Chapter 5 Normal Probability Distributions

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem A.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. 1.If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means. a.Every sample statistic has a sampling distribution. 2.Remember that sample means can vary from one another and can also vary from the population mean. a.This type of variation is to be expected and is called sampling error. 3.Properties of sampling distributions of sample means: a.The mean of the sample means is equal to the population mean.

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem b.The standard deviation of the sample means is equal to the population standard deviation divided by the square root of n. 1) The standard deviation of the sampling distribution of the sample means is called the standard error of the mean. B. The Central Limit Theorem 1.The Central Limit Theorem forms the foundation for the inferential branch of statistics. a.It describes the relationship between the sampling distribution of sample means and the population that the samples are taken from. b.It is an important tool that provides the information you’ll need to use sample statistics to make inferences about a population mean.

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem 2.The Central Limit Theorem says: a.If samples of size n, where n ≥ 30, are drawn from any population with a mean μ and a standard deviation σ, then the sampling distribution of sample means approximates a normal distribution. 1)The greater the sample size (the larger number n is), the better the approximation. b.If the population itself is normally distributed, the sampling distribution of sample means is normally distributed for any sample size n. 3.Whether the original population distribution is normal or not, the sampling distribution of sample means has a mean equal to the population mean.

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem a.In real life words, this means that if we take the average of all of the means, from all of the samples that are done on one population, the mean of those averages will equal the mean of the population. 4.The sampling distribution of sample means has a variance equal to 1/n times the variance of the population. a.The standard deviation of sample means will be smaller than the standard deviation of the population. 5.The sampling distribution of sample means has a standard deviation equal to the population standard deviation divided by the square root of n. a.The distribution of sample means has the same center as the population, but it is not as spread out.

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem 1)The bigger n (the sample size) gets, the smaller the standard deviation will get. a)The more times we take a sample of the same population, the more tightly grouped the results will be. b.The standard deviation of the sampling distribution of the sample means, σₓ, is also called the standard error of the mean. C.Probability and the Central Limit Theorem 1.Using what we’ve learned in Section 5-2, and what we’ve been told here in Section 5-4, we can find the probability that a sample mean will fall in a given interval of the sampling distribution.

Chapter 5 Normal Probability Distributions Section 5-4 – Sampling Distributions and the Central Limit Theorem a.To find a z-score of a random variable x, we took the value minus the mean and divided by the standard deviation. b.To convert the sample mean to a z-score, we alter that slightly. 1)Instead of dividing by the standard deviation, we divide by the sample error. a)Remember, this is the standard deviation of the population divided by the square root of n (the sample size).

Chapter 5 Normal Probability Distributions Section 5-4 – Example 4 Page 275

Chapter 5 Normal Probability Distributions Section 5-4 – Example 4 Page 275

Chapter 5 Normal Probability Distributions Section 5-4 – Example 6 Page 277 A bank auditor claims that credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. 1)What is the probability that a randomly selected credit card holder has a credit card balance less than $2500? 2)You randomly select 25 credit card holders. What is the probability that their mean credit card balance is less than $2500? 3)Compare the probabilities from (1) and (2) and interpret your answer in terms of the auditor’s claim.

Chapter 5 Normal Probability Distributions Section 5-4 – Example 6 Page 277 A bank auditor claims that credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. 3)Compare the probabilities from (1) and (2) and interpret your answer in terms of the auditor’s claim. The probability of a single card holder owing less than $2500 is 34%, but the probability of the average of 25 card holders balances is less than $2500 is only 2%. Either the auditor is wrong about the distribution being normal, or your sample is unusual and needs to be done again, more carefully.

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