An “app” thought!

VC question: How much is this worth as a killer app?

GAUSS, Carl Friedrich

f(X) = Where  = and e =  2  e -(X -  ) / 2  22

Normal Distribution Unimodal Symmetrical 34.13% of area under curve is between µ and +1  34.13% of area under curve is between µ and -1  68.26% of area under curve is within 1  of µ % of area under curve is within 2  of µ.

Some Problems If z = 1, what % of the normal curve lies above it? Below it? If z = -1.7, what % of the normal curve lies below it? What % of the curve lies between z = -.75 and z =.75? What is the z-score such that only 5% of the curve lies above it? In the SAT with µ=500 and  =100, what % of the population do you expect to score above 600? Above 750?

X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ X _ μ

Population Sample A X A µ _ Sample B X B Sample E X E Sample D X D Sample C X C _ _ _ _ In reality, the sample mean is just one of many possible sample means drawn from the population, and is rarely equal to µ.  sasa sbsb scsc sdsd sese n n n nn

Population Sample A X A µ _ Sample B X B Sample E X E Sample D X D Sample C X C _ _ _ _ In reality, the sample sd is also just one of many possible sample sd’s drawn from the population, and is rarely equal to σ.  sasa sbsb scsc sdsd sese n n n nn

SS (N - 1) s2s2 = SS N 22 = What’s the difference?

SS (N - 1) s2s2 = SS N 22 = What’s the difference? ^ (occasionally you will see this little “hat” on the symbol to clearly indicate that this is a variance estimate) – I like this because it is a reminder that we are usually just making estimates, and estimates are always accompanied by error and bias, and that’s one of the enduring lessons of statistics)

Standard deviation. SS (N - 1) s =

As sample size increases, the magnitude of the sampling error decreases; at a certain point, there are diminishing returns of increasing sample size to decrease sampling error.

Central Limit Theorem The sampling distribution of means from random samples of n observations approaches a normal distribution regardless of the shape of the parent population. Just for fun, go check out the Khan Academy

_ z = X -  XX - Wow! We can use the z-distribution to test a hypothesis.

Step 1. State the statistical hypothesis H 0 to be tested (e.g., H 0 :  = 100) Step 2. Specify the degree of risk of a type-I error, that is, the risk of incorrectly concluding that H 0 is false when it is true. This risk, stated as a probability, is denoted by , the probability of a Type I error. Step 3. Assuming H 0 to be correct, find the probability of obtaining a sample mean that differs from  by an amount as large or larger than what was observed. Step 4. Make a decision regarding H 0, whether to reject or not to reject it.

An Example You draw a sample of 25 adopted children. You are interested in whether they are different from the general population on an IQ test (  = 100,  = 15). The mean from your sample is 108. What is the null hypothesis?

An Example You draw a sample of 25 adopted children. You are interested in whether they are different from the general population on an IQ test (  = 100,  = 15). The mean from your sample is 108. What is the null hypothesis? H 0 :  = 100

An Example You draw a sample of 25 adopted children. You are interested in whether they are different from the general population on an IQ test (  = 100,  = 15). The mean from your sample is 108. What is the null hypothesis? H 0 :  = 100 Test this hypothesis at  =.05

An Example You draw a sample of 25 adopted children. You are interested in whether they are different from the general population on an IQ test (  = 100,  = 15). The mean from your sample is 108. What is the null hypothesis? H 0 :  = 100 Test this hypothesis at  =.05 Step 3. Assuming H 0 to be correct, find the probability of obtaining a sample mean that differs from  by an amount as large or larger than what was observed. Step 4. Make a decision regarding H 0, whether to reject or not to reject it.

GOSSET, William Sealy

The t-distribution is a family of distributions varying by degrees of freedom (d.f., where d.f.=n-1). At d.f. = , but at smaller than that, the tails are fatter.

_ z = X -  XX - _ t = X -  sXsX - s X = s  N N -

The t-distribution is a family of distributions varying by degrees of freedom (d.f., where d.f.=n-1). At d.f. = , but at smaller than that, the tails are fatter.

df = N - 1 Degrees of Freedom

Problem Sample: Mean = 54.2 SD = 2.4 N = 16 Do you think that this sample could have been drawn from a population with  = 50?

Problem Sample: Mean = 54.2 SD = 2.4 N = 16 Do you think that this sample could have been drawn from a population with  = 50? _ t = X -  sXsX -

The mean for the sample of 54.2 (sd = 2.4) was significantly different from a hypothesized population mean of 50, t(15) = 7.0, p <.001.

The mean for the sample of 54.2 (sd = 2.4) was significantly reliably different from a hypothesized population mean of 50, t(15) = 7.0, p <.001.

Population Sample A Sample B Sample E Sample D Sample C _  XY r XY

The t distribution, at N-2 degrees of freedom, can be used to test the probability that the statistic r was drawn from a population with  = 0. Table C. H 0 :  XY = 0 H 1 :  XY  0 where r N r 2 t =