Warm Up- Class work Activity 10 Pg.534 -each person tosses the pin 50 times

Warm Up- Class work Activity 10 Pg.534 -each pair tosses the pin 50 times total

Warm Up #2 A SRS of 500 America Adults is collected to find the average score on a game. Is this an independent sample? Would you feel comfortable “inferring” about the population using the above sample? Why? Name the 4 topics on AP Test and describe each. 3

Warm Up 1. How much does the fat content of Bran X hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard deviation σ is 2.84 to 7.55 a) Interpret the confidence interval. b) Interpret the confidence level. c) True or False: The interval from 2.84 to 7.55 has 95% chance of containing the actual population standard deviation σ. Justify your answer. 4

Answers to Warm Up a) We are 95% confident that the interval from 2.84 to 7.55 g captures the population standard deviation of the fat content of Brand X hot dogs. b) If this sampling process were repeated many times, approximately 95% of the resulting confidence intervals would capture the population standard deviation of the fat content Brand X hot dogs. c) False. Once the interval is calculated, it either contains σ or it does not contain σ. 5

6 Section 10.1 Estimating with Confidence AP Statistics

7 We say: Check for independence.

Four Main Topics on AP Test 1. Descriptive Statistics (given data describe shape, spread, max, min, etc) 2. Data Collection (unbiased sampling, types of sampling, designing experiments, etc) 3. Probability( what do we mean when something has the probability of blank? What are the rules of probability?) 4. Statistical Inference 8

9 An introduction to statistical inference Statistical Inference provides methods for drawing conclusions about a population from sample data. In other words, from looking at a sample, how much can we “infer” about the population. We may only make inferences about the population if our samples unbiased. This happens when we get our data from SRS or well-designed experiments.

10 Example A SRS of 500 California high school seniors finds their mean on the SAT Math is 461. The standard deviation of all California high school seniors on this test is 100. What can you say about the mean of all California high school seniors on this exam?

11 Example (What we know) Data comes from SRS, therefore unbiased. There are approximately 350,000 California high school seniors. 350,000>10*500. The sample mean 461 one value in the distribution of sample means.

12 Example (What we know) The mean of the distribution of sample means is the same as the population mean. Because the n>25, the distribution of sample means is approximately normal. (Central Limit Theorem)

13 Our sample is just one value in a distribution with unknown mean…

14 Confidence Interval A level C confidence interval for a parameter has two parts.  An interval calculated from the data, usually in the form (estimate plus or minus margin of error)  A confidence level C, which gives the long term proportion that the interval will capture the true parameter value in repeated samples.

15 95% (2 std.devs) of the time the true mean is captured in your interval. 95% of the intervals capture the true mean.

16 Conditions for Confidence Intervals 1. the data come from an SRS or well designed experiment from the population of interest 2. the sample distribution is approximately normal  Central Limit Theorem (n>25)  Normal Population 3. Is our sampling independent?  Population is >10n

17 Upper critical value 90 th percentile

Using the Calculator We want the Z-core associated with the lower 90%  2 nd  VARS  #3 .9  Enter You enter a percentile and it gives you a Z-value 18

19 Confidence Interval Formulas Lower end of the confidence interval Upper end of the confidence interval

20 Confidence Interval Formulas on Formula Sheets Confidence int=statistic + critical value * std. dev. of stat

21 Using the z table… Confidence level Tail Areaz* 90% % %

Four Step Process: (Inference Toolbox) Step 1 (Pop and para)  Define the population and parameter you are investigating Step 2 (Conditions)  Do we have biased data? If SRS, we’re good. Otherwise PWC.  Do we have independent sampling? If pop>10n, we’re good. Otherwise PWC.  Do we have a normal distribution? If pop is normal or n>30 using central limits theorem, we’re good. Otherwise, PWC. Step 1  California Seniors  Average SAT Scores Step 2  We have a SRS  Certainly there are more than 5000 Cali. Seniors.  n>30 22

23 Four Step Process (Inference Toolbox) Step 3 (Calculations)  Find z* based on your confidence level. If you are not given a confidence level, use 95%  Calculate CI. Step 4 (Interpretation)  “With ___% confidence, we believe that the true mean of_____ is between (lower, upper)”

Writing Inference 1.pop: para: 3.z*= (, ) 2.Bias? Independent? Normal? 4. “With ___% confidence, we believe that the true mean _______________ is between ______and _____.” 24 Margin of Error

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26 Confidence interval behavior To make the margin of error smaller…  make z* smaller, which means you have lower confidence  make n bigger

27 Confidence interval behavior If you know a particular confidence level and ME, you can solve for your sample size.

28 Example Company management wants a report screen tensions which have standard deviation of 43 megaVolts. They would like to know how big the sample has to be to be within 5 mV with 95% confidence? You need a sample size of at least 285.

29 Mantras “Interpret 80% confidence interval of (454,467)”  With 80% confidence we believe that the true mean of California senior SAT-M scores is between 454 and 467. “Interpret 80% confidence”  If we use these methods repeatly, 80% of the time our confidence interval captures the true mean.  Probability

30 Assignment Exercises 10.1 to every other odd