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Section 5.4

Section 5.4. Sampling Distributions and the Central Limit Theorem. Sampling Distributions.

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Section 5.4

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  1. Section 5.4 Sampling Distributions and the Central Limit Theorem

  2. Sampling Distributions • Definition 1:A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means.

  3. Sampling Distributions • Properties of Sampling Distributions of Sample Means: • The mean of the sample means is equal to the population mean µ. • The standard deviation of the same means is equal to the population standard deviation divided by the square root of n. The standard deviation of the sampling distribution of the sample means is called the standard error of the mean.

  4. The Central Limit Theorem • If samples of size n, where n ≥ 30, are drawn from any population with a mean and a standard deviation , then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation. • If the population itself is normally distributed, the sampling distribution of sample means is normally distributed for any sample size n. • In either case, the sampling distribution of sample means has a mean equal to the population mean. • The sampling distribution of sample means has a variance equal to 1/n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n. • The standard deviation of the sampling distribution of the sample means is also called the standard error of the mean.

  5. Example 1 Phone bills for residents of a city have a mean of $64 and a standard deviation of $9, as shown in the following graph. Random samples of 36 phone bills are drawn from this population, and the mean of each sample is determined. Find the mean and the standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution of sample means.

  6. Example 2 The heights of fully grown white oak trees are normally distributed, with a mean of 90 feet and standard deviation of 3.5 feet, as shown in the following graph. Random samples of size 4 are drawn from this population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch the graph of the sampling distribution of sample means.

  7. Example 3 • The mean room and board expense per year at a four-year college is $5850. You randomly select 9 four-year colleges. What is the probability that the mean room and board is less than $6180? Assume that the room and board expenses are normally distributed, with a standard deviation of $1125. • There is an 81.06% chance that a sample will have a mean less than $6180.

  8. Example 4 • The average sales price of a single-family house in the U.S. is $243,756. You randomly select 12 single-family houses. What is the probability that the mean sales price is more than $200,000? Assume that the sales prices are normally distributed with a standard deviation of $44,000. • There is a 99.97% chance that a sample of homes will have a mean price of more than $200,000.

  9. Example 5 • A bank auditor claims that credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. • What is the probability that a randomly selected credit card holder has a credit card balance less than $2500? • The probability that a card holder’s balance is less than $2500 is 34.09%. • You randomly select 25 credit card holders. What is the probability that their mean credit card balance is less than $2500? • The probability that the mean credit card balance of 25 card holders is less than $2500 is 1.97%. • Compare probabilities from (1) and (2) and interpret your answer in terms of the auditor’s claim. • Although there is a 34% chance that an individual will have a balance less than $2500, there is only a 2% chance that the mean of a sample of 25 will have a balance less than $2500. Because there is only a 2% chance that the mean of a sample of 25 will have a balance less than $2500, this is a rare occurrence. So, it is possible that the sample is unusual, or it is possible that the auditor’s claim that the mean is $2870 is incorrect.

  10. TOTD • A consumer price analyst claim that prices for sound-system receivers are normally distributed, with a mean of $625 and a standard deviation of $150. (1) What is the probability that a randomly selected receiver costs less than $700? (2) You randomly select 10 receivers. What is the probability that their mean cost is less than $700? (3) Compare these two probabilities.

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