# Solution sampling error 19 you believe that the true

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Solution: Sampling error. 19. You believe that the true mean weight of statistics books is 7.0 lbs. A study of 8 randomly selected statistics books weights (shown below) was conducted to test this claim. Use a signif- icance level of 0.025 and assume that the weights are normally distributed. 6.4, 7.1, 6.1, 6.9, 5.6, 7, 7.7, 8 (a) (1 point) What type of hypothesis test will you use? Solution: Use a one sample mean test with σ unknown. (b) (2 points) What are the test’s requirements? Solution: (1) Simple random samples, (2) CLT applies. (c) (1 point) Are the requirements satisfied? State how they are satisfied. Solution: Yes. Simple random samples used, and population was normally dis- tributed. (d) (2 points) What are the hypothesis? Solution: H 0 : μ = 6 . 0, H a : μ 6 = 6 . 0 (e) (1 point) What α will you use? Solution: α = 0 . 025 (f) (2 points) Conduct the hypothesis test. What is the p -value? Solution: > x [ 1 ] 6.4 7.1 6.1 6.9 5.6 7.0 7.7 8.0 > res = t . t e s t (x , mu = 7 , a l t e r n a t i v e = ”two . sided ”) > res One Sample t - t e s t data : x t = - 0.5315 , df = 7 , p - value = 0.6115 a l t e r n a t i v e hypothesis : true mean i s not equal to 7 95 percent confidence i n t e r v a l : Instructor: Anthony Tanbakuchi Points earned: / 10 points
MAT 167: Statistics, Test II SOLUTIONS p. 12 of 13 6.182678 7.517322 sample estimates : mean of x 6.85 The p -value is 0.612. Or, if you prefer to do it manually, find the p -value with the test statistic t = ¯ x - μ 0 s/ n , df = n - 1 > x [ 1 ] 6.4 7.1 6.1 6.9 5.6 7.0 7.7 8.0 > x . bar = mean(x) > x . bar [ 1 ] 6.85 > s = sd (x) > s [ 1 ] 0.7982123 > mu0 = 7 > n = length (x) > n [ 1 ] 8 > t = (x . bar - mu0)/( s / sqrt (n )) > s i g n i f ( t , 3) [ 1 ] - 0.532 > p . val = 2 * pt ( t , df = n - 1) > s i g n i f (p . val , 3) [ 1 ] 0.612 The p -value is 0.612. (g) (1 point) What is your formal decision? Solution: Since p -val α , fail to reject H 0 . (h) (2 points) State your final conclusion in words. Solution: The sample data does not contradict the claim that statistics text book weight is equal to 7.0 lbs. (i) (1 point) If we reject H 0 , what is the actual probability of a Type I error for this study data? Solution: The p -value = 0.612. Instructor: Anthony Tanbakuchi Points earned: / 4 points
MAT 167: Statistics, Test II SOLUTIONS p. 13 of 13 ************ End of exam. Reference sheets follow. ************ Instructor: Anthony Tanbakuchi Points earned: / ?? points
Statistics Quick Reference Card & R Commands by Anthony Tanbakuchi. Version 1.8.2 ANTHONY @ TANBAKUCHI · COM Get R at: R commands: bold typewriter text 1 Misc R To make a vector / store data: x=c(x1, x2, ...) Help: general RSiteSearch("Search Phrase") Help: function ?functionName Get column of data from table: tableName\$columnName List all variables: ls() Delete all variables: rm(list=ls()) x = sqrt(x) (1) x n = x n (2) n = length(x) (3) T = table(x) (4) 2 Descriptive Statistics 2.1 N UMERICAL Let x=c(x1, x2, x3, ...) total = n i = 1 x i = sum(x) (5) min = min(x) (6) max = max(x) (7) six number summary : summary(x) (8) μ = x i N = mean(x) (9) ¯ x = x i n = mean(x) (10) ˜ x = P 50 = median(x) (11) σ = ( x i - μ ) 2 N (12) s = ( x i - ¯ x ) 2 n - 1 = sd(x) (13) CV = σ μ = s ¯ x (14) 2.2 R ELATIVE STANDING z = x - μ σ = x - ¯ x s (15) Percentiles: P k = x i , (sorted x ) k = i - 0 . 5 n · 100% (16) To find x i given P k , i is: 1. L = ( k / 100% ) n 2. if L is an integer: i = L + 0 . 5; otherwise i=L and round up.