Welcome to . Week 10 Thurs . MAT135 Statistics

In-Class Project Is normal human temperature REALLY 98.6° ?

Normal Distribution The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

Normal Distribution We use normal distributions a lot in statistics because lots of things have graphs this shape! heights weights IQ test scores bull’s eyes

Sampling Distributions Even data which are not normally distributed have sample averages which DO have normal distributions

Sampling Distributions As “n” increases, the distributions of the sample means become closer and closer to normal

Sampling Distributions We usually say the sample mean will be normally distributed if n is ≥ 20or30 (the “good-enuff” value)

Sampling Distributions We will use the sample mean 𝒙 to estimate the unknown population mean µ

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

Inferences about μ If you can assume the distribution of the sample means is normal, you can use the normal distribution probabilities for making probability statements about µ

as “n” increases, variability (spread) also decreases Inferences about μ as “n” increases, variability (spread) also decreases

Inferences about μ We use: s/ n for the measure of variability in the new population of 𝒙 s

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

Inferences about μ 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se So our normal curve for the true value of the population mean µ is: 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

Inferences about μ About 95% of the possible values for μ will be within 2 SE of 𝒙

This allows us to create a “confidence interval” for values of μ Confidence Intervals This allows us to create a “confidence interval” for values of μ

Confidence Intervals Confidence interval formula: 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n or 𝒙 - 2se ≤ μ ≤ 𝒙 + 2se With a confidence level of 95%

The “2” in the equations is called the “critical value” Confidence Intervals The “2” in the equations is called the “critical value”

It comes from the normal curve, which gives us the 95% Confidence Intervals It comes from the normal curve, which gives us the 95%

2s/ n or 2se is called the “margin of error” Confidence Intervals 2s/ n or 2se is called the “margin of error”

What if we wanted a confidence level of 99% Confidence Intervals PROJECT QUESTION What if we wanted a confidence level of 99%

Confidence Intervals PROJECT QUESTION What if we wanted a confidence level of 99% We’d use a value of “3” rather than 2

Confidence Intervals For most scientific purposes, 95% is “good-enuff” In the law, 98% is required for a criminal case In medicine, 99% is required

Confidence Intervals For a 95% confidence interval, 95% of the values of μ will be within 2se of 𝒙

Confidence Intervals If we use the confidence interval to estimate a likely range for true values of μ, we will be right 95% of the time

For a 95% confidence interval, we will be WRONG 5% of the time Confidence Intervals For a 95% confidence interval, we will be WRONG 5% of the time

For a 99% confidence interval, how much of the time will we be wrong? Confidence Intervals PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong?

Confidence Intervals PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong? we will be wrong 1% of the time

Confidence Intervals The percent of time we are willing to be wrong is called “α” (“alpha”) or “the α-level”

Confidence Intervals Everyday use of confidence intervals: You will frequently hear that a poll has a candidate ahead by 10 points with a margin of error of 3 points

Confidence Intervals This means: 10-3 ≤ true difference ≤ 10+3 Or, the true difference is between 7 and 13 points (with 95% likelihood)

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality? yes, because n>20

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level? 5%

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value? 2, because we want a 95% confidence interval

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error? s/ n = 5/ 25 = 5/5 = 1

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error? 2se = 2(1) = 2

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2 ≤ μ ≤ 7 + 2 5 ≤ μ ≤ 9 with 95% confidence

Confidence Intervals PROJECT QUESTION Interpreting confidence intervals: If the 95% confidence interval is: 5 ≤ µ ≤ 9 Is it likely that µ = 10?

Confidence Intervals PROJECT QUESTION No, because it’s outside of the interval That would only happen 5% of the time

Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49 Confidence Intervals PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49

Confidence Intervals PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49 49 ≤ µ ≤ 57

Can you say with 95% confidence that µ ≠ 55? Confidence Intervals PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55?

Confidence Intervals PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55? Nope… it’s in the interval It IS a likely value for µ

Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121 Confidence Intervals PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121

Confidence Intervals PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121 453 ≤ µ ≤ 509

Can you say with 95% confidence that µ might be 450? Confidence Intervals PROJECT QUESTION Can you say with 95% confidence that µ might be 450?

Confidence Intervals PROJECT QUESTION Can you say with 95% confidence that µ might be 450? µ is unlikely to be 450 – that value is outside of the confidence interval and would only happen 5% of the time

You will have a smaller interval if you have a larger value for n Confidence Intervals You will have a smaller interval if you have a larger value for n

So you want to take the LARGEST sample you can Confidence Intervals So you want to take the LARGEST sample you can

This is called the “LAW OF LARGE NUMBERS” Confidence Intervals This is called the “LAW OF LARGE NUMBERS”

Confidence Intervals LAW OF LARGE NUMBERS The larger your sample size, the better your estimate

What if you have a sample size smaller than 20??? Confidence Intervals What if you have a sample size smaller than 20???

Confidence Intervals What if you have a sample size smaller than 20??? You must use a different (bigger) critical value W.S. Gosset 1908

Questions?

“Tailed” Tests A two-tailed test will reject H0 either if the experimental values we get are too high or too low

“Tailed” Tests α is split between the upper and lower tails

“Tailed” Tests A one-tailed test will reject H0 only on the side we think is likely to be true

“Tailed” Tests You will be able to reject H0 more often for a one-tailed test – if you pick the right tail!

“TAILED” TESTS PROJECT QUESTION Your owner's manual says you should be getting 30 mpg highway After owning the car for six months, you are only getting 27 mpg highway

“TAILED” TESTS PROJECT QUESTION Is that different enough to reject the company's claim? What is your α-level? What is Ha? What is H0?

“TAILED” TESTS PROJECT QUESTION Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? What is H0?

“TAILED” TESTS PROJECT QUESTION Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? μ < 30 mpg What is H0?

“TAILED” TESTS PROJECT QUESTION Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? μ < 30 mpg What is H0? μ ≥ 30 mpg

We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg “TAILED” TESTS PROJECT QUESTION We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg

Is this a one-tailed or a two- tailed test? “TAILED” TESTS PROJECT QUESTION Is this a one-tailed or a two- tailed test?

“TAILED” TESTS PROJECT QUESTION Is this a one-tailed or a two- tailed test? one-tailed Is it right-tailed or left-tailed?

Is it right-tailed or left-tailed? left-tailed “TAILED” TESTS PROJECT QUESTION Is it right-tailed or left-tailed? left-tailed

“Tailed” Tests Remember our two type of errors: Type 1 error: reject a true H0 (α) Type 2 error: fail to reject a false H0 (β)

“Tailed” Tests The likelihood of making the right decision and rejecting the (false) null hypothesis is: 1 - β

“Tailed” Tests The likelihood of making the right decision and rejecting the (false) null hypothesis is: 1 - β called the “power of the test”

“Tailed” Tests For a given α value, we would like the test to be as "powerful" as possible, give us the best chance of rejecting a false null hypothesis

Which is more powerful, a one-tailed or a two-tailed test? “TAILED” TESTS PROJECT QUESTION Which is more powerful, a one-tailed or a two-tailed test?

“TAILED” TESTS PROJECT QUESTION Which is more powerful, a one-tailed or a two-tailed test? one-tailed (if you guess the correct side)

Questions? http://i.imgur.com/aliTlT3.jpg

Hypothesis Tests Now we will create a confidence interval about 𝒙 and see if our hypothesized value for μ falls in it

Hypothesis Tests How to do it!

Hypothesis Tests How to do it! Set your α-level (how often you are willing to be wrong)

Hypothesis Tests How to do it! Set your α-level Define your Ha and H0

Hypothesis Tests How to do it! Set your α-level Define your Ha and H0 Get your data (for a confidence interval, you need the hypothesized μ, s and n (or se)

Hypothesis Tests How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value (for α=5% two-sided it is ≈2 for one-sided it is ≈1.64)

Hypothesis Tests How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval for μ

Hypothesis Tests How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval for μ The test will be: Is x in it?

Hypothesis Tests PROJECT QUESTION Back to our mpg! H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 And suppose we know that: se = 4 mpg

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg Are we going to reject H0 for values of x greater than 30 or less than 30?

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg The 1-tailed critical value is: 1.64

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the margin of error?

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the margin of error? 1.64×4 = 6.56

H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI? Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI?

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI? 95% of x values from a population with μ ≥ 30 will fall above: 30 – 6.56 = 23.44

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg So reject H0 if x < 23.44 What is our conclusion?

Hypothesis Tests PROJECT QUESTION H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg So reject H0 if x < 23.44 What is our conclusion? fail to reject H0

Hypothesis Tests If you reject H0 with an α-level of 0.05, we also say our x value is “significant at the .05 level” or we say we found a “significant difference”

Questions? http://i.imgur.com/aliTlT3.jpg

CI for Proportions We’ll use the normal curve for proportions: p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

If p = .4 and n = 30 find the 95% CI for p CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p

If p = .4 and n = 30 find the 95% CI for p se = CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se =

If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089 CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089

If p = .4 and n = 30 find the 95% CI for p me = CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me =

If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178 CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178

If p = .4 and n = 30 find the 95% CI for p CI: CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI:

CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI: .4 - .178 ≤ p ≤ .4 + .178 .222 ≤ p ≤ .578

Questions? http://i.imgur.com/aliTlT3.jpg

Hypothesis Tests We can make our x more likely to be significant by (as usual): TAKING A LARGER SAMPLE SIZE

Hypothesis Tests Because we can “cheat the system” by taking a huge sample size that will find any teeny, tiny difference to be significant, we have a backup plan

Hypothesis Tests We also set levels of “practical significance” - what numerical difference would convincingly show a significant difference

Hypothesis Tests These levels of practical significance come from our knowledge of the variables we are measuring

Hypothesis Tests If we had taken a sample of 10,000,000 to calculate our mpg average and se, we could easily have had an se of 0.1 mpg Probably we wouldn’t really think that was a significant difference in mileage

Hypothesis Tests A practically significant difference would be the amount in mpg that you would think is different enough from 30 mpg to be important

Hypothesis Tests We set a level of practical significance at the same time we set the α-level

Hypothesis Tests PROJECT QUESTION What would be your level of practically significant difference for mpg?

Questions? http://i.imgur.com/aliTlT3.jpg

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