutes an exercise break for its workers to see if this will improve job satisfaction, as measured by a questionnaire that assesses workers satisfaction. Scores of 10 randomly selected workers before and after the implementation of the exercise program are shown.

111 or Indep? Job Satisfaction Index Worker Number Before After

112 or Indep? Worker (e.g.) 5 s job satisfaction after the program clearly has some extra association with his job satisfaction before. We have a clear matching here and MP is appropriate.

113 or Indep? Summary of the Data Mean Std. Dev. before -after

114 or Indep? Notation: Let µ b be the mean job satisfaction before. and µ a be the mean job satisfaction after.

115 or Indep? Notation: Let µ b be the mean job satisfaction before. and µ a be the mean job satisfaction after. Hypotheses: H 0 : µ b = µ a H a : µ a > µ b

116 or Indep? Hypotheses: H 0 : µ b = µ a H a : µ a > µ b n b = n a = 10, so we use df = 9 for hand calculation.

117 or Indep? Hypotheses: H 0 : µ b = µ a H a : µ a > µ b n b = n a = 10, so we use df = 9 for hand calculation. SE = s n = = 2.36

118 or Indep? Hypotheses: H 0 : µ b = µ a H a : µ a > µ b n b = n a = 10, so we use df = 9 for hand calculation. SE = s n = = 2.36 t = =

119 or Indep? Hypotheses: H 0 : µ b = µ a H a : µ a > µ b n b = n a = 10, so we use df = 9 for hand calculation. SE = s = 7.47 = 2.36 n 10 t = = Our P-value is very tiny (<.005 from the tables) so we reject the null; The program appears to improve job satisfaction.

120 or Indep? Chapter 25 # 21: Do these data suggest that there is a significant difference in calories between servings of strawberry and vanilla yogurt?

121 or Indep? Calories per Serving Brand Strawberry Vanilla America s Choice Breyer s Lowfat Columbo Dannon Light n Fit Dannon Lowfat Dannon lacreme Great Value La Yogurt Mountain High Stonyfield Farm Yoplait Custard Yoplait Light

122 or Indep? Since the two flavors are the same brand, MP is appropriate. There is also an outlier Great Value which we will exclude from our analysis.

123 or Indep? Summary of the Data (w/o Great Value) Mean Std. Dev. strawberry-vanilla

124 or Indep? Notation: Let µ s be the mean calories in a strawberry serving. and µ v be the mean calories in a vanilla serving.

125 or Indep? Notation: Let µ s be the mean calories in a strawberry serving. and µ v be the mean calories in a vanilla serving. Hypotheses: H 0 : µ s = µ v H a : µ s µ v

126 or Indep? Hypotheses: H 0 : µ s = µ v H a : µ s µ v n s = n v = 11, so we use df = 10 for hand calculation.

127 or Indep? Hypotheses: H 0 : µ s = µ v H a : µ s µ v n s = n v = 11, so we use df = 10 for hand calculation. SE = s n = = 5.45

128 or Indep? Hypotheses: H 0 : µ s = µ v H a : µ s µ v n s = n v = 11, so we use df = 10 for hand calculation. SE = s n = = 5.45 t = =.835

129 or Indep? Hypotheses: H 0 : µ s = µ v H a : µ s µ v n s = n v = 11, so we use df = 10 for hand calculation. SE = s = = 5.45 n 11 t = =.835 Our tail probability is above.1 from the tables and so our P-value is above.2. We retain the null. We haven t proven there are any differences in calories between strawberry and vanilla of a give brand.

130 MP as more Sensitive Sales figures for two car brands by district.

131 MP as more Sensitive

132 MP as more Sensitive

133 MP as more Sensitive

134 MP as more Sensitive

135 MP as more Sensitive

136 MP as more Sensitive

137 MP as more Sensitive