1342 Notes Chapter 8 Answers 2019

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1 Chapter 8 - Sampling Distributions Answers Section 8.1 - Distribution of the Sample Mean Statistics such as x are random variables since their value varies from sample to sample. As such, they have probability distributions associated with them. Sampling Distribution of a Statistic is a probability distribution for all possible values of the statistic computed from a sample of size n . Example: The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 31, 18, 5, 26, 43 Population Mean: μ = 26 Samples of size 3: 5, 18, 23 5, 23, 31 18, 26, 36 31, 36, 43 Sample means: x = 15.7 x = 19.7 x = 26.7 x = 36.7 Sampling Distribution of the Sample Mean, x , is a probability distribution of all possible values of the sample mean computed from a sample of size n from a population with mean μ and standard deviation . Properties of the sampling distribution of the mean: 1. The sample means target the value of the population mean. (That is, the mean of the sample means is the population mean. The expected value of the sample mean is equal to the population mean.) 2. The distribution of sample means tends to be a normal distribution. (The distribution tends to become closer to a normal distribution as the sample size increases.) 3. As the size of the sample increases, the standard deviation of the distribution of the sample mean decreases. The Mean and Standard Deviation of the Sampling Distribution of x : Suppose that a simple random sample of size n is drawn from a large population with mean μ and standard deviation . The sampling distribution of x will have mean x = , standard deviation x n = . The standard deviation of the sampling distribution of x , X , is called the standard error of the mean .
2 Example: The weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 grams and standard deviation 0.02 grams. Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 5 from this population. The data represent the sample means for the 200 simple random samples of size n = 5. For example, the first sample of n = 5 had the following data: 2.493 2.466 2.473 2.492 2.471 Note: x = 2.479 for this sample Sample Means for Samples of Size n = 5 The mean of the 200 sample means is 2.46, the same as the mean of the population. The standard deviation of the sample mean is 0.0086, which is smaller than the standard deviation of the population. What role does n , the sample size, play in the standard deviation of the distribution of the sample mean? Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n = 20 from the population of weights of pennies minted after 1982. ( μ = 2.46 grams and σ = 0.02 grams) The mean of the 200 sample means for n = 20 is still 2.46, but the standard deviation is now 0.0045 (s = 0.0086 for n = 5). As expected, there is less variability in the distribution of the sample means with n = 20 than with n = 5.
3 Example: Determine x x and from the given parameters of the population and the sample size. = 64, = 18, n = 36 3 36 18 and 64 x x = = = •When working with an individual value from a normally distributed population, use . •When working with a mean for some sample (group) , use x n = . Example: For women aged 18-24, the systolic blood pressures are normally distributed with mean 114.8 and standard deviation 13.1. a. If you select one woman, find the probability that her systolic blood pressure is greater than 120. P(x > 120) normalcdf(120, 1E99, 114.8, 13.1) = 0.3457 b. If you select 36 women, find the probability that their mean systolic blood pressure is greater than 120. P( x > 120) normalcdf(120, 1E99, 114.8, 13.1/ 36 ) = 0.0086 What about the shape of the distribution? If the original population is normally distributed, then the distribution of the sample mean will be normally distributed, regardless of the sample size, n . What if the original population is not normally distributed or the shape is unknown ? The Central Limit Theorem: the shape of the distribution of the sample mean becomes approximately normal as the sample size n increases, regardless of the shape of the population. In other words, if the sample size is large enough, the distribution of sample means can be approximated by a normal distribution, even if the original population is not normally distributed. The Central Limit Theorem involves two different distributions: the distribution of the original population and the distribution of the sample means.
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