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Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

How do I find P-value?Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

Today’s Schedule: Feb 16th and 17thRecap on 10.1 (5 min)Assessment Opportunity on Confidence Intervals (15 min)Ms. Liu’s Real World Experience with Statistical Inference, and Intro to 10.2 Discussion (30 minutes)Go over HW (30 min)/Fun problem(if time allows)Debrief, Questions?

Use Table A or Table C to find z*!

Say this: „ I took a SRS of 100 kids and am 95% confident that the true population proportion of kids who like cotton candy is between 0.66 and 0.89.

Mean this: „If I repeated my method many times, 95% of the CI’s I would create would contain the true population proportion of kids who like cotton candy. BUT I don’t know if my specific interval (0.66, 0.89) contains it or not!

Assessment Opportunity on Confidence Intervals! (15 minutes)

(Everyone get out a blank sheet of paper)

Instructions: Answer the questions on the next slide with your group members. Be sure to EXPLAIN clearly your answers with correct notation, formulas and context. I will grade one person’s paper randomly from your group, and your entire group gets the same grade.

Help each other out! Show me that you understand.

(15 minutes)

Farfle asked a SRS of 200 children if they like cotton candy. 58% of them said

yes.

- What is the 95% Confidence Interval for the proportion of all kids who like cotton candy?
- Interpret the interval you calculated. What does being ’95% confident’ mean?
- How many kids should Farfle ask if he wants to estimate the true population proportion of kids who like cotton candy within +-2%?

Intelligent Spacecraft Interface Laboratory (ISIS) at NASA Ames Research Center

Goal: to enhance crew efficiency and mission safety

Human Occulomotor Performance Model

Goal:

To develop a model that mimics human eye fixations on display panel during spacecraft malfunction

Statistical Inference: Significance Tests (10.2)

Think back to last class: What is the goal of creating confidence intervals?

A: To estimate a population parameter!

Today we are going to talk about Tests of Significance. What is the goal of a significance test?

A: To assess the evidence provided by data about some claim concerning a population

Example: I claim that I make 90% of my free throws. To test my claim, you ask me to shoot 100 free throws. I make 40 of them. Do you believe my claim?

Probably not. The probability that I make as few as 40 out of 100, if i really make 90% in the long run, is 0.000000000001. This probability convinces you to reject my claim.

There is a vocabulary word for the claim that I made about my basketball skills. We call this the null hypothesis (H0), or the statement we make about our parameter as its currently claimed to be.

H0: p= 90%

The suspicion or what we think is true is described as the alternative hypothesis.

Ha: p<90%

KEY POINT: A significance tests works by asking how unlikely the observed outcome would be if the null hypothesis were really true.

Let’s Practice writing Null and Alternate Hypotheses!

Example: You want to test if the new gerbil food your company just manufactured is causing weight gain in gerbils. Normally, gerbils average 2 pounds. A sample of gerbils you took weigh on average 6 pounds.

H0:

Ha:

Do this one with your group: Ice makers in fridges should make on average 10 pieces of ice per hour. However, you are suspicious about the ice maker in your fridge since you observed over 5 hours, an average of only 2 pieces per hour.

H0:

Ha:

Look at the picture on the left.

Let’s pretend the parameter we are estimating is the difference between the daily number of hours high schoolers spend on facebook and the hours they spend reading their math textbook...

Q: If the population mean were truly 0, is it more likely that I get a sample mean of 0.3 or 1.02?

A: 0.3! Why? Becuase the P-value is greater. That is, the probability that we get a sample with a mean of 0.3 hours is more likely than obtaining a sample with a 1.02 mean.

So, if I got a sample mean of 1.02 when the population mean was supposed to be 0.....I will question the null hypothesis.

Q: How small must the P-value be for us to reject the null hypthesis?

0.05 is a common rule of thumb. If the P-value is less than 0.05, the observed result is said to be statistically significant at the 0.05 level, and you would reject the null hypothesis.

Practice: Is my result statistically significant?

Remember: A result is statistically significant if its P-value is less than the specified significance level!

1) The probabiltiy that I only make 88 out of 100 tosses of gum wrappers into the trashcan (if I claim my trashketball shooting percentage to be 90%) is 0.23. Is this significant at the 0.05 level?

No! Because P-value 0.23 is NOT less than 0.05

2) I opened 100 cans of olives and got an average of 284 per can. A can of olives is supposed to contain 400. The P-value of my observation is 0.003. Is this significant at the 0.01 level?

YES! 0.003<0.01

3) The average adult can drink 8 oz of water in 3 seconds. The average time it takes a SRS of 70 adults to drink 8 oz of water is 2.7 seconds. The probability of obtaining this sample mean is 0.04. Is the sample result statistically significant at the 0.01 level?

No! 0.04 is NOT less than 0.01

Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

How do I find P-value?First, write down your null and alternative hypotheses!

Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

Then, calculate the one sample z statistic! (H0: )

Finally, choose the appropriate calculation depending on your alternative hypothesis!

Example: There are supposed to be 200 grams of sugar in a bag of cotton candy, with a standard deviation of 5 grams. I took an SRS of 50 bags and found the average amount of sugar to be 189 grams. Is there good evidence that cotton candy companies are trying to rip me off?

We want to calculate P (Z<z), where z is our one sample z statistic!

Approximately ZERO. My result is significant at the 0.01 level. I have good evidence to reject my null hypothesis and believe the average grams of sugar in a bag of cotton candy is not 200, but rather less.

P(Z<-15.55) =

Today’s Schedule FRIDAY Feb 18th/:

Go over Last class’ Group assessment opportunity (10 min)

Math Symbols Bingo (20 min)

Reflect Review Relearn and Learn (10.1, 10.2) (15 min)

Novel Research Activity! (15 min)

Let’s assess our knowledge (HW discussion #35, 39, 44, 48, 53) (30 min)

Debrief

Today’s Schedule MONDAYFeb 21tst:

Go over Last class’ Group assessment opportunity (10 min)

Math Symbols Bingo (20 min)

Reflect Review Relearn and Learn (10.1, 10.2) (15 min)

Novel Research Activity! (15 min)

Let’s assess our knowledge (HW discussion #35, 39, 44, 48, 53) (30 min)

Debrief

Farfle asked a SRS of 200 children if they like cotton candy. 58% of them said

yes.

- What is the 95% Confidence Interval for the proportion of all kids who like cotton candy?
- Interpret the interval you calculated. What does being ’95% confident’ mean?
- How many kids should Farfle ask if he wants to estimate the true population proportion of kids who like cotton candy within +-2%?

Draw a 4x4 grid on a piece of paper. Place 16 of the following math symbols/formulas anywhere you want in your grid.

Use Table A or Table C to find z*!

Say this: „ I took a SRS of 100 college students and am 95% confident that the true population proportion of college students who use Twitter is between 0.66 and 0.89.

Mean this: „If I repeated my method many times, 95% of the CI’s I would create would contain the true population proportion of college students who use Twitter. BUT I don’t know if my specific interval (0.66, 0.89) contains it or not!

Statistical Inference: Significance Tests

Q: Review: What is the goal of creating confidence intervals?

A: To estimate a population parameter!

Q: What is the goal of a significance test?

A: To assess the evidence provided by data about some claim concerning a population

Example: I claim that I average 11 ground balls per lacrosse game I play with a standard deviation of 1 GB. To test my claim, you went to my next 10 games. I averaged 3 ground balls per game. Do you believe my claim?

Definitely not..! The probability that I average as few as 3 GBs per game if I really average 11, is 0.0000000043. This probability convinces you to reject my claim.

There is a vocabulary word for the claim that I made about my lacrosse ground ball average. We call this the null hypothesis (H0), or the statement we make about our parameter as its currently claimed to be.

H0: u=11

The suspicion or what we think is true is described as the alternative hypothesis.

Ha: u<11

KEY POINT: A significance tests works by asking how unlikely the observed outcome would be if the null hypothesis were really true.

Let’s Practice writing Null and Alternate Hypotheses!

Example: Ludacris is coming to Duke for LDOC on April 27th. A poll done two years ago showed the proportion of students who listen to his music to be 32%. You suspect his popularity would be higher now.

H0:

Ha:

p=0.32

p>0.32

Do this one with your group: You read that the percentage of Americans that buy girl scout cookies every year is 15%. But when you sampled 100 of friends, 88 of them said they buy girl scout cookies every year.

H0:

Ha:

p=.15

p>.15

Look at the picture on the left.

Let’s pretend the parameter we are estimating is the difference between the daily number of hours high schoolers spend on facebook and the hours they spend reading their math textbook...

Q: If the population mean were truly 0, is it more likely that I get a sample mean of 0.3 or 1.02?

A: 0.3! Why? Becuase the P-value is greater. The P-value is the probability you would get our observed outcome if the null hypothesis were really true.

So, if I got a sample mean of 1.02 when the population mean was supposed to be 0.....I will question the null hypothesis.

Q: How small must the P-value be for us to reject the null hypthesis?

0.05 is a common rule of thumb. If the P-value is less than 0.05, the observed result is said to be statistically significant at the 0.05 level, and you would reject the null hypothesis.

Practice: Is my result statistically significant?

Remember: A result is statistically significant if its P-value is less than the specified significance level!

1) The probabiltiy that I only make 88 out of 100 tosses of gum wrappers into the trashcan (if I claim my trashketball shooting percentage to be 90%) is 0.23. Is this significant at the 0.05 level?

No! Because P-value 0.23 is NOT less than 0.05

2) I opened 100 cans of olives and got an average of 284 per can. A can of olives is supposed to contain 400. The P-value of my observation is 0.003. Is this significant at the 0.01 level?

YES! 0.003<0.01

3) The average adult can drink 8 oz of water in 3 seconds. The average time it takes a SRS of 70 adults to drink 8 oz of water is 2.7 seconds. The probability of obtaining this sample mean is 0.04. Is the sample result statistically significant at the 0.01 level?

No! 0.04 is NOT less than 0.01

First, write down your null and alternative hypotheses!

Then, calculate the one sample z statistic! (H0: )

Finally, choose the appropriate calculation depending on your alternative hypothesis!

We want to calculate P (Z<z), where z is our one sample z statistic!

Approximately ZERO. My result is significant at the 0.01 level. I have good evidence to reject my null hypothesis and believe the average grams of sugar in a bag of cotton candy is not 200, but rather less.

P(Z<-15.55) =

There has been no research done on the heights of students taking AP Statistics. Is there a difference in AP Stat students’ heights and the national average? If we find a significant result, this could be the beginning of an exciting research project!

„Studies show that male and female students in AP Statistics courses are shorter than the national average, possibly due to lack of sleep”-MathJournal

„Interesting find: Students in AP Statistics classrooms tower over the average American in height”-NCTimes

It is reported that the average height of an American female is 64 inches with a standard deviation of 2.6 inches. The average height of an American male is 69 inches with a standard deviation of 2.6 inches.

Questions to Think about..

Why does higher confidence change the length of the interval?

Why does a bigger sample size change the length of the interval?

Does lack of significance imply that H0 is true?

How do we find the P-value?

This is best explained by doing an example problem. Let’s make up our own example!

Group #1: Name of a Person

Group #2: Location

Group #3: Product you might want to buy

Group #4: Product you can sell

Group #5: Name of a Company

Group #6: Emotion

Group #7: Integer between 10 and 1000, Integer from 50-100

Group #8: Time Period (day, week, year, hour etc)

______________ lives in _______________, where the demand for ___________ is very high. That is why s/he started a _______________ business called ____________. Recently, ____________ has been very __________. S/he has only sold on average ____ ______________ in the past ____ __________! Normally s/he sells ___ _______________per _________, with a standard deviation of ____. Is this evidence that average sales are now lower?

Today’s Schedule TUESDAYFeb 22nd, PRACTICE PRACTICE PRACTICE:

Go Over Ch. 8 Quiz (15 min)

Go Over Ch. 9 Quiz (30 min) (If you haven’t taken the quiz, you are taking it while we do this)

The Inference Toolbox (30 min)

Questions?

Homework: Do Inference Toolbox Worksheet and Read 10.3

Turn to page 557 in your textbook, and look at #22. We are going to model how to use the Inference Toolbox for creating confidence intervals.

Inference Toolbox (Confidence Intervals)

Step 1: Identify the population of interest and the parameter you want to draw conclusions about.

Q: What is meant by „Population of Interest”?

A: Population of interest refers to the group you are trying to study. Your SRS comes from this population.

Examples of „Populations of Interest”: American women, hotel general managers, high school students, current NBA players, etc...

Q: What is a „parameter”?

A: A number that describes a population. We have talked about population means and population proportions.

Examples: average height, average years working in current company, proportion who use facebook, percentage who have higher than 75% shooting percentage, etc...

Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

Q: What are the inference procedures (for CIs) we have talked about so far?

A: Confidence intervals for population means and proportions

Q: What conditions do we need to ensure that we can use the normal approximation for x-bar? (Think back to Ch. 9!)

A: Data must come frome an SRS. Either the population distribution has to be normal or the sample size has to be big (CLT).

Q: What conditions do we need to ensure we can use the normal approximation and standard deviation formula for the sampling distribution of p-hat?

A: Data must come from an SRS. The two rules of thumbs from Ch.9! np>=10, n(1-p)>=10, and population must be 10x the sample

Step 3: If the conditions are met, create your confidence interval!

Step 4: Interpret your results in the context of the problem.

Aka, say „ I am 95% confident that....”, and then say what that means!

Let’s Model the Inference Toolbox for Significance Tests by doing #48.

Step 1: Identify the Population of Interest and the Parameter you want to draw conclusions about. State the null and alternative hypotheses in words and symbols.

Step 2: Choose the appropriate inference procedure for significance tests. Verifiy the conditions for using the selected procedure.

So far we’ve only learned the one sample z test for means. We have to check that we can use the normal approximation and that the data is from a SRS

Step 3: If conditions are met, carry out the inference procedure (calculate the test statistic and find the P-value)

Step 4: Interpret your results in the context of the problem. (is it statistically significant? Can we reject the null hypothesis?)

Instructions: Take 10 min in your group to discuss your answers for #44. Turn in one copy to me, making sure to follow the Inference Toolbox procedures!

Today’s Schedule THURSDAYFeb 24th

Warmup (30 minutes)

Inference Toolbox Worksheet (35 min)

Ask me questions!

Multiple Choice Practice

Statistical significance is often used to persuade people into buying products, believe a certain claim, cause controversy, instigate discussion... And should always be looked at with caution

Ask me any last questions you have about 10.1 or 10.2! Take 5 minutes to write your questions on a piece of paper.

Example questions:

I still don’t get what being „95% confident” means!

How do you carry out a two sided z test?

Can you draw graphically what it means to have a P-value of 0.023?

I’m not sure when to use __________ formula!

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