Remember when determining concavity, you use the same method as finding maximum and minimum using the first derivative test, except now you used the second derivative.

For example, suppose the function if f(x) = x^3 + 2x^2 - 5x

f'(x) = 3x^2 +4x - 5

f''(x) = 6x + 4

6x + 4 = 0

6x = -4

x = -2/3

Now test a point on both sides of -2/3, we'll use -1 and 0

f''(-2) = 6(-2) + 4 = -8

f''(0) = 6(0) + 4 = 4

that means we have concave down from (-infinity, -2/3) and concave up from (-2/3, infinity)

## Thursday, June 25, 2015

## Sunday, June 21, 2015

For the exponential distribution Mean (Mu) = B and variance (sigma squared) = B^2

Examine these and you can see the difference.. The means are basically the same except gamma has a and exponential does not. Variance for gamma also includes a, while exponential does not.

If a = 0 and B = 0 then Mu for exponential is 0 and sigma^2 = 0.

If a = 0 and B = B then Mu for exponential is B and sigma^2 = B^2

If a = 1 and B = B then Mu for exponential is B and sigma^2 = B^2, but note that the Gamma Distribution will also have Mu = B and sigma^2 = B^2.

That's the key, the exponential distribution is EQUAL to the Gamma distribution when a = 1 and B = B.

## Thursday, June 18, 2015

For example, if the population standard deviation is 10 and we have a sample of size 50 from the population, the standard deviation of the sample mean is 10/sqrt(50)

## Wednesday, June 10, 2015

## Wednesday, June 3, 2015

The way to figure out the highest reasonable outcome, take the 1.96(which is 1.96 standard deviations above the mean) and multiply by the standard deviation and add to the mean.

70,000 + 1.96(10,000) = highest maximum reasonable value.

For part c, we need to take (80,000 - 70,000)/(5,000/sqrt(100)) to get the z value. The formula is (value - mean)/(standard deviation/square root of n)

If this falls between -1.96 and 1.96, it is a reasonable value. It does not, as you will see, so it's not reasonable.

d) For this, we know the z-value that will give the maximum reasonable mean salary is 1.96

so use the formula

1.96 = (x - 70,000)/(5,000/sqrt(100))

The x is the value we are solving for, which is the mean salary for a random sample.

If you do this correctly, you will get 70,980

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