Let \(\displaystyle{x}_{{{1}}},\cdots,{x}_{{{n}}}\) be a sample from a

Leroy Davidson

Leroy Davidson

Answered question

2022-03-24

Let x1,,xn be a sample from a normal population having unknown mean and variance. Let x be the average of the first n of them.
What is the distribution of xn+1x?
If x=4, give an interval that, with 90% confidence, will contain the value of xn+1.
The distribution of x would be normal with mean equal to the population mean and standard deviation equal to the population standard deviation divided by n. But how do we find the distribution of xn+1?

Answer & Explanation

Alejandra Hanna

Alejandra Hanna

Beginner2022-03-25Added 10 answers

Step 1
xn+1 itself is normally distributed with mean μ and variance σ (two unknown numbers). Therefore, assuming it is also independent of the others, xn+1x is normally distributed. Recall a few useful rules:
Expectation is linear
Var[X+Y]=Var[X]+Var[Y] for independent X,Y
Var[cX]=c2Var[X] for a real number c
Using these rules, you can calculate E[xn+1x]=0 and Var[xn+1-x¯]=σ2+nn2σ2=n+1nσ2 This answers part 1.
If you knew σ, then you would use the above work to conclude that xn+1=x+y, where y is a normal random variable independent of x1,,xn with mean zero and variance n+1nσ2. (Note that not knowing the mean enlarges the uncertainty a little bit: if we knew the mean then we would write xn+1=μ+y, where y is normal with mean zero and variance σ2.)
Again, if you knew σ, the above would allow you to construct a confidence interval for part 2. However, σ is not known, so it needs to be estimated. Doing this estimation effectively enlarges the uncertainty somewhat. We correct this using the Student's t distribution.

paganizaxpo3

paganizaxpo3

Beginner2022-03-26Added 8 answers

Step 1
Here is the first couple of steps I would make. Assume the population's mean and variance are μ and σ
Then xkN(μ,σ2) for any kN. As you noted, it is not hard to see that x is normally distributed with mean μ. Find the standard deviation.
Then subtract xn+1x to get a 0-mean random variable. What is the variance, what is it's distribution?
Hope you can take it from here.

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