Let X 1 , X 2 , ... , X 11 be a sample of observations from a normal population
ID: 3250194 • Letter: L
Question
Let
X1,X2, ... ,X11
be a sample of observations from a normal population with unknown mean and unknown standard deviation . Suppose the sample mean is x and the sample standard deviation is s. We wish to test the null hypothesis H0:=9 against the alternative Ha:>9. Our test statistic will be T =
. Our sample yields x = 9.75 and s = 1.51. Answer the following using R code.
a)If xbar = 9.75 and s = 1.51 then what is the value of T? 1
b) What type of distribution does T have?
Binomial t Normal exponential
c) If we test at the 4% level then what is the critical value? 3
d) What is the p-value based upon our sample? 4
e) Do we reject the null hypothesis at the 4% level ?(Y/N)?
N Y
f) If we repeat a 4% level test 300 times then about how many times do we expect to commit a type(I) error? 6
Explanation / Answer
a) T has value 1.64733
b) T has t distribution with 10 degrees of freedom
c) critical value is 19.02074
d) p value is 0.5611
e) T is below critical value so we do not reject null hypothesis at 4% level
f) A 4% level test has a type 1 error probability of 4%.So in 300 times we expect to have a type 1 error 12 times.
R-Code
xbar=9.75
mu=9
s=1.51
n=11
T=(xbar-mu)/(s/sqrt(n))
df=10
level=0.04
critical=qchisq(1-0.04,10)
pvalue=pchisq(T,2)
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