# point estimates and endpoints of confidence interval

• Jun 10th 2009, 06:45 PM
chrissy72
point estimates and endpoints of confidence interval
distribution of x is N(mu,4) n=10:
55.95
56.54
57.58
55.13
57.48
56.06
59.93
58.30
52.57
58.46

point estimate?
endpoints for 95% confidence interval for mu?
probability when one is selected at random that it is less than 52?

i'm lost and would loooove some help..... thankyou
• Jun 10th 2009, 07:15 PM
Random Variable
$\bar{x} \pm z_{0.025} \ \Big( \frac {\sigma}{\sqrt{n}}\Big)$

where $\bar{x}$ is the sample mean (which you have to calculate), $\sigma$ is the standard deviation, and $n$ is the sample size
• Jun 10th 2009, 07:30 PM
chrissy72
i got a mean of 56.83, a variance of 3.814 and a standard deviation of 1.95295 but don't know what to do with them. where do you get the z0.025 from? thankyou for helping me!
• Jun 10th 2009, 08:07 PM
Random Variable
$\sigma$ is the standard deviation of the population, not the sample. So in this problem $\sigma = \sqrt{4} = 2$.

You get $z_{0.025}$ from a standard normal distribution table. There is one in just about every statistics textbook. It's the value of z such that P(Z>z) = 0.025. The value is 1.96.

Have you gone over this stuff in class?
• Jun 10th 2009, 09:33 PM
matheagle
Chrissy, plug in 1.96 into Free One Tailed Area Under the Standard Normal Curve Calculator
and see that the area to the right of 1.96 is .0249978.
• Jun 13th 2009, 02:15 PM
chrissy72
All I have is a textbook. No class. Would the average of the 10 values be considered a point estimate of $\mu$? Also, if the area to the right of 1.96 is 0.249978, would the endpoints of the 95% confidence interval be 1.9849978 on the right and 1.96 or 1.9350022 on the left?

Thanks. I really don't know what I am doing and my text is no help. I need an actual example to learn from but it is very hard to find. I don't have enough calculus OR statistic background to solve these problems without a better book OR an actual teacher. Any tutors out there? =)
• Jun 13th 2009, 02:45 PM
Random Variable
From the data you calculated the sample mean ( $\bar{x}$) to be 56.83.

So a 95% confidence inteveral for the population mean ( $\mu$) is

$\Big[56.83 - 1.96* \frac{2}{\sqrt{10}}, \ 56.83 + 1.96*\frac{2}{\sqrt{10}}\Big]$

[55.59, 58.07]

Now either this interval contains $\mu$ or it doesn't. But if you constructed many such intervals by taking different samples of the same size, 95% of them would contain $\mu$.
• Jun 13th 2009, 02:59 PM
chrissy72
you are wonderful! Thankyou. So would that in fact be considered a point estimate (56.83)? I'm not clear on exactly what that is. My book only mentions that it is a point rather than an interval, but I don't know which point. Also, would the probability of randomly choosing a variable less than 52 be simply 0? Thanks once again! you have made this so much easier to understand.
• Jun 13th 2009, 03:32 PM
Random Variable
EDIT:

If $X$ is $N(\mu, \sigma^{2})$, then $\frac {X-\mu}{\sigma}$ is $N(0,1)$ (or just $Z$) .

So $P(X<52) = P\Big(\frac{X -\mu}{\sigma} < \frac{52 - \mu}{2} \Big) = P(Z

But that can't be answered unless you know $\mu$
• Jun 13th 2009, 03:40 PM
chrissy72
I feel so silly, I think I can just do the old (52-56.83)/2=-2.415, but 2.415 is not on my table so 2.42 becomes 0.0078. I'm not sure how to write that out. Is it $\Phi2.42=0.0078$?
• Jun 13th 2009, 10:19 PM
matheagle
you can plug 2.415 into....
Free One Tailed Area Under the Standard Normal Curve Calculator
and get .0078676
• Jun 13th 2009, 10:37 PM
Random Variable
But I'm fairly certain (which probably means I'm wrong) that $\frac {X-\bar{x}}{\sigma}$ is not N(0,1).
• Jun 13th 2009, 10:55 PM
matheagle
It has mean zero and it is a normal.
I doubt the variance is 1.
Is this X part of the sample, where $\bar X$ was derived from or a different observaton?

If it's part of the sample, then ....

$V\biggl({X_1-\bar X\over \sigma}\biggr)={V\biggl(X_1-\bar X\biggr) \over \sigma^2}$

$={V\biggl(X_1\bigl(1-{1\over n}\bigr)-{\sum_{k=2}^n X_k\over n}\biggr) \over \sigma^2}$

$={\bigl(1-{1\over n}\bigr)^2\sigma^2 +{n-1\over n^2}\sigma^2 \over \sigma^2}$

$=1-{1\over n}$

Otherwise ...

$V\biggl({X_{n+1}-\bar X\over \sigma}\biggr)={\sigma^2+{\sigma^2\over n} \over \sigma^2}=1+{1\over n}$

which is exactly what you get in a prediction interval, inside the square root of course.