Suppose X is a normally distributed random variable with unknown mean μ and unknown standard deviation σ. Assume a random sample of X values, of size n = 25, is available. Let s^2 denote the sample variance. If σ^2=40, what is P(S^2≤23.08)?

Question

Suppose X is a normally distributed random variable with unknown mean μ and unknown standard deviation σ. Assume a random sample of X values, of size n = 25, is available. Let s^2 denote the sample variance.  If σ^2=40, what is P(S^2≤23.08)?

amistre64 · Answer

$$z=\frac{x-\mu}{\sigma}\sqrt{n}$$ and i believe that if sigma is unknown then $$\sigma=\sqrt{.5*.5}=.5$$but my memory on that is fuzzy

amistre64 · Answer

well,$$\sigma=\sqrt{\frac{p\cdot q}{n}}$$

anonymous · Answer

The graph approachs a normal distribution for the variance, so all I really need is a way to calculate the variance of the original function

anonymous · Answer

Like the variance becomes the mean of a normal distribution, so do you know anyway to figure out the variance of the first, cause everything after that is just prob from a bell curve, which is easy

amistre64 · Answer

im doing a little reviewing at the moment to refresh some stuff :)

amistre64 · Answer

so we are trynig to estimate the population mean and standard deviation with the sample statistics .... or am i misreading that?

anonymous · Answer

I think what the question wants is an estimation of the population variance. From that, it wants the probability that the variance will be greater than 23.08

amistre64 · Answer

my elementary stats textbook says:

Procedure for constructing a confidence interval for $\sigma$ or $\sigma^2$:

1: verify that requirements are satisfied

2:  using n-1 df, find $\Large \chi_R^2$ and $\Large \chi_L^2$ that correspond to a desired confidence interval

3: evaluate the upper and lower limits$$\frac{(n-1)s^2}{\chi_R^2}<\sigma^2<\frac{(n-1)s^2}{\chi_L^2}$$

4:  if confidence interval estimate for $\sigma$, then take the sqrt of the limits and change the sigma^2 to just sigma

5:  round the results as appropriate

amistre64 · Answer

we seem to have not covered this when i took the class.  so im not sure how much of that is pertinent

anonymous · Answer

very pertinent. I just found this in my stats book too. Thanks!! and lucky you for not going over this mathematical devilry

amistre64 · Answer

im contemplating taking a higher stats class in the fall :)

anonymous · Answer

Im a BSCE major, so this is the last I will probably need to take when it comes to pure stats. Probably nothing but binomial and normal distributions from here on out. Good luck to you

amistre64 · Answer

same to you ;)