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Sampling Distributions

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  1. SamplingDistributions Martina Litschmannová martina.litschmannova@vsb.cz K210

  2. 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. Exploratory Data Analysis sampling sample population InferentialStatistics

  3. Characteristicof a populationvs. characteristicof 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.

  4. 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…

  5. Therollofonedie • A fair die is thrown infinitely many times,with the random variable X = # of spots on any throw. • The probability distribution of X is: • The mean, variance and standard deviation are calculated as: ,

  6. Theroll of Two DicesTheSamplingDistributionofMean • A sampling distribution is created by looking atall samples of size n=2 (i.e. two dice) and their means.

  7. Theroll of Two DicesTheSamplingDistributionofMean • A sampling distribution is created by looking atall samples of size n=2 (i.e. two dice) and their means. • While there are 36 possible samples of size 2, there are only 11 values for , and some (e.g. ) occur more frequently than others (e.g. ).

  8. Theroll of Two DicesTheSamplingDistributionofMean • A sampling distribution is created by looking atall samples of size n=2 (i.e. two dice) and their means. ,

  9. Theroll of Two DicesTheSamplingDistributionofMean • A sampling distribution is created by looking atall samples of size n=2 (i.e. two dice) and their means. • = # of spots on i-thdice, • ,

  10. Compare • = # of spots on i-thdice, , Notethat: , Distributionof X Sampling Distributionof

  11. Generalize - Central Limit Theorem • The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size. , , • The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution.

  12. Central Limit Theorem • … randomvariable, , • Notethat: , , … standard error SameDistributionofall Sampling Distributionof

  13. Generalize - Central Limit Theorem • The sampling distribution of drawn from any population is approximately normal for a sufficiently large sample size. • In many practical situations, a sample size of 30 may be sufficiently large to allow us to use the normal distribution as an approximation for the sampling distribution of . • Note: If X is normal, is normal. We don’tneed Central Limit Theorem in this case.

  14. 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?

  15. 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?

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

  17. 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-monthincomes.

  18. 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?

  19. A sample of size n=16 is drawn from a normally distributed population with and . Find (Do we need the Central Limit Theorem to solve ?)

  20. Battery life . Guarantee: avg. battery life in a case of 24 exceeds 16 hrs. Find the probability that a randomly selected case meets the guarantee.

  21. 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. Since , eitheryou observed a “rare” event (recall: 5,97 oz is 2,67 stand. dev. below the mean) and the mean fill is in fact 6,05 oz. (the value claimed by the canner), the true mean fill is less than 6,05 oz., (the canner is lying ).

  22. Sampling Distribution of a Proportion • The estimator of a population proportion of successes is the sample proportion . That is, we count the number of successes in a sample and compute: • X is the number of successes, n is the sample size.

  23. Normal Approximation to Binomial • Binomial distribution with n=20 and with a normal approximation superimposed ( and).

  24. Normal Approximation to Binomial • Normal approximation to the binomial works best when the number of experimentsn(sample size) is large, and the probability of successis close to 0,5. • For the approximation to provide good results one condition should be met: .

  25. Sampling Distribution of a Sample Proportion • Using the laws of expected value and variance, we can determine the mean, variance, and standard deviation of . , , • Sample proportions can be standardized to a standard normal distribution using this formulation:. standard error of the proportion

  26. 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.

  27. 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%?

  28. Sampling Distribution: Difference of two means Assumption: Independent random samples be drawn from each of twonormal populations. • If this condition is met, then the sampling distribution of the difference between the two sample means will be normally distributed if the populations are both normal. • Note: If the two populations are not both normally distributed, but the sample sizes are “large” (>30), the distribution of is approximately normal – Central Limit Theorem.

  29. Sampling Distribution: Difference of two means , standard errorofthedifferencebetweentwomeans

  30. Sampling Distribution: Difference of two proportions Assumption: Central Limit Theorem: ,

  31. Sampling Distribution: Difference of two means , standard errorofthedifferencebetweentwoproportions

  32. SpecialContinousDistribution

  33. Distribution , pak DegreesofFreedom UsingofDistribution

  34. 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. Whatis probability, that the standard deviation of the selected batteries is greatherthan6 minutes?

  35. Student's t Distribution , and are independent variables If, thenhas Student‘s t Distributionwithdegreesoffreedom, . UsingofStudent‘stDistribution The t distribution should be used with small samples from populations that are not approximately normal.

  36. 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?

  37. 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 n1 from a normal population, having a standard deviation equal to σ1. • Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2. • The f statistic is the ratio of s12/σ12 and s22/σ22.

  38. F Distribution Here are the steps required to compute an f statistic: • Select a random sample of size n1 from a normal population, having a standard deviation equal to σ1. • Select an independent random sample of size n2 from a normal population, having a standard deviation equal to σ2. • The f statistic is the ratio of s12/σ12 and s22/σ22. Degreesoffreedom

  39. Supposeyourandomlyselect 7 womenfrom a populationofwomen, and 12 menfrom a populationofmen. The table belowshowsthe standard deviation in each sample and in eachpopulation. • Find probability, that sample standard deviationofmenisgreatherthantwice sample standard deviationofwomen.

  40. Study materials : • http://homel.vsb.cz/~bri10/Teaching/Bris%20Prob%20&%20Stat.pdf (p. 93 - p.104) • http://stattrek.com/tutorials/statistics-tutorial.aspx?Tutorial=Stat (Distributions – Continous (Students, F Distribution) + Estimation