H 0 </msub> : &#x03BC;<!-- μ --> = 0 against H A </

Abram Boyd 2022-06-26 Answered
H 0 : μ = 0 against H A : μ > 0.
Had the following question on my exam today and I'm just wondering if I did this correctly.
From a normally distributed population with mean μ and variance σ 2 a sample has been drawn:
X = ( 0.05 , 4.35 , 0.48 , 0.63 , 1.17 , 2.01 ) ..
a) Test H 0 : μ = 0 against H A : μ > 0 on a 5% significance level under the assumption that σ2 is unknown.
c) Do the same test, given that σ = 1.5..
Solution: I know that H 0 is to be rejected if T c ,, where T is the teststatistic and F t n 1 = 1 α ,, where α is the significance level.
So lets first compute them for a), where we need to use a t-distribution since σ is unknown, so we have that c = F t 5 1 ( 0.95 ) 2.015.. Also, the estimator for σ 2 is
s 2 = 1 n 1 ( k = 1 n X k 2 1 n ( k = 1 n X k ) 2 ) = 1 5 ( 24.961 1 6 ( 41.861 ) ) = 3.405 ,
so, s = 3.405 = 1.845.. We also have that X ¯ = 6.47 / 6 = 1.078..
The teststatistic is
T = X ¯ μ 0 s / n = 1.078 0 1.845 / 6 = 1.078 0.753 = 1.43 c ,
which means that we can not reject H0.
For b) we do an identical approach but wtith s = σ = 1.5 and we use a z-test instead. Here we have that c = Φ 1 ( 0.95 ) = 1.644. The teststatistic in this case is
Z = X ¯ μ 0 σ / n = 1.078 1.5 / 6 = 1.76 c ,
thus we reject H 0 ..
Is this correct?
What my professor does in the solutions is he just makes a 95% confidenceinterval and just arrives to the same conclusions as I've done. Is his method correct without using the teststatistics?
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Answers (1)

Colin Moran
Answered 2022-06-27 Author has 21 answers
Yes, your solution is indeed correct. You are doing exactly what is required to, while your professor is solving a bit more general problem, namely constructing the 95% confidence interval and then checks whether given values are inside this interval or no.
This, I think, is just a manner of habit. I, personally, am used to the one presented in your post.

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