Welcome to . Week 08 Thurs . MAT135 Statistics

We use the sample mean 𝒙 to estimate the unknown population mean µ Estimation We use the sample mean 𝒙 to estimate the unknown population mean µ

Estimation Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

Estimation The sample standard deviation “s” is the best estimate we have for the unknown population standard deviation “σ”

Using s to estimate σ is also an inference Estimation Using s to estimate σ is also an inference

s IS NOT the measure of variability in the new population of 𝒙 s Estimation s IS NOT the measure of variability in the new population of 𝒙 s

It needs to be decreased to take sample size into account! Estimation It needs to be decreased to take sample size into account!

Estimation We use: s/ n for the measure of variability in the new population of 𝒙 s

Estimation The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

Estimation So our curve is: 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

Binomial Proportions All of these activities can be done when you have measured data (continuous)

Binomial Proportions What if you had counts in two categories and wanted to estimate the population proportion?

Binomial Proportions Weep? Throw things? Give up? http://www.pickthebrain.com/blog/wp-content/uploads/2011/07/Screen-shot-2011-07-17-at-11.04.16-AM.png

Binomial Proportions Naw… we’ve got an easy solution! http://www.shutterstock.com/s/whew/search.html

Binomial Proportions Suppose you have a random sample of size “n”

Binomial Proportions Suppose you have a random sample of size “n” Suppose you want the proportion of successes p

Binomial Proportions Suppose you have a random sample of size “n” Suppose you know the number of successes in your sample is “x”

Binomial Proportions The sample proportion (symbolized: p , called: “p-hat”) is: p = x n

You have a random sample with frequencies: Is it binomial? BINOMIAL PROPORTIONS IN-CLASS PROBLEM 1 You have a random sample with frequencies: Is it binomial? Blue Red 35 23

You have a random sample with frequencies: Is it binomial? yep BINOMIAL PROPORTIONS IN-CLASS PROBLEM 1 You have a random sample with frequencies: Is it binomial? yep Blue Red 35 23

You have a random sample with frequencies: What is n? BINOMIAL PROPORTIONS IN-CLASS PROBLEM 2 You have a random sample with frequencies: What is n? Blue Red 35 23

You have a random sample with frequencies: What is n? 58 BINOMIAL PROPORTIONS IN-CLASS PROBLEM 2 You have a random sample with frequencies: What is n? 58 Blue Red 35 23

You have a random sample with frequencies: What is xred? BINOMIAL PROPORTIONS IN-CLASS PROBLEM 3 You have a random sample with frequencies: What is xred? Blue Red 35 23

You have a random sample with frequencies: What is xred? 23 BINOMIAL PROPORTIONS IN-CLASS PROBLEM 3 You have a random sample with frequencies: What is xred? 23 Blue Red 35 23

You have a random sample with frequencies: What is p red? BINOMIAL PROPORTIONS IN-CLASS PROBLEM 4 You have a random sample with frequencies: What is p red? Blue Red 35 23

BINOMIAL PROPORTIONS IN-CLASS PROBLEM 4 You have a random sample with frequencies: What is p red? x n = 23 58 ≈ 40.0% Blue Red 35 23

Questions?

Binomial Estimation Just like the sample mean 𝒙 is the best estimate of the true population mean μ…

Binomial Estimation …there is a true population proportion “p” that is best estimated by the sample proportion p

Binomial Estimation In symbols: µ p = p

BINOMIAL ESTIMATION IN-CLASS PROBLEM 5 You have a random sample with frequencies: What is the best estimate for pred? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 5 You have a random sample with frequencies: What is the best estimate for pred? 40.0% Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 6 You have a random sample with frequencies: What is the best estimate for qred? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 6 You have a random sample with frequencies: What is the best estimate for qred? 60.0% Blue Red 35 23

Binomial Estimation Because p-values distributions are often skewed, you need to be sure your sample size is large before assigning probabilities to this estimate using a normal distribution

Binomial Estimation You can assume normality if: np ≥ 5or10 and n(1-p) ≥ 5or10 (nq ≥ 5or10)

This is known as the Rule of Sample Proportions Binomial Estimation This is known as the Rule of Sample Proportions

Binomial Estimation Note: your book says the shape of the sampling distribution of p-hat is approximately normal provided np(1-p)≥10 (I think it’s a typo)

Binomial Sampling What if your sample size is too small?

Binomial Sampling The normal approximation can always be used, but if the conditions are not met, then the approximation may not be that good of an approximation.

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Is np ≥ 5or10? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Is np ≥ 5or10? np ≈ 58 × .400 ≈ 23.2 Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Is nq ≥ 5or10? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Is nq ≥ 5or10? nq ≈ 58 ×.400 ≈ 23.2 Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? Blue Red 35 23

BINOMIAL ESTIMATION IN-CLASS PROBLEM 7 You have a random sample with frequencies: Is p red likely to be normally-distributed? yep, close enough Blue Red 35 23

Questions?

Binomial Sampling Just as a gazillion samples each with its own mean gives a new population: the gazillion means

Binomial Sampling …a gazillion samples each with its own proportion gives a new population: the gazillion p s

Binomial Sampling And, again if you plotted the frequency of the gazillion p values, it would be called a SAMPLING DISTRIBUTION

Binomial Sampling And, again, the shape of the plot of the gazillion sample means would have a normal-ish distribution NO MATTER WHAT THE ORIGINAL DATA LOOKED LIKE

Binomial Sampling Very non- normal population

Binomial Sampling Very non- normal population Normal-er

Binomial Sampling Very non- normal population Normal-ish

Binomial Sampling And, again, as “n” increases, the variability (spread) decreases

Binomial Sampling The standard deviation of the p s is: σ p = p(1−p) n = pq n

Binomial Sampling The standard deviation of the p s is: σ p = p(1−p) n = pq n also called the standard error of p-hat: se p

Binomial Sampling So a normal curve would be: p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

Binomial Sampling So a normal curve would be: p -3 se p p -2 se p p -se p p p +se p p +2 se p p +3 se p

BINOMIAL SAMPLING IN-CLASS PROBLEM 8 You have a random sample with frequencies: What is the standard error of p-hat? Blue Red 35 23

BINOMIAL SAMPLING IN-CLASS PROBLEM 8 You have a random sample with frequencies: What is the standard error of p-hat? × se p = pq n = .6×.4 58 ≈ .064 Blue Red 35 23

BINOMIAL SAMPLING IN-CLASS PROBLEM 9 You have a random sample with frequencies: What would the normal curve look like? Blue Red 35 23

BINOMIAL SAMPLING IN-CLASS PROBLEM 9 p -3 se p p -2 se p p -se p p p +se p p +2 se p p +3 se p

BINOMIAL SAMPLING IN-CLASS PROBLEM 9 .6-3(.064) .6-2(.064) .6-.064 .6 .6+.064 .6+2(.064) .6+3(.064)

BINOMIAL SAMPLING IN-CLASS PROBLEM 9 .408 .472 .536 .6 .664 .728 .792

What is P(0.536<p<.664)? BINOMIAL SAMPLING IN-CLASS PROBLEM 10 .408 .472 .536 .6 .664 .728 .792

What is P(0.536<p<.664)? 68% BINOMIAL SAMPLING IN-CLASS PROBLEM 10 What is P(0.536<p<.664)? 68% .408 .472 .536 .6 .664 .728 .792

BINOMIAL SAMPLING IN-CLASS PROBLEM 11 What is the range of values for p that would with 95% certainty include the true p? .408 .472 .536 .6 .664 .728 .792

BINOMIAL SAMPLING IN-CLASS PROBLEM 11 What is the range of values for p that would with 95% certainty include the true p? .472<p<.728 .408 .472 .536 .6 .664 .728 .792

What is the probability the true p lies between 0.47 and .65? BINOMIAL SAMPLING IN-CLASS PROBLEM 12 What is the probability the true p lies between 0.47 and .65? .408 .472 .536 .6 .664 .728 .792

What is the probability the true p lies between 0.47 and .65? 76.2% BINOMIAL SAMPLING IN-CLASS PROBLEM 12 What is the probability the true p lies between 0.47 and .65? 76.2% .408 .472 .536 .6 .664 .728 .792

Binomial Sampling A study I did for my science class…

Binomial Sampling Some diamonds fluoresce under UV light

Binomial Sampling What percentage of diamonds fluoresce?

Binomial Sampling I inherited a bunch of jewelry Some of it has diamonds in it

Binomial Sampling Is this a binomial variable?

Binomial Sampling What is n? Fluoresce Don’t 8 114

Binomial Sampling What is x? Fluoresce Don’t 8 114

Binomial Sampling What is your best estimate of p? Fluoresce Don’t 8 114

Binomial Sampling Is it normal? Fluoresce Don’t 8 114

Binomial Sampling To find out what sample size you need to: Estimate p Use the min of p and q Algebra magic: n > 5or10/min(p,q)

Binomial Sampling p ≈ .066 so q ≈ .934 so we’ll use p n > 10/.066 or 151.5 Fluoresce Don’t 8 114

Binomial Sampling p ≈ .066 so q ≈ .934 so we’ll use p n > 10/.066 or 151.5 In other words n > 152 Fluoresce Don’t 8 114

Questions?

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