AP StatisticsFriday, 29 January 2016

• OBJECTIVE TSW determine confidence intervals.

• Yesterday’s tests are not graded.

• TEST: Continuous Distributions tests are graded.

Test Addendum: Continuous Distributions

• Up to 5 points will be awarded on the Continuous Distributions test.

• Directions must be followed.– 5 decimals for probabilities– 3 decimals for other values– Complete sentence answers when indicated.– All work – including calculator key strokes – shown.

• This is due on Tuesday, 02 February 2016.

AP Exam Registration• Registration must be done on line.

www.TotalRegistration.net/AP/443381

• Regular registration: 01/05/2016 – 03/04/2016.

• Late registration: 03/05/2016 – 03/20/2016– Additional $10 per test fee added on.

• Cost per test:– $96/test

• Can apply for a Cy-Hope scholarship (reduction of $25/test, up to 3 tests).

• Pick up applications at Counselors’ Corner or at Mr. Hernandez’s office or Ms. Lewis’ office.

– $11/test (free/reduced lunch program)• Not eligible for Cy-Hope scholarship.

Confidence Intervals

Rate your confidence0 - 100

• Name my age – within 10 years.– within 5 years.– within 1 year.

• Shoot a basketball at a wading pool and make a basket.

• Shoot the ball at a large trash can and make a basket.

• Shoot the ball at a carnival game of chance and make a basket.

What happens to your confidence as the interval gets smaller?

As the interval gets smaller, your confidence goes down.The larger your confidence, the wider the interval.

Point Estimate• Use a single statistic based on

sample data to estimate a population parameter

• Simplest approach• But not always very precise due to

variation in the sampling distribution

Confidence intervals

• Are used to estimate the unknown population mean

• Formula:

estimate + margin of error

Margin of error• Shows how accurate we believe our estimate

is• The smaller the margin of error, the more

precise our estimate of the true parameter• Formula:

statistic theofdeviation standard

value

criticalm

Confidence level• Is the success rate of the method

used to construct the interval

• “Using this method, ____% of the time the intervals constructed will contain the true population parameter.”

What does it mean to be 95% confident?

• 95% chance that m is contained in the confidence interval. ( ? ? ? )

• The probability that the interval contains m is 95%. ( ? ? ? )

• The method used to construct the interval will produce intervals that contain m 95% of the time. ( ? ? ? )

• Which is correct?

AP StatisticsMonday, 01 February 2016

• OBJECTIVE TSW determine confidence intervals.

• ASSIGNMENT DUE DATES– WS Confidence Intervals #1 tomorrow, 02/02/16– WS Confidence Intervals #2 Wednesday, 02/03/16

• Sampling Distributions tests are not graded.

• AP EXAM REGISTRATION (through 03/04/16)

www.TotalRegistration.net/AP/443381

• Found from the confidence level• The upper z-score with probability p lying to

its right under the standard normal curve

Confidence level tail area z*.05 1.645.025 1.96.005 2.576

Critical value (z*)

.05

z*=1.645

.025

z*=1.96

.005

z*=2.57690%95%99%

Confidence interval for a population mean (Formula):

n

zx *estimate

Critical value

Standard deviation of the statistic

Margin of error

Steps for doing a confidence interval:1) State the assumptions –

• SRS taken from population• Sampling distribution is normal (or approximately

normal)• Given (normal)• Large sample size (approximately normal)• Graph data (approximately normal)

• is known2) Calculate the interval3) Write a statement about the interval in the

context of the problem (complete sentence).

Statement: (memorize!!)

We are ________% confident that the true mean of context lies within the interval ______ and ______.

Assumptions:1) Have an SRS of blood measurements2) Potassium level is normally distributed (given)3) known

We are 90% confident that the true mean potassium level is between 3.010 and 3.390.

Example 1: A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

0.23.2 1.645 3.0101, 3.38993

Assumptions:1) Have an SRS of blood measurements2) Potassium level is normally distributed (given)3) known

We are 95% confident that the true mean potassium level is between 2.974 and 3.426.

95% confidence interval?

0.23.2 1.96 2.9737, 3.42633

99% confidence interval?

Assumptions:1) Have an SRS of blood measurements2) Potassium level is normally distributed (given)3) known

We are 99% confident that the true mean potassium level is between 2.903 and 3.497.

0.23.2 2.576 2.9026,3.49743

What happens to the interval as the confidence level increases?

the interval gets wider as the confidence level increases

How can you make the margin of error smaller?• z* smaller

(lower confidence level)

• smaller(less variation in the population)

• n larger(to cut the margin of error in half, n

must be 4 times as big)

Really cannot change!

Example 2: A random sample of 50 JVHS students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of JVHS students?

We are 95% confident that the true mean SAT score for JVHS students is between 1220.9 and 1279.1

How do you find a critical value (z*) for a given confidence level?• Use invNorm on the calculator.

– Example: For a 90% confidence level,invNorm(0.95) = 1.644853626 . . .

– For an 84% confidence level,invNorm(0.92) = 1.405071561 . . .

Assumptions:• SRS (given)• The distribution is approximately normal (“boxplot is symmetrical” or “quantile plot is linear”).

• σ is given.

We are 95% confident that the true mean SAT score for JVHS students is between 1115.072 and 1270.642.

Suppose that we have this random sample of SAT scores for JVHS students:950 1130 1260 1090 1310 1420 1190What is a 95% confidence interval for the true mean SAT score? (Assume = 105)

Example 2.5:

1051192.857 1.967

AP StatisticsTuesday, 02 February 2016

• OBJECTIVE TSW (1) finish viewing the presentation on confidence intervals, (2) turn in WS #1, and (3) work on WS #2.

• ASSIGNMENTS DUE– WS Confidence Intervals #1 wire basket– WS Test Addendum: Continuous Distributions

black tray

• ASSIGNMENT DUE TOMORROW– WS Confidence Intervals #2

• QUIZ: Confidence Intervals is tomorrow.

• If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:

Finding a sample size:

n

zm *

Always round up to the nearest person!

Example 3: The heights of JVHS male students is normally distributed with = 2.5 inches. How large a sample is necessary to be accurate within + 0.75 inches with a 95% confidence interval?

n = 43

2.5Solve 0.75 1.9 o6 f r .n

n

Example 4: In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, giving them either calcium or a placebo. The paper reports a mean seated systolic blood pressure of 114.9 with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed.Can you find a z-interval for this problem? Why or why not?

No – the population standard deviation (σ) is not known.

Student’s t- distribution• Developed by William Gosset• Continuous distribution• Unimodal, symmetrical, bell-shaped density

curve• Above the horizontal axis• Area under the curve equals 1• Based on degrees of freedom

How does t compare to normal?• Shorter & more spread out• More area under the tails• As n increases, t-distributions

become more like a standard normal distribution

How to find t*• Use Table B for t distributions (green chart)• Look up confidence level at bottom & df on

the sides• df = n – 1

Find these t*90% confidence when n = 595% confidence when n = 15

t* =2.132t* =2.145

Can also use invT on the calculator!

Need upper t* value with 5% is above – so 95% is below

invT(p,df)

Assumptions for t-inference• Have an SRS from population• unknown• Normal distribution

– Given– Large sample size– Check graph of data

Formula:

Confi dence I nterval: * sx tn

estimate

Critical value

Standard deviation of statistic

Margin of error

Assumptions:• Have an SRS of healthy, white males• Systolic blood pressure is normally distributed

(given).• is unknown

We are 95% confident that the true mean systolic blood pressure is between 111.220 and 118.580.

For Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group.

9.395% CI 114.9 2.056 (111.220, 118.580)27

* invT 0.975, 27 1 2.056t

Robust• An inference procedure is ROBUST if the

confidence level or p-value doesn’t change much if the assumptions are violated.

• t-procedures can be used with some skewness, as long as there are no outliers.

• Larger n can have more skewness.

Example 5: A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults. (Just find the interval.)

(70.8834, 74.4965)

3.8695% CI 72.69 2.093020

* invT 0.975, 20 1t 2.0930

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain.

The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Example 6: Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:

160 200 220 230 120 180 140130 170 190 80 120 100 170

Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. (Just find the interval.)

(126.1618, 189.5524)

157.8571x 44.7521s

* invT 0.99,14 1t 2.650344.752198% CI 157.8571 2.6503

14

A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate?

Since 120 calories is not contained within the 98% confidence interval, the evidence suggests that the average calories per serving does not equal 120 calories.

Note: confidence intervals tell us if something is NOT EQUAL –

never less or greater than!

Some Cautions:• The data MUST be a SRS from the

population• The formula is not correct for more

complex sampling designs, i.e., stratified, etc.

• No way to correct for bias in data

Some Cautions (continued):• Outliers can have a large effect on

confidence interval• Must know to do a z-interval –

which is unrealistic in practice