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Sampling Distributions Martina Litschmannová martina.litschmannova@vsb.cz K210

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Populations vs. Sample A population includes each element from the set of observations that can be made. A sample consists only of observations drawn from the population. population sample Inferential Statistics Exploratory Data Analysis sampling

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Characteristic of a population vs. characteristic of a sample A a measurable characteristic of a population, such as a mean or standard deviation, is called a parameter, but a measurable characteristic of a sample is called a statistic. Population Median x 0,5 Std. deviation σ Probability π Sample Sample variance S 2 Sample std. deviation S Relative frequency p

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Sampling Distributions A sampling distribution is created by, as the name suggests, sampling. The method we will employ on the rules of probability and the laws of expected value and variance to derive the sampling distribution. For example, consider the roll of one and two dices…

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123456 1/6 The roll of one die

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A sampling distribution is created by looking at all samples of size n=2 (i.e. two dice) and their means. The roll of Two Dices The Sampling Distribution of Mean SampleMeanSampleMeanSampleMean {1, 1}1,0{3, 1}2,0{3, 1}3,0 {1, 2}1,5{3, 2}2,5{3, 2}3,5 {1, 3}2,0{3, 3}3,0{3, 3}4,0 {1, 4}2,5{3, 4}3,5{3, 4}4,5 {1, 5}3,0{3, 5}4,0{3, 5}5,0 {1, 6}3,5{3, 6}4,5{3, 6}5,5 {2, 1}1,5{4, 1}2,5{4, 1}3,5 {2, 2}2,0{4, 2}3,0{4, 2}4,0 {2, 3}2,5{4, 3}3,5{4, 3}4,5 {2, 4}3,0{4, 4}4,0{4, 4}5,0 {2, 5}3,5{4, 5}4,5{4, 5}5,5 {2, 6}4,0{4, 6}5,0{4, 6}6,0

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The roll of Two Dices The Sampling Distribution of Mean 1,01/36 1,52/36 2,03/36 2,54/36 3,05/36 3,56/36 4,05/36 4,54/36 5,03/36 5,52/36 6,01/36

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The roll of Two Dices The Sampling Distribution of Mean 1,01/36 1,52/36 2,03/36 2,54/36 3,05/36 3,56/36 4,05/36 4,54/36 5,03/36 5,52/36 6,01/36

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The roll of Two Dices The Sampling Distribution of Mean 1,01/36 1,52/36 2,03/36 2,54/36 3,05/36 3,56/36 4,05/36 4,54/36 5,03/36 5,52/36 6,01/36

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Compare Distribution of X

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Generalize - Central Limit Theorem

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Central Limit Theorem

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Generalize - Central Limit Theorem

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1.The foreman of a bottling plant has observed that the amount of soda in each “32-ounce” bottle is actually a normally distributed random variable, with a mean of 32,2 ounces and a standard deviation of 0,3 ounce. A) If a customer buys one bottle, what is the probability that the bottle will contain more than 32 ounces?

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1.The foreman of a bottling plant has observed that the amount of soda in each “32-ounce” bottle is actually a normally distributed random variable, with a mean of 32,2 ounces and a standard deviation of 0,3 ounce. B) If a customer buys a carton of four bottles, what is the probability that the mean amount of the four bottles will be greater than 32 ounces?

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Graphically Speaking What is the probability that one bottle will contain more than 32 ounces? What is the probability that the mean of four bottles will exceed 32 oz?

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2.The probability distribution of 6-month incomes of account executives has mean $20,000 and standard deviation $5,000. A) A single executive’s income is $20,000. Can it be said that this executive’s income exceeds 50% of all account executive incomes? Answer: No information given about shape of distribution of X; we do not know the median of 6-month incomes.

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2.The probability distribution of 6-month incomes of account executives has mean $20,000 and standard deviation $5,000. B) n=64 account executives are randomly selected. What is the probability that the sample mean exceeds $20,500?

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5.Cans of salmon are supposed to have a net weight of 6 oz. The canner says that the net weight is a random variable with mean =6,05 oz. and stand. dev. =0,18 oz. Suppose you take a random sample of 36 cans and calculate the sample mean weight to be 5.97 oz. Find the probability that the mean weight of the sample is less than or equal to 5.97 oz.

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Sampling Distribution of a Proportion

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Normal Approximation to Binomial

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Sampling Distribution of a Sample Proportion standard error of the proportion

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6.Find the probability that of the next 120 births, no more than 40% will be boys. Assume equal probabilities for the births of boys and girls.

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7.12% of students at NCSU are left-handed. What is the probability that in a sample of 50 students, the sample proportion that are left-handed is less than 11%?

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Sampling Distribution: Difference of two means

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standard error of the difference between two means

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Sampling Distribution: Difference of two proportions

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Sampling Distribution: Difference of two means standard error of the difference between two proportions

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Special Continous Distribution

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Degrees of Freedom

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8.The Acme Battery Company has developed a new cell phone battery. On average, the battery lasts 60 minutes on a single charge. The standard deviation is 4 minutes. Suppose the manufacturing department runs a quality control test. They randomly select 7 batteries. What is probability, that the standard deviation of the selected batteries is greather than 6 minutes?

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Student's t Distribution

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9.Acme Corporation manufactures light bulbs. The CEO claims that an average Acme light bulb lasts 300 days. A researcher randomly selects 15 bulbs for testing. The sampled bulbs last an average of 290 days, with a standard deviation of 50 days. If the CEO's claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than 290 days?

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F Distribution The f Statistic The f statistic, also known as an f value, is a random variable that has an F distribution. Here are the steps required to compute an f statistic: Select a random sample of size n 1 from a normal population, having a standard deviation equal to σ 1. Select an independent random sample of size n 2 from a normal population, having a standard deviation equal to σ 2. The f statistic is the ratio of s 1 2 /σ 1 2 and s 2 2 /σ 2 2.

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F Distribution Degrees of freedom

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10.Suppose you randomly select 7 women from a population of women, and 12 men from a population of men. The table below shows the standard deviation in each sample and in each population. Find probability, that sample standard deviation of men is greather than twice sample standard deviation of women. Population Population standard deviation Sample standard deviation Women3035 Men5045

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