Statistics I MTH160 Chapter 7 Sample Variability 7.1 Sampling Distribution 7.2 The Sampling Distribution of Sample Means 7.3 Application of the Sampling Distribution of Sample Means MTH 160 Brigitte Martineau Statistics I Chapter 7 7.1 Sampling Distribution Do you think the sample mean varies from sample to sample? What is a Sampling Distribution of a Sample Statistics? The distribution of values for a sample statistics obtained from ____________ samples, all of the same __________ and all drawn from the same _____________________. Example: Consider the population of even digit on a die: {2, 4, 6} Find all samples of size 2 with their respective mean. Samples Mean What is the probability of selecting any of these samples? We say that each of these samples is ___________ ________________ Page 2 of 8 MTH 160 Brigitte Martineau Statistics I Chapter 7 Construct the sampling distribution for the sample mean of samples of size 2. Sample Means Probabilities Construct a histogram of the probability distribution. We just created a sampling distribution of sample means also called the sampling distribution of x . We could also create a sampling distribution of sample range, or sample minimum,… Let’s look at the sampling distribution of sample range Sample Ranges Probabilities Page 3 of 8 MTH 160 Brigitte Martineau 7.2 Statistics I Chapter 7 The Sampling Distribution of Sample Means The Sampling Distribution of Sample Means (SDSM) If all possible random samples, each of size n, are taken from any population with mean μ and a standard deviation σ, The mean of the sampling distribution of x is equal to The standard deviation of the sampling distribution of x is equal to If the sampled population has a normal distribution then the sampling distribution of x will The Central Limit Theorem (CLT) If the sampled population is not normal or unknown, the sampling distribution of sample means will more closely resemble… Look at p. 373 and 374 for illustrations of the CLT Page 4 of 8 MTH 160 Brigitte Martineau Statistics I Chapter 7 Examples: Suppose x has a normal distribution with mean 18 and standard deviation 3 . If we draw random samples of size 5 from the x distribution and x represents the sample mean, what can you say about the x distribution? Suppose x has a mean 75 and a standard deviation 12 but we have no information as to whether or not the x distribution is normal. If we draw random samples of size 30 from the x distribution and x represents the sample mean, what can you say about the x distribution? Suppose you did not know that x had a normal distribution. Would you be justified in saying that the x distribution is approximately normal if the sample size was n 8 Page 5 of 8 MTH 160 Brigitte Martineau 7.3 Statistics I Chapter 7 Applications of the Sampling Distribution of Sample Means When the sampling distribution of sample means is normally distributed (or approximately normal by CLT) we can answer probability question using the standard normal distribution. But how can we transform our x distribution into a z distribution? z x x x Examples A random sample of size 36 is to be selected from a population that has a mean of 50 and a standard deviation of 10. o This sample of 36 has a mean value of x that belongs to a sampling distribution. Find the shape of this sampling distribution. o Find the mean of this sampling distribution. o Find the standard error of this sampling distribution. o What is the probability that this sample mean will be between 45 and 55? Page 6 of 8 MTH 160 Brigitte Martineau Statistics I Chapter 7 o What is the probability that this sample mean will have a value greater than 48? Consider a normal population with 100 and 20 . Suppose a sample of size 16 is selected at random. o P(90 x 110) o P( x 115) o P( x 115) Page 7 of 8 MTH 160 Brigitte Martineau Statistics I Chapter 7 The amount of fill (weight of contents) put into a glass jar of spaghetti sauce is normally distributed with mean 850g and standard deviation 8g o Describe the distribution of x, the amount of fill per jar. o Describe the distribution of x , the mean weight for a sample of 24 such jars of sauce. o Find the probability that one jar selected at random contains between 848g and 855g. o Find the probability that a random sample of 24 jars has a mean weight between 848g and 855g Page 8 of 8

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