Problem 1: The average ACT composite score for all applicants to Upper Midwest U
ID: 3053854 • Letter: P
Question
Problem 1: The average ACT composite score for all applicants to Upper Midwest University
(UMU) in 2008 was 23.4. An administrator at UMU wants to know whether the mean ACT
composite score of applicants for the 2014-2015 school year is higher. (don't use minitab and show your work)
a) Using a 5% significance level, find the sample size needed to achieve 80% power for
detecting a 1 point increase in the mean assuming that the population standard deviation is
approximately 2.8 points. Explicitly state the values of Ho, Ha, Lowercase Alpha, Lowercase Delta, and Lowercase sigma.
b) Suppose instead of detecting an increase in the mean ACT score, the administrator wants to be
able to detect any change of more than 1.5 points. Using a 5% significance level, find the
sample size need to achieve 90% power for detecting a 1.5 point change (an increase or
decrease) in the mean assuming that the population standard deviation is approximately 2.8
points. Explicitly state the values of Ho, Ha, Lowercase Alpha, Lowercase Delta, and Lowercase sigma.
Explanation / Answer
a)
H0 : µ= 23.4 Ha : µ> 23.4
?=0.05
? =2.8
?=1-0.8=0.2
µ0 =23.4 , µ =23.4 +1=24.4
?= diff(24.4 -23.4)/ ? = 1/2.8 =0.36
Sample size can be found from Operating Characteristics Curve for ? 0.36 and ? 0.2 comes close to 50 (use one tail test for ?=0.05
b)
H0 : µ= 23.4 Ha : µ<>23.4
?=0.05
? =2.8
?=1-0.9=0.1
µ0 =23.4 ,
?= diff/ ? = 1.5/2.8 =0.54
Sample size can be found from Operating Characteristics Curve for ? 0.54 and ? 0.1 comes close to 40 (use two tail test for ?=0.05)