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+1−x―?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?

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Alejandra Hanna · Accepted Answer

Step 1xn+1&amp;nbsp;itself is normally distributed with mean&amp;nbsp;μ&amp;nbsp;and variance&amp;nbsp;σ&amp;nbsp;(two unknown numbers). Therefore, assuming it is also independent of the others,&amp;nbsp;xn+1−x―&amp;nbsp;is normally distributed. Recall a few useful rules:Expectation is linearVar[X+Y]=Var[X]+Var[Y]&amp;nbsp;for independent X,YVar[cX]=c2Var[X]&amp;nbsp;for a real number cUsing these rules, you can calculate&amp;nbsp;E[xn+1−x―]=0&amp;nbsp;and&amp;nbsp;Var[xn+1-x¯]=σ2+nn2σ2=n+1nσ2&amp;nbsp;This answers part 1.If you knew&amp;nbsp;σ, then you would use the above work to conclude that&amp;nbsp;xn+1=x―+y, where y is a normal random variable independent of&amp;nbsp;x1,⋯,xn&amp;nbsp;with mean zero and variance&amp;nbsp;n+1nσ2. (Note that not knowing the mean enlarges the uncertainty a little bit: if we knew the mean then we would write&amp;nbsp;xn+1=μ+y, where y is normal with mean zero and variance&amp;nbsp;σ2.)Again, if you knew&amp;nbsp;σ, the above would allow you to construct a confidence interval for part 2. However,&amp;nbsp;σ&amp;nbsp;is not known, so it needs to be estimated. Doing this estimation effectively enlarges the uncertainty somewhat. We correct this using the Student&#039;s t distribution.

paganizaxpo3 · Accepted Answer

Step 1Here is the first couple of steps I would make. Assume the population&#039;s mean and variance are μ and σThen xk∼N(μ,σ2) for any k∈N. As you noted, it is not hard to see that x― is normally distributed with mean μ. Find the standard deviation.Then subtract xn+1−x― to get a 0-mean random variable. What is the variance, what is it&#039;s distribution?Hope you can take it from here.

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

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