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MATH 2400 Chapter 11 Notes

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Parameters & Statistics A parameter is a number that describes the population. The exact value of the parameter is usually unknown because we cannot examine the entire population. A statistic is a number that can be computed from the sample data. We often use a statistic to estimate a parameter.

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Parameters vs Statistics ParameterStatistic Meanμx̄ Standard DeviationσS When studying a population, we collect a sample and use S to approximate σ and use x̄ to approximate μ.

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Example 1 Determine whether each bolded number is a parameter or a statistic. At a local school, student birthdays were analyzed and it was determined that 34% of the students have summer birthdays and 3% had birthdays within 5 days of Christmas. In a random sample of 100 children, 28% of them had summer birthdays and 4% had birthdays within 5 days of Christmas.

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Statistical Inference Average GPA of UNG students. Average age of vehicles on the road. Average amount spent on Valentine’s Day.

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Law of Large Numbers Easy explanation… As n ∞, x̄ μ andS σ

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Example 2 Use you Ti-84 to simulate rolling a six-sided number cube 10 times. Compute your average roll. MATH, PRB tab, 5:randInt( randInt(1,6)

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Population Distributions Sampling Distributions The population distribution of a variable is the distribution of values of the variable among all the individuals in the population The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population

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Example 3 Consider the 12,000 able-bodied male undergraduates at the University of Illinois that participated in physical training during WWII. Their times for a mile run followed the Normal distribution with mean 7.11 minutes and standard deviation 0.74 minutes. A SRS of size 100 has mean 6.97 minutes. After many SRSs, the many values of the mean follow the Normal distribution with mean 7.11 minutes and standard deviation 0.074 minutes.

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Example 3 continued 1)What is the population? 2)What values does the population distribution describe? 3)What is this distribution 4)What values does the sampling distribution of x̄ describe? 5)What is the sampling distribution?

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Sampling Distribution of x̄ Suppose that x̄ is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of x̄ has mean μ, but the standard deviation is. (Note, this is the standard deviation of x̄.)

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Implications for Statistical Inference x̄ is considered to be an unbiased estimator of the parameter μ because there is no systematic tendency to overestimate or underestimate the parameter. Averages are less variable than individual observations. As the sample size of x̄ gets larger, its standard deviation gets smaller.

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Example 3 (11.9) Suppose that the blood cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean μ = 186 mg/dl and standard deviation σ = 41 mg/dl. 1) An SRS of 100 men from this population is chosen. What is the sampling distribution of x̄? 2) What is the probability that x̄ takes a value between 183 and 189 mg/dl? This is the probability that x̄ estimates μ within ±3 mg/dl.

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Example 3 continued… 3) An SRS of 1000 men is chosen from this population. What is the probability that x̄ falls within ±3 mg/dl of μ? The larger sample is much more likely to give an accurate estimate of μ.

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Hey, is this Normal? When the sample size is large enough, the distribution of x̄ is very close to Normal. This is true no mater what shape the population distribution has.

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Central Limit Theorem The following represents the annual income of 75,310 households. The mean was $71,305.

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Central Limit Theorem The following histogram represents the distribution of x̄ of many SRSs.

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Central Limit Theorem This is an enlarged version of the previous distribution to better show its shape.

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Example 4 The time (in hours) it takes a technician to perform maintenance on an air- conditioning unit is governed by the exponential distribution to the right. We know μ = 1 hour and σ = 1 hour.

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Example 4 continued… Your company is under contract to maintain 70 of these units in an apartment building. You must schedule technicians’ time for avisit to this building. Is it safe to budget an average of 1.1 hours for each unit? Or should you budget an average of 1.25 hours? 1) What is the probability that the average maintenance time for 70 units exceeds 1.1 hours? 2) 1.25 hours?

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