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)

For the gamma distribution Mean (Mu) = aB and variance (sigma squared) = aB^2
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.

Remember for the normal distribution, the standard deviation of the sample mean is the standard deviation of the population divided by the square root of n.

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)

Suppose you have two triangles, ABC and DEF and want to determine if they are congruent. The Side Angle Side theorem (known as SAS) says that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. So if side AB is length 5 and AC is length 6 with angle A of 50 degrees, and side DE is also 5 and DF is 6 with angle D of 50 degrees, the triangles are congruent by the SAS theorem.

What you need to do for these is figure out how many standard deviations from the mean the value is. the farther away from the mean, the less likely it is for that value to occur. For a graduate to have a salary of 80,000, that is 10,000 above the mean, which is 2 standard deviations above the mean, since (80,000 - 70,000)/5,000 = 2. It's value minus mean, all divided by standard deviation. If you check on a z chart for normal probability distribution, you will see that .9772 will be less than a salary of 80,000 and only 2.228 percent have more than 80,000. Suppose we test that the mean salaries are indeed 70,000, the null hypothesis, against a hypothesis that the mean salaries are not 70,000 at alpha = .05. The rejection region for this would be when z > 1.96 or z <-1.96. So any value of z in between the -1.96 and 1.96 would be considered a "reasonable" value for z, which would be a reasonable outcome. Since 80,000 gives a z-value of 2, it falls just outside of 1.96 which is not a reasonable outcome.
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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