Warm Up Exercise

Warm Up Write down the four steps to confidence interval. 2

Warm Up Write down the four steps to confidence interval. Write down the four steps to test of Significance. 3

Warm Up 1) Many teens have posted profiles on sites such as Facebook. A sample survey asked random samples of teens with online profiles if they included false information in their profiles. Of 170 younger teens(ages 12 to 14) polled, 117 said “Yes.” Of 317 older teens (ages 15 to 17) polled, 152 said “Yes.” A 95% confidence interval for the difference in the population proportions(younger teens-older teens) is to Interpret the confidence interval and the confidence level. Interval: We are 95% confident that the interval from to captures the true difference (younger-older) in the proportions of younger teens and older teens who include false information on their profiles. Level: If this sampling process were repeated many times, approximately 95% of the resulting confidence intervals would capture the true difference(younger-older) in the proportions of young teens and older teens who include false information on their profiles. 4

A study conducted by the US Census found that in their sample of 50,000 households, the average income was $33,500 with a margin of error of $3,000 for 95% confidence. 1.What is the confidence interval for the average income of the US population? (Show steps 1-4) (Hint: you don’t need standard deviation since you have the margin or error. Use x-bar + M.E.) 2.What is the equation for margin of error? 3.Explain why this doesn’t mean that the average family in the US makes $33,500. We can be 95% confident that the mean income for the US population is between 30,500 and 36,500. $33,500 is the average for our sample and is a statistic, the parameter, μ, we expect to be between 30,500 and 36,500 with 95% confidence (meaning it should be there 95% of the time but 5% of the time it will fall outside our interval). Warm Up-

Warm Up 6

Warm Up Exercise a) See page 106 (do in calculator) b) Use tool box c) Share with your partner 7

Section 10.2 Tests of Significance AP Statistics 8

Coin Flipping Example 9

Why did you doubt Ms. Mendonca’s truthfulness? Because the outcome of the coin flipping experiment is very unlikely. How unlikely?  where k is the number of flips before you caught me. 10

Supposition (a fancy way of saying “unsupported”) Built into the argument that “Ms. Mendonca is lying to us” is a supposition What is that supposition?  “We suppose that the coin is fair and that I was telling the truth.” Where does the supposition show up? .5^k or 11

Two common statistical inference 1. Confidence intervals (sect10.1)  Use to estimate the population parameter 2. Test of significance  To assess the evidence provided by data about some claim concerning a population. 12

The Test of Significance The test of significance asks the question: 1. “Does the statistic result from a real difference from the supposition” or 2. Does the statistic result from just chance variation?” 13

Coin Flipping scenario Population – All the coin flipping I could do in the world. Sample-the 6(or 9) coin flips I did in class. Statistic – coming out of that sample was a p-hat. (a proportion that ended in heads was 6/6=1 or 9/9=1)  Did that statistic result from a real difference from supposition?  Supposition-was we assumed that I was telling the truth and that the coin was fair. 14

 We didn’t expect our statistic of 1 we expected a statistic around 0.5  So going back to “test of significance” question  Would you have been willing to say I was a liar if I got 4 heads and 2 tails?  Maybe no since this was close to the expected ½. ->Can be chance variation  (5H & 1T)? So when? At some point in our sampling our statistic seems outrageously different than the expected. When the result from the sampling is outrageously different from the supposition we call that being statistically significant. 15

Significance Test Procedure 16

Significance Test Procedure 3. If the conditions are met, carry out the inference procedure.  Calculate the test statistic. Z-value for a Z-test  Find the P-value Calculate the probability of our outcome happening by chance under the assumption that the null hypothesis was true. 4. Interpret your results in the context of the problem P-value is small enough which would imply something was wrong with the null hypothesis.(we will reject null hypothesis.) P-value is not so small then we fail to reject the null hypothesis. 17

Example Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a month at high temperature to imitate the effect of four months’ storage. Each taster scores the cola again after storage. What kind of experiment is this? Match pair design-the taster tastes the cola once then tastes the cola at a later date and then the tasters compares their tastings. 18

Example Here’s the data: 2.0,.4,.7, 2.0, -.4, 2.2, -1.3, 1.2, 1.1, 2.3 Positive scores indicate a loss of sweetness.(Old-New) What is x-bar? (the mean of our sample) Are these data good evidence that the cola lost sweetness in storage? 19

A test of significance ask Does the sample result x-bar=1.02 reflect real loss of sweetness OR Could we easily get the outcome x- bar=1.02 just by chance? 20

Significance Test Procedure Step 1: Define the population and parameter of interest. State null and alternative hypotheses in words and symbols.  Population: Diet cola.  Parameter of interest: mean sweetness loss.  Suppose there is no sweetness loss (Nothing special going on). H 0: µ=0.  You are trying to find if there was sweetness loss. Your alternate hypothesis is: H a : µ>0. 21

Significance Test Procedure Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.  We are going to use sample mean distribution Z-Test:  Do the samples come from an SRS? We don’t know. PWC.  Is the population at least ten times the sample size? Yes.  Is the population normally distributed or is the sample size at least 30. We don’t know if the population is normally distributed, and the sample is not big enough for CLT to come into play. PWC 22

Step 3: Calculate the test static and the P- value. The P-value is the probability that our sample statistics is that extreme assuming that H 0 is true.  µ=0, x-bar=1.02, σ=1  Look at H a to calculate “What is the probability of having a sample mean greater than 1.02?”  z=(1.02-0)/(1/root(10))=3.226,  P(Z>3.226) = =normalcdf(3.226,1E99) from Significance Test Procedure 23 Z-test Old New

Step 3: Calculate the test static and the P-value. The P-value is the probability that our sample statistics is that extreme assuming that H 0 is true.  µ=0, x-bar=1.02, σ=1  Look at H a to calculate “What is the probability of having a sample mean greater than 1.02?”  z=(1.02-0)/(1/root(10))=3.226,  P(Z>3.226) = =normalcdf(3.226,1E99) from Significance Test Procedure 24 Z-test Old New

Significance Test Procedure Step 4: Interpret the results in the context of the problem.  Since the P-value is so low, we reject H 0. There is significant evidence that the diet cola lost sweetness. If we get a high P-value  Since the P-value is so high, we fail to reject H 0. There is insufficient evidence that the diet cola lost sweetness. 25

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Example: Given the normal distribution… SAT Math scores w/out preparation 100 students go through a rigorous math program to increase their test scores. Is this result significant at the.01 level? (if level not given assume.05) 27

Step 3: Step 4: Because the P- value is too high, we fail to reject the null hypothesis. We have insufficient evidence to suggest that the mean SAT score were higher 28

If we know a Confidence Interval and want to do a significance test you just need to check if the interval captures the parameter. 29

Assignment Exercises odd, odd 30

Assignment Exercises odd, odd Against All Odds Video Episode

Example I say I make 80% of basketball free throws. To test my claim you ask me to shoot 20 free throws. I only make 8/20. What is the probability of this happening?  Hint: binompdf(n,p,k) 32