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Today in Class. Last time we discussed statistical reasoning and Type I and Type II errors Today we’ll discuss Type I and Type II errors in more depth We’ll also discuss the necessity of sampling distributions and how to find the sampling distribution for a sample proportion.

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Today in Class

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Today in class

Today in Class

  • Last time we discussed statistical reasoning and Type I and Type II errors

  • Today we’ll discuss Type I and Type II errors in more depth

  • We’ll also discuss the necessity of sampling distributions and how to find the sampling distribution for a sample proportion


Today in class

Hypothesis Testing Example

  • I know I have 5 eggs, but I don’t know if they’re good or bad.

  • I’ll make a guess that 3 are good.

  • Then I can get all possible samples of 3 from that scenario.

  • I note that for this hypothetical pop, it is impossible to get 3 bad eggs out of 3.

  • It is also unlikely (but still possible) to get 3 good eggs out of 3.

  • I’ll take a real sample, if I get either of these cases, I won’t believe the hypothesized pop.


Type i and type ii errors

Type I and Type II Errors

  • Recall that a Type I error is rejecting a true null hypothesis.

  • If the null hypothesis (3/5 good eggs) is true, my decision rule will reject this hypothesis for 1/10 samples. Therefore, the probability of a Type I error is 0.10.

  • Type II errors depend on what the true population is.


Type i and type ii errors1

Type I and Type II Errors

  • If there are no bad eggs in the pop of 5, then all sample of 3 will have all bad eggs. I’ll reject the null hypothesis - correct decision. In this case, I can’t make a Type II error.

  • If there is 1 bad egg in the pop of 5, then of the 10 possible samples, 6 samples have at least one bad egg and at least one good egg. I’ll fail to reject the false null hypothesis, and make a Type II error. Thus for this case, I have a 0.6 probability of a Type II error.


Type i and type ii errors2

Type I and Type II Errors

  • If there are really 3 bad eggs in the pop of 5, then there is one sample (of 10 possible samples) for which I reject the null hypothesis. Thus, the probability of a Type II error is 0.90.

  • If there are really 4 bad eggs in the pop of 5, then there are 4 samples (of 10) for which I will reject the null hypothesis. Probability of a Type II is 0.60.


Type i and type ii errors3

Type I and Type II errors

  • If there are 5 bad eggs out of 5 in the pop, then every sample has 3 bad eggs and I reject the null hypothesis. Thus, the probability of a Type II error is 0 for this case.

  • I’ll demonstrate this with the coin-flip challenge.


Coin flip challenge

Coin Flip Challenge

  • I make the real flips my null hypothesis, because I can characterize all the possible sets of 200 flips and their probabilities for real flips

  • I’ll make a decision rule to decide whether a set of 200 flips is real or not.


Statistical reasoning

Statistical Reasoning

  • Since we must rely on samples to make inference about the population, we want to consider every possible sample from a hypothetical population.

  • The sampling distribution is the characterization of a sample statistic based on every possible sample from a hypothetical population.

  • Finding sampling distributions is central to statistics.


Finding sampling distributions

Mathematical

Use of mathematics and systematic reasoning to derive sampling distribution

Results in normal, t, c2, and F distributions (which we will study later)

Simulation

Uses a computer to mimick sampling process

Take 1000’s of samples

Relies on a sample of samples

Mathematical approach should be used whenever possible

Finding Sampling Distributions


An example of a simulation

An Example of a Simulation

  • To determine the distribution of the longest run in 200 coin flips, I used a simulation

  • Program to simulate flipping a fair coin 200 times

  • Repeat the 200 flips 1000 times

  • Note how often each run occurs.


Sampling distribution of a proportion

Sampling Distribution of a Proportion

  • Suppose we’re drawing from a very large population and asking person if they’re a Democrat

  • Suppose 50% are Democrats

  • If we ask just one person, then we’ll get either a “yes” or “no”

  • Ask 2 people: (Y,Y), (Y,N), (N,Y), (N,N)


Sampling dist for proportion

Sampling Dist. for Proportion

  • Ask 3 people, you get (YYY), (YYN), (YNY), (YNN), (NYY), (NYN), (NNY), (NNN)

  • Ask 4 people, continue

  • Keep going and for a large enough sample you get a bell-shaped curve!


The normal distribution

The Normal Distribution

  • Symmetric and Bell-Shaped

  • Total Area = 1 since it covers all possible samples

  • Characterized by two quantities: the mean m and the standard deviation s

  • Represents all possible samples for hypothetical population

  • The mean m is the center

  • The sd s is how spread the curve is

_

s

m

Increasing s makes the curve shorter and fatter

Increasing m moves the curve to the right

Areas represent probabilities of certain samples for the hypothetical population


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