4- ProbabilityDistributionAnswers

Probability Distribution: Review Answers Jeremy Biesanz 2020-09-24 Normal Distribution: Review and Consolidation 1. What is the probability of observing a normal score above 1.4? pnorm ( 1.4 , lower.tail= FALSE ) ## [1] 0.08075666 2. What is the probability of observing a normal score below -0.7? pnorm ( - 0.70 , lower.tail= TRUE ) ## [1] 0.2419637 3. Compute the probability of observing a normal score between - 0 . 7 and 1 . 0 ?. #There are several ways to get to this answer. pnorm ( 1.00 , lower.tail= TRUE ) - pnorm ( - 0.70 , lower.tail= TRUE ) ## [1] 0.5993811 4. What is the critical z -score, two-tailed, associated with α = . 05 ? qnorm (. 025 , lower.tail= FALSE ) ## [1] 1.959964 5. What is the critical z -score, two-tailed, associated with α = . 01 ? qnorm (. 005 , lower.tail= FALSE ) ## [1] 2.575829 6. What is the critical z -score, two-tailed, associated with α = . 005 ? qnorm (. 0025 , lower.tail= FALSE ) ## [1] 2.807034 7. What is the p -value (always assume two-tailed) that corresponds to an observed test statistic z = 2.93? 2 * pnorm ( 2.93 , lower.tail= FALSE ) ## [1] 0.00338962

probability distribution: review answers 2 χ 2 ( df ) -Distribution: Review and Consolidation 1. What is the probability of observing a χ 2 (3) score above 4.0? pchisq ( 4.0 , df = 3 , lower.tail= FALSE ) ## [1] 0.2614641 2. What is the probability of observing a χ 2 (1) score below 2? pchisq ( 2 , df = 1 , lower.tail= TRUE ) ## [1] 0.8427008 3. What is the critical χ 2 (1) -score associated with α = . 05 ? qchisq (. 05 , df = 1 , lower.tail= FALSE ) ## [1] 3.841459 4. What is the square root of the answer above? Have you seen this number before and does this make sense? #This answer corresponds to the critical value for the normal distribution #with alpha = .05, two-tailed. sqrt ( qchisq (. 05 , df = 1 , lower.tail= FALSE )) ## [1] 1.959964 5. What is the critical χ 2 (1) -score, associated with α = . 01 ? qchisq (. 01 , df = 1 , lower.tail= FALSE ) ## [1] 6.634897 6. What is the critical χ 2 (12) -score associated with α = . 005 ? qchisq (. 005 , df = 12 , lower.tail= FALSE ) ## [1] 28.29952 7. What is the p -value that corresponds to an observed test statistic χ 2 (2) = 3 . 66 ? pchisq ( 3.66 , df = 2 , lower.tail= FALSE ) ## [1] 0.1604136

probability distribution: review answers 4 F -Distribution: Review and Consolidation 1. What is the probability of observing an F (4, 80) score above 2.5? pf ( 2.5 , df1= 4 , df2= 80 , lower.tail= FALSE ) ## [1] 0.04896127 2. What is the probability of observing an F (2, 200) score above 1.5? pf ( 1.5 , df1= 2 , df2= 200 , lower.tail= FALSE ) ## [1] 0.2256294 3. What is the critical F (1, 200)-score associated with α = . 05 ? Is this really the square of the t (200) critical value for the same α ? qf (. 05 , df1= 1 , df2= 200 , lower.tail= FALSE ) ## [1] 3.888375 #Yes it is. sqrt ( qf (. 05 , df1= 1 , df2= 200 , lower.tail= FALSE )) ## [1] 1.971896 4. What is the critical F (1, 200) value associated with α = . 005 ? qf (. 005 , df1= 1 , df2= 200 , lower.tail= FALSE ) ## [1] 8.05716 5. What is the p -value that corresponds to F (1, 200) = 3.90? pf ( 3.90 , df1= 1 , df2= 200 , lower.tail= FALSE ) ## [1] 0.04966202 6. What is the p -value that corresponds to F (3, 160) = 2.54? pf ( 2.54 , df1= 3 , df2= 160 , lower.tail= FALSE ) ## [1] 0.05840662 Probability Distribution Review and Consolidation No computations are required to answer the following questions. 1. Which test statistic has a lower p -value: z = 2 . 0 or t (16) = 2 . 0 ? z = 2 . 0 will have the lower p -value.

probability distribution: review answers 5 2. Which test statistics have the same p -values? z = 2 . 0 , t (16) = 2 . 0 , χ 2 (1) = 4 . 0 , F (1 , 16) = 4 . 0 z = 2 . 0 and χ 2 (1) = 4 . 0 will have the same p -value, as will t (16) = 2 . 0 and F (1 , 16) = 4 . 0 3. Why does pnorm(2.00, lower.tail=FALSE) not give us the p - value for the question above, but pchisq(4.0, df=1, lower.tail=FALSE) does? pnorm(2.00, lower.tail=FALSE) provides the probability of z ≥ 2 whereas pchisq(4.0, df=1, lower.tail=FALSE) provides the two- tailed probability (probablity z ≥ 2 + probability z ≤ - 2 ). 4. What do we have to the answer to pnorm(2.00, lower.tail=FALSE) to give us the correct p -value? To get the two-tailed p -value we need to multiply this answer by 2. 5. For α = . 05 under the normal distribution, is the critical value of 1.96 (a) a p -value, (b) a quantile, (c) a parameter, or (d) a test- statistic? This is (b) a quantile. 6. What do the df for a t -test indicate? Does this tell us about the amount of information in the means or the variance? The df inform us about the amount of information (and thus preci- sion) associated with the estimate of the variance that is being used in computing the t -test. 7. Which statistic’s sampling distribution is the χ 2 ( df ) -distribution related to? The χ 2 ( df ) -distribution is related to the sampling distribution of the sample variance.