Question

help with stats!
A variable X has a normal distribution with mean 100. It is known that about 47.5% of
the values of X fall between 85 and 100. What is the approximate value of the standard
deviation
?

Answer 1

see attachment:

Answer 2

@kropot72 @agent0smith

Answer 3

A recycling plant compresses aluminum cans into bales. The weights of the bales are
known to follow a normal distribution with standard deviation eight pounds. In a ran-
dom sample of 64 bales, what is the probability that the sample mean diers from the
population mean by no more than one pound?
see attachment

Answer 4

A variable X has a normal distribution with mean 100\[
\Pr(X\leq 100) = 0.5
\]
It is known that about 47.5% of the values of X fall between 85 and 100.\[
\Pr(X> 85, X \leq 100) = 0.457
\]

Answer 5

When we simplify we get: \[
1 - \bigg (\Pr(X\leq 85) + \Pr(X>100) \bigg) = 0.475 \\
1 - \Pr(X \leq 85) - 0.5 = 0.475 \\
\Pr(X\leq 85) = 0.5 - 0.475 = 0.025
\]

Answer 6

\[
\large \Phi \left(\frac{85-100}{\sigma}\right) = 0.025\\

\large \Phi \left(\frac{-15}{\sigma}\right) = 0.025
\]Or: \[
\sigma = \frac{-15}{\Phi^{-1}(0.025)}
\]

Answer 7

Recycling plant question:
According to the Central Limit Theorem the distribution of the sample means is approximately normal, with standard deviation
\[\frac{\sigma}{\sqrt{n}}=\frac{8}{\sqrt{64}}=1\ pound\]
In this case the required probability is found by first finding the cumulative probability for z = 1 using a standard normal distribution table. Then subtract from that value the cumulative probability for z = -1.

Answer 8

thanks@kropot72 and @wio

Answer 9

You're welcome :)