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Probability Question - Nonstandard Normal Distributions

  • #1

Homework Statement

The weight of eggs produced by a certain type of hen varies according to a distribution that is approximately normal with mean 6.5 grams and standard deviation 2 grams.

What is the probability that the average of a random sample of the weights of 25 eggs will be less than 6 grams



Homework Equations


P(X<6)=P((6-6.5)/σ)



The Attempt at a Solution

- The part I can't figure out is how to arrive at sigma. This is a problem from a practice exam, so I already know that sigma is 0.40. If I'm understanding correctly, then that would make V(X) = 4/25. I just can't figure out how to arrive at these conclusion from the data that is given. I'm pretty stumped.
 
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Answers and Replies

  • #2
statdad
Homework Helper
1,495
35
your problem asks about the probability the MEAN of a sample will be a certain size. what do you know about distributions of sample means?
 
  • #3
your problem asks about the probability the MEAN of a sample will be a certain size. what do you know about distributions of sample means?
Not much.

I think I figured out why sigma is what it is though. Since the SD is 2 grams, it follows that V(X) = 4. Since I'm trying to find out what the average of X is, I divide 4 by 25. that is how I get the 4/25. From there sigma is easy. I think that is kind of close anyway.
 
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  • #4
statdad
Homework Helper
1,495
35
You've essentially got it. If you take a sample of size [tex] n [/tex] from a normally distributed population, then [tex] \overline X [/tex] has a normal distribution. For the
distribution of [tex] \overline X [/tex],

[tex]
\mu = \text{ original population mean}
[/tex]

and

[tex]
\sigma = \frac{\text{Original standard deviation}}{\sqrt n}
[/tex]

As long as the original population itself has a normal distribution, this is true
for any sample size.
 

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