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Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution). Using a subscript that matches the random variable, suppose:

  • μ X = the mean of X
  • σ X = the standard deviation of X
If you draw random samples of size n , then as n increases, the random variable X which consists of sample means, tends to be normally distributed and

X ~ N ( μ X , σ X n )

The Central Limit Theorem for Sample Means says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by n , the sample size. n is the number of values that are averaged together not the number of times the experiment is done.

To put it more formally, if you draw random samples of size n ,the distribution of the random variable X , which consists of sample means, is called the sampling distribution of the mean . The sampling distribution of the mean approaches a normal distribution as n , the sample size, increases.

The random variable X has a different z-score associated with it than the random variable X . x is the value of X in one sample.

z = x - μ X ( σ X n )

μ X is both the average of X and of X .

σ X = σ X n = standard deviation of X and is called the standard error of the mean.

An unknown distribution has a mean of 90 and a standard deviation of 15. Samples of size n = 25 are drawn randomly from the population.

Find the probability that the sample mean is between 85 and 92.

Let X = one value from the original unknown population. The probability question asks you to find a probability for the sample mean .

Let X = the mean of a sample of size 25. Since μ X = 90 , σ X = 15 , and n = 25 ;

then X ~ N ( 90 , 15 25 )

Find P ( 85 x 92 ) Draw a graph.

P ( 85 x 92 ) = 0.6997

The probability that the sample mean is between 85 and 92 is 0.6997.

Normal distribution curve from -∞ to ∞ and an x-axis with the values of 85, 90, and 92. The x-axis is equal to the mean of a sample size of 25. A vertical upward line extends from points 85 and 92 to the curve. The probability area is between 85 and 92.

TI-83 or 84: normalcdf (lower value, upper value, mean, standard error of the mean)

The parameter list is abbreviated (lower value, upper value, μ , σ n )

normalcdf (85,92,90, 15 25 ) = 0.6997

Find the value that is 2 standard deviations above the expected value (it is 90) of the sample mean.

To find the value that is 2 standard deviations above the expected value 90, use the formula

value = μ X + (#ofSTDEVs) ( σ X n )

value = 90 + 2 15 25 = 96

So, the value that is 2 standard deviations above the expected value is 96.

The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours . A sample of size n = 50 is drawn randomly from the population.

Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

Let X = the time, in hours, it takes to play one soccer match.

The probability question asks you to find a probability for the sample mean time, in hours , it takes to play one soccer match.

Let X = the mean time, in hours, it takes to play one soccer match.

If μ X = _________, σ X = __________, and n = ___________, then X ~ N (______, ______) by the Central Limit Theorem for Means.

μ X = 2 , σ X = 0.5 , n = 50 , and X ~ N ( 2 , 0.5 50 )

Find P ( 1.8 x 2.3 ) . Draw a graph.

P ( 1.8 x 2.3 ) = 0.9977

normalcdf (1.8,2.3,2, .5 50 ) = 0.9977

The probability that the mean time is between 1.8 hours and 2.3 hours is ______.

Questions & Answers

what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
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lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
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12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
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Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
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Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
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I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
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is it a question of log
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Commplementary angles
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Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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What is the expressiin for seven less than four times the number of nickels
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why surface tension is zero at critical temperature
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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