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7.2a HW: pg 431: 21 – 24; pg 439: 27, 29, 33, 35, 37, 41

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## 7.2a HW: pg 431: 21 – 24; pg 439: 27, 29, 33, 35, 37, 41

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**Sample ProportionsTarget Goal: I canFIND the mean and**standard deviation of the sampling distribution of a sample proportion.DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion.CALCULATE probabilities involving the sample proportion.EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion. 7.2a HW: pg 431: 21 – 24; pg 439: 27, 29, 33, 35, 37, 41**What is the proportion of teens that know that Columbus**discovered America in 1492? Gallop pollfound that 210 out of a random sample of 501 teens knew this. = 210/501 = 0.42**Recall:**• The sample proportion is the statistic that we use to gain information about the unknown parameter p. • The parameter is the population proportion. In practice, this value is always unknown. (If we know the population proportion, then there is no need for a sample.) • The statistic is the sample proportion. • We use to estimate the value of .**The value of the statistic changesas the sample changes.**We describe the sampling model for by • Shape • Outliers • Center • Spread**If our sample is an SRS of size n, then the following**statements could describe the sampling model for : • The Shape is approximately normal. ASSUMPTION: Sample size is sufficiently large. CONDITION: “Rule of Thumb 2”– Use the normal approximation when**The center is described by the mean. Is**• The spread is described by standard deviation. is or ASSUMPTION:Sample size is sufficiently large. CONDITION: “Rule of Thumb 1”– Use only when the population is at least 10 times as large as the sample. As sample size increases, the spread decreases.**Ex: Applying to Colleges**A SRS of 1500 first year college students were asked if they applied to any other college. 35% of all first year students said they applied to other colleges. Determine the probability that the SRS of 1500 will give a result with in 2 percentage points of this true value.**What does this mean “with in 2 percentage points of its**true value” ? We want find the probability that falls between 0.33 and 0.37. n = 1500, = 0.35 The sampling distribution has a mean = = 0.35**Find :“rule of thumb 1” says we must have;**Test: Is thepopulation is at least 10 times as large as the sample? 10(1500) = 15,000 Yes, the population of first year students is over 1.7 million**is =**= 0.0123**Is the distribution approximately normal?**“Rule of Thumb 2” – Use the normal approximation when Test: = 525= 975 Yes, approximately normal.**Step 1:Standardize(create new statistic)**P(0.33 ≤≤ 0.37)**Step2:Find z-scoresof= 0.33 and = 0.37**P(-1.63 ≤ z≤ 1.63)**Step3:Draw a picture.Use Table A or normcdf to find desired**area. normcdf(-1.63,1.63) = .8968**State results:**Almost 90% of all samples will give a result w/in 2 percentage points of the truth about the population. In Context: 90% of samples of first year college students stated they applied to other colleges will give a result within 2 % points of the true proportion.**Exercise:Do You Drink the Cereal Milk?**A USA Today poll asked a random sample of 1012 U.S. adults what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 67% said that they drink it. Suppose that 70% of U.S. adults actually drink the milk. n = 1012 p = 0.70**a) Find the mean and standard deviation of the proportion**of the sample that say they drink the cereal milk. μ = p = 0.70 = 0.0144 Why did we use .70 instead of .67? This time we knew the population proportion.**Explain why we can use the formula for the standard**deviation of in this setting.(rule of thumb 1). Clearly the U.S. population is 10 times greater than 1012. • Can we use the normal approximation?(rule of thumb 2) Check: np = 1012(0.70) ≥ 10? n(1-p) = 1012(0.3) ≥ 10? Yes.**Find the probability of obtaining a sample of 1012 adults in**which 67% or fewer say they drink the milk.Do we have any doubts about the result of this poll? 1) Find P( ≤ 0.67) Draw a picture 2) Find the z-score = 0.0188 This is a fairly unusual result if 70% of the population actually drinks the milk.**e) What sample size would be required to reduce the**standard deviation of the sample proportion to one-half the value we found in a)? Just multiply the sample size by 4 or**sx**If multiplying sample size by 4: 4n = 4(1012) ½ = and solve for n = 4048 adults As the sample size increases, thestandard deviation of the sampling distribution decreases. as n**f) If the pollsters had polled teenagers instead of**adults,do you think the sample proportion greater than , equal to, or less than 0.67? Explain: It should be greater than since children tend to drink more milk than adults.