STAT 301, Spring 2013 Name Lec 1, MWF 9:55 - Ismor Fischer Discussion Section: Please circle one! TA: Shixue Li...... 311 (M 4:35) / 312 (M 12:05) / 315 (T 4:00) Xinyu Song... 313 (M 2:25) / 316 (T 12:05) Yi Liu 314 (M 3:30) EXAM # 2 Please show all work! Problem Points Grade 1 30 2 30 3 40 Total 100
1. A hexagonal dartboard consists of six equally-sized regions, each of which bears a score X, as shown. In a typical game, a blindfolded player throws a dart at random, which lands inside one of the regions. Answer each of the following. Show all work! 10 7 10 13 13 13 (a) Suppose the game is played indefinitely (in principle). Below, construct the corresponding probability table and clearly labeled probability histogram for the variable X. (10 pts) x f(x) (b) Calculate the expected value of X, as well its variance. (8 pts) Suppose the game is independently played n = 3 times, resulting in an outcome in the form of an ordered triple of scores (X 1, X 2, X 3 ). (c) Explicitly list the outcomes that correspond to the event (6 pts) E = All three scores are identical = { and calculate its probability P(E). (d) Let the sum of the three scores be denoted by Y = X 1 + X 2 + X 3. What is the expected value of the random variable Y? Hint: Use part (b). (2 pts) (e) Suppose that, if the sum of the three scores is exactly equal to 30, the player wins a prize. With what probability will this occur? (4 pts)
2. Three of the most commonly used variables in human studies are age, sex, and race. Suppose the Venn diagram below shows the demographic breakdown of a certain population. Answer each of the following; assume independent outcomes within each intersection. Show all work! Adult Male.03.09.06.36.12.24.08.02 White Suppose a random sample of n = 10 individuals is to be selected from this population. (6 pts ea) (a) Set up BUT DO NOT EVALUATE an expression for the probability of exactly 6 boys. (b) Set up BUT DO NOT EVALUATE an expression for the probability of exactly 6 whites. (c) Set up BUT DO NOT EVALUATE an expression for the probability of exactly 6 white boys. (d) Calculate the probability that the sample contains at least one boy (i.e., one or more). (e) Calculate the probability that if ten boys are selected, they are all white. That is, the probability of ten white boys, given ten boys.
3. Assume it is known that the variable X = Age at first birth in a certain population of women is normally distributed, with mean µ = 30.0 years, and standard deviation σ = 4.0 years, i.e., X ~ N(30, 4). Answer each of the following. Show all work! (a) Suppose that an individual woman is to be randomly selected from this population. Calculate the probability that her age at first birth is older than or equal to 31 years. (3 pts) (b) Suppose that a random sample of n = 100 women is to be selected from this population. Calculate the probability that the mean age at first birth is older than or equal to 31 years. (5 pts) Suppose it is also known that the ages X of several other populations are normally distributed with the same standard deviation σ = 4.0 years, but each perhaps with a significantly different mean µ. In each case, a sample of n = 100 women is randomly selected for formal hypothesis testing at the α =.05 level. Answer the following for each of the scenarios described in (c), (d), (e), and (f) below. Show all work! Calculate the p-value of the sample. (Tip: Draw a diagram; use (b) when possible.) (3 pts) Can the null hypothesis formally be rejected in favor of the alternative hypothesis? Why? (2 pts) Is the result considered to be statistically significant (Yes / No)? (3 pts) If yes, does the sample provide strong, moderate, or weak statistical evidence that the mean age is significantly different from 30 years? In addition, is the mean age significantly younger or older than 30 years? If no, are there any informal conclusions that are suggested by this analysis, and if so, what? (c) Null hypothesis H 0 : µ = 30 versus Alternative hypothesis H A : µ 30; sample mean age x = 30.8 years. (d) Null hypothesis H 0 : µ = 30 versus Alternative hypothesis H A : µ 30; sample mean age x = 31.0 years. (e) Null hypothesis H 0 : µ 30 versus Alternative hypothesis H A : µ > 30; sample mean age x = 31.0 years. (f) Null hypothesis H 0 : µ 30 versus Alternative hypothesis H A : µ > 30; sample mean age x = 29.0 years.
One Sample POPULATION PARAMETER Null Hypothesis H 0 : θ = θ 0 SAMPLE STATISTIC Point Estimate ˆ θ = f(x 1,, x n ) Mean 2 H 0 : µ = µ 0 ˆµ = x = x i n Proportion H 0 : π = π 0 ˆ π (= p) = X n, where X = # Successes CRITICAL VALUE (2-sided) 1 n 30: t n 1, α /2 or z α /2 n < 30: t n 1, α /2 only MARGIN OF ERROR = product of these two factors: Any n: STANDARD ERROR (estimate) 2 s / n n 30 : z α /2 ~ N(0, 1) n 30 also, nπ 15 and n(1 π) 15: For Confidence Interval: ˆ π (1 ˆ π) n n < 30: Use X ~ Bin(n,π). For Acceptance Region, p-value: (not explicitly covered) π (1 π ) n 0 0 Two Independent Samples Two Paired Samples 3 Null Hypothesis H 0 : θ 1 θ 2 = 0 Point Estimate ˆ θ ˆ θ 1 2 Means 2 H 0 : µ 1 µ 2 = 0 x1 x2 Proportions H 0 : π 1 π 2 = 0 ˆ π ˆ 1 π2 CRITICAL VALUE (2-sided) 1 n 1, n 2 30: t + α or z α /2 n1 n2 2, / 2 n 1, n 2 < 30: Is σ 1 2 = σ 2 2? Informal: 1/4 < s 1 2 /s 2 2 < 4? Yes t n1+ n2 2, α / 2 No Satterwaithe s Test n 1, n 2 30 : z α /2 (or use Chi-squared Test) n 1, n 2 < 30: Fisher s Exact Test (not explicitly covered) STANDARD ERROR (estimate) 2 n 1, n 2 30: s 2 1 / n 1 + s 2 2 / n 2 n 1, n 2 < 30: 2 s pooled 1 / n 1 + 1 / n 2 where s pooled 2 = (n 1 1) s 1 2 + (n 2 1) s 2 2 n 1 + n 2 2 n 1, n 2 30 also, (see criteria above): For Confidence Interval: ˆ π (1 ˆ π ) n + ˆ π (1 ˆ π ) n 1 1 1 2 2 2 For Acceptance Region, p-value: ˆ π where pooled (1 ˆ π ) 1 n + 1 n pooled 1 2 ˆ π pooled = (X 1 + X 2 ) / (n 1 + n 2 ) k samples (k 2) Null Hypothesis H 0 : θ 1 = θ 2 = = θ k Independent Dependent (not covered) Means H 0 : µ 1 = µ 2 = = µ k F-test (ANOVA) Repeated Measures, Blocks Proportions H 0 : π 1 = π 2 = = π k Chi-squared Test Other techniques 1 For 1-sided hypothesis tests, replace α /2 by α. 2 For Mean(s): If normality is established, use the true standard error if known either σ / n or σ 2 2 / n σ / n 1 1 2 2 + with the Z-distribution. If normality is not established, then use a transformation, or a nonparametric Wilcoxon Test on the median(s). 3 For Paired Means: Apply the appropriate one sample test to the pairwise differences D = X Y. For Paired Proportions: Apply McNemar s Test, a matched version of the 2 2 Chi-squared Test.