Help with confidence interval problem (screenshot attached)

- anonymous

Help with confidence interval problem (screenshot attached)

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- anonymous

@ganeshie8 @tkhunny @jim_thompson5910 X)

- anonymous

Would I just simply use\[\huge \text{Z}_\alpha=\frac{\bar{x}-\mu}{\sigma}\]and solve for \(\mu\)?

- ganeshie8

whats the z* value for 99% confidence level ?

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## More answers

- anonymous

-2.33

- anonymous

Oh, so it'd just simply have to be greater than 2.33?

- ganeshie8

|dw:1449638671166:dw|

- ganeshie8

that table says \(Z^{\star} = 2.576\) ?

- anonymous

Ah. On the chart I have it's between -2.32 and -2.33

- ganeshie8

Make sure you're not looking at single tail sheet

- anonymous

Wouldn't it be single tailed though? Because it only mentioned that it's greater than certain value.

- ganeshie8

Oh, thats right !

- anonymous

I just used that \(\alpha=0.01\)

- anonymous

Ok haha I thought I was going crazy. Where can I go from here with this information?

- ganeshie8

find the margin of error by multiplying "standard error" and "Z*"

- anonymous

Okay, I got -0.8621

- ganeshie8

margin of error = 0.37*2.33 = 0.86

- anonymous

Yay X)

- ganeshie8

margin of error is always positive

- anonymous

Ah, okay.

- ganeshie8

|dw:1449639292614:dw|

- ganeshie8

subtract margin of error from the mean to get the lower bound of true mean

- anonymous

Alright, I have 134.5279.

- ganeshie8

looks good to me!

- anonymous

Just curious, why is it the lower bound?

- anonymous

Having a hard time in general with this material with picturing what's going on

- ganeshie8

That is due to central limit theorem

- ganeshie8

what does CLT say about the "sampling distribution" ?

- ganeshie8

let me ask this question
what is a sampling distribution ?

- anonymous

The probability of a statistic based on a sample from a population, yeah?

- anonymous

We can use it to make inferences about the population directly from the sample

- ganeshie8

"sampling distribution"
as the name says, it is a distribution

- ganeshie8

distribution of what ?

- anonymous

I'm not sure :S

- ganeshie8

suppose you want to know the average height of ppl in your country

- ganeshie8

it is impossible to collect data of each and everyone in your country

- anonymous

True, that makes sense. So we try to take multiple samples right?

- ganeshie8

so you just select "n" people randombly and take their average height

- ganeshie8

Now is the real question

- ganeshie8

How confident are you about your sample ?

- ganeshie8

How do you know that the "average height in your sample" is same as the "average height of all the people in your country" ?

- anonymous

Definitely not 100% because you don't know ALL of the heights

- ganeshie8

right, that is where we use central limit theorem

- anonymous

Which means that we can approximate the sample as normal, right?

- ganeshie8

what do you mean the sampel is normal ?

- anonymous

I mean, it will follow a normal distribution

- ganeshie8

CLT talks about this :
Suppose you have one sample of size \(n\) with average height = \(\overline{x}\) and standard deviation of \(\sigma\).
Now take multiple samples of the same size \(n\) and find the average height in each of those samples.

- ganeshie8

You have a collection of average height from samples.
CLT says that this collection of average height from samples is a normal distribution

- anonymous

Ahhh, I guess I was kinda close X)

- anonymous

So how does that lower bound come into play here?

- ganeshie8

As you can see, there is a very less chance for the average height of your single sampel to be same as the average height of the total population

- ganeshie8

Would you agree that there are more chances for the population mean to lie somewhere around your sample mean ?

- ganeshie8

Using confidence intervals, we give some range of values around the sample mean in which the population mean lies with certain confidence

- ganeshie8

Let me ask you another question
If we want to be more confident about the population mean to lie in our interval, do we need a wider interval or a shorter interval ?

- anonymous

We'd want a wider interval, correct?

- ganeshie8

Yes, CLT also gives you the \(\overline{x}\) and \(\sigma\) of the sampling distribution :
1) mean of sampling distribution is same as the mean of a sample
2) standard deviation of sampling distribution depends on size of sample : \(\dfrac{\sigma}{\sqrt{n}}\)

- ganeshie8

Once you have the "sampling" distribution, you can find the confidence intervals using zscores

- anonymous

Ohh, I think I'm getting the idea now

- ganeshie8

Under normal curve, you know that 99% of the area lies after the zscore of -2.33
therefore, to say that the true population mean is greater than some X with 99% confidence, you need to take the observation(X) corresponding to the zscore of -2.33

- anonymous

Yeah that makes sense now!

- ganeshie8

single tailed 99% confidence interval looks as below :
\[(\overline{x}-2.33*\dfrac{\sigma}{\sqrt{n}},~~\infty)\]

- ganeshie8

double tailed 99% confidence interval looks as below :
\[(\overline{x}-2.576*\dfrac{\sigma}{\sqrt{n}},~~\overline{x}+2.576*\dfrac{\sigma}{\sqrt{n}})\]

- anonymous

Ahh, it's starting to look more familiar now :P

- ganeshie8

Either case, the interval captures 99% of the area under the normal curve

- anonymous

That's why it's only \(\alpha\) for a single tail and \(\alpha/2\) for a double tail.

- anonymous

Hehe, that makes more sense now. Thank you!! X)

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