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Stat 217 – Day 15 Statistical Inference (Topics 17 and 18)

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Previously – Central Limit Theorem If taking random samples from a population with proportion , and the sample size is large, then the sampling distribution of the sample proportions will follow a normal distribution with mean equal to the population proportion and standard deviation equal to

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Activity 15-1 (p. 295) Probability of sample proportion at least.25 =.156 In many, many random samples of American adults, about 16% of samples have >.25

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Activity 15-1 (g) What if sample size is 400 instead of 100? A sample proportion.25 or larger is even more unlikely to come from a larger sample.

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Activity 15-1 (i) What about the population size? DOESN’T MATTER! As long as it is much, much bigger than the size of the sample (j) Virginia, =.209, probability of sample proportion exceeding.25? SAME!

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Preliminary Questions (p. 333) Which tire would you pick? Left frontRight front Left rearRight rear (driver)

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Activity 17-1

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1. Define parameter Identify the symbol and specify the (unknown) parameter in words Make sure the type of number (mean, proportion), the population of interest and variable are clear

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2. Stating hypotheses Ho: Ho-hum hypothesis Ho: parameter = hypothesized value Ha: Aha! Hypothesis Ha: parameter, ≠ hypothesized value Good practice: always state in symbols and in words

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Stating Hypotheses Direction of alternative hypothesis depends on research question Lab 1: Do infants prefer the helper? (a) Ho: =.5 (b) Ha: >.5

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3. Checking technical conditions Sample size condition Randomness condition If sample size condition is not met? If randomness condition is not met? Good practice: Include shaded and labeled sketch of the relevant sampling distribution

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4. Test statistic Standardize the observed statistic comparing it to the hypothesized value, dividing by the standard deviation of the statistic (amount of sampling variability)

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5. p-value Row-Row-Row your boat It is key to know What p-value means -- It’s the chance (with the null) you obtain data that’s At least that extreme! NOT the chance the null hypothesis is true!!

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6. Stating conclusions If you ask Ben a question he either says “no” or he says nothing. So if he says nothing, does this “prove” he agrees?

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Activity 17-1: Flat Tires (p. 334) (a) If nothing special, people will pick the right front tire 25% of the time (proportion.25) (b) parameter, (c) >.25 (d) 73(.25)=18.25>10; 73(.75)=54.75)>10.051.25 =34/73=.466 Less than a.02% chance would get a sample proportion this large if =.25 Strong evidence for >.25 2. H 0 : =.25 1. Let represent the population proportion that would pick right front H a : >.25 4. Test statistic Table II: probability above <.0002 5. p-value 6. Reject H 0 3. Technical conditions

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In conclusion 6. We have strong evidence from this sample proportion, that more than 25% of the population would choose the right front tire in this situation Caution: This was not a random sample. We may consider this sample representative of Stat 217 students in general at Cal Poly

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Test of Significance (p. 338) 1) Define population parameter in words 2) State 2 competing claims about parameter null hypothesis H 0 : parameter = value alternative hypothesis H a : parameter or ≠ value 3) Check “technical conditions” for procedure 4) Calculate test statistic (assuming H 0 true) comparing what observed to what conjectured in H 0 5) Calculate p-value (see Ha for “more extreme”) how often see sample data this extreme when H 0 true 6) Make a decision to either reject or fail to reject H 0 Is p-value small? State conclusion in context Compare p-value to “level of significance, ”

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So what next? If reject.25 as a plausible value for the , the proportion of all CP students who would pick the right front, next question might be what are plausible values for ?!

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What about.3?

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What about.4?

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Observations So we fail to reject.25 and.3 as a plausible value for Another reminder that when you fail to reject the null, that doesn’t mean you have proven the parameter has that value. but we reject.4 as a plausible value What is the set of all plausible values? Which values of the parameter will we fail to reject based on our observed sample statistic?

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The main idea Plausible values of the population parameter are values that are not “too far” from the observed sample statistic Say within 2 standard deviations When CLT applies, 95% of sample proportions should fall within 2 standard deviations of the population proportion

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Activity 16-1 (p. 312) (a) Observational units Youths (b) Variable Whether or not a television in their room (c) Parameter or statistic Statistic, p-hat (d) Want to know proportion of all American youth that have a TV in their room (e) Can’t determine it exactly (not a census) (f) But should be in the ball park (large random sample)

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Activity 16-1 To estimate the standard deviation, use the sample proportion Standard error (h) (.68(1-.68)/2032) =.0103 If we use p-hat (sample proportion) in place of (population proportion), we obtain an approximate confidence interval for .68 – 2(.0103) to.68 + 2(.0103) (.659,.701)

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Ta dah! I am 95% confident that between 65.9% and 70.1% of all American youths have a television set in their bedroom Why am I allowed to say this? Empirical rule Normal distribution Central Limit Theorem Random sample and n > 10 and n(1- )>10 Ok, use p-hat here too…

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Test of Significance Calculator

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To Turn in, with partner (n) boys or girls CI (p. 318) Calculation and interpretation (I’m 95% confident that…) For Tuesday Activities 16-3, 16-6 (with technology)

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