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The Sampling Distribution of the Sample Mean Section 6.5 How Close Are Sample Means to Population Means? The sample mean, X, is a random variable. The sample mean varies from sample to sample. By contrast, the population mean, µ, is a single fixed number. Agresti/Franklin Statistics, 1e, 1 of 19 Agresti/Franklin Statistics, 1e, 2 of 19 Mean and Standard Error of the Sampling Distribution of the Sample Mean Example: How Much Do Mean Sales Vary From Week to Week? For a random sample of size n from a population having mean µ and standard deviation , the sampling distribution of the sample mean X has: σ • Center described by the mean µ (the same as the mean of the population). • Spread described by the standard error, which equals the population standard deviation divided by the square root of the sample size: σ n Daily sales at a pizza restaurant vary from day to day. The sales figures fluctuate around a mean µ = $900 with a standard deviation = $300. σ Agresti/Franklin Statistics, 1e, 3 of 19 Agresti/Franklin Statistics, 1e, 4 of 19 Example: How Much Do Mean Sales Vary From Week to Week? Example: How Much Do Mean Sales Vary From Week to Week? The mean sales for the seven days in a week are computed each week. The weekly means are plotted over time. These weekly means form a sampling distribution. Agresti/Franklin Statistics, 1e, 5 of 19 What are the center and spread (standard error) of the sampling distribution? µ = $900 σ= 300 = 113 7 Agresti/Franklin Statistics, 1e, 6 of 19 1 Sampling Distribution vs. Population Distribution Standard Error Agresti/Franklin Statistics, 1e, 7 of 19 Agresti/Franklin Statistics, 1e, 8 of 19 Standard Error Central Limit Theorem The standard error of the sample mean: As the sample size n increases, the denominator increase, so the standard error decreases. With larger samples, the sample mean is more likely to fall close to the population mean. This property applies to the sampling distribution of the sample proportion too. σ n Knowing how to find a standard error gives us a mechanism for understanding how much variability to expect in sample statistics “just by chance.” Question: How does the sampling distribution of the sample mean relate with respect to shape, center, and spread to the probability distribution from which the samples were taken? Agresti/Franklin Statistics, 1e, 9 of 19 Agresti/Franklin Statistics, 1e, 10 of 19 Central Limit Theorem Central Limit Theorem: How Large a Sample? For random sampling with a large sample size n (at least 30), the sampling distribution of the sample mean is approximately a normal 2 distribution: X ~ N µ , σ , if n ≥ 30. n This result applies no matter what the shape of the probability distribution from which the samples are taken. Agresti/Franklin Statistics, 1e, 11 of 19 The sampling distribution of the sample mean takes more of a bell shape as the random sample size n increases. • • For large n, the sampling distribution is approximately normal even if the population distribution is not. This enables us to make inferences about population means regardless of the shape of the population distribution. The more skewed the population distribution, the larger n must be before the shape of the sampling distribution is close to normal. In practice, the sampling distribution is usually close to normal when the sample size n is at least about 30. Agresti/Franklin Statistics, 1e, 12 of 19 2 Notation A Normal Population Distribution and the Sampling Distribution If the population distribution is approximately normal, then the sampling distribution is approximately normal for all sample sizes. the mean of the sample means µx = µ the standard deviation of sample mean σx = nσ (often called standard error of the mean) Agresti/Franklin Statistics, 1e, 13 of 19 Agresti/Franklin Statistics, 1e, 14 of 19 Example: Given the population of men has normally distributed weights (X) with a mean of 172 lb and a standard deviation of 29 lb, Example: Note µ=172, σ=29. a) if one man is randomly selected, find the probability that his weight is greater than 167 lb. P(X>167)=? a) if one man is randomly selected, find the probability that his weight is greater than 167 lb. P(X>167)=? b) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb. Step 1: Compute z-score of 167 P(X>167)=? z = 167 – 172 = –0.17 29 Agresti/Franklin Statistics, 1e, 15 of 19 Example: Note µ=172, σ=29. x x. / n = σ x Step 2: From table A-2, of 12 men. So z-score of 167 is z = 167 – 172 = –0.60 29 12 Example: Summary: x σ Step 1: Note that 167 is the mean We get P(X>167)=P(Z>-.17) =1-.4325 =.5675 Agresti/Franklin Statistics, 1e, 16 of 19 z − score for x is z= (b) Find P(X>167) Step 2: From table A-2. P(X>167)=P(Z>-.60) =1-.2743 =.7257 a) if one man is randomly selected, find the probability that his weight is greater than 167 lb. P(x > 167) = 0.5675 b) if 12 different men are randomly selected, their mean weight is greater than 167 lb. P(x > 167) = 0.7257 It is much easier for an individual to deviate from the mean than it is for a group of 12 to deviate from the mean. Agresti/Franklin Statistics, 1e, 17 of 19 Agresti/Franklin Statistics, 1e, 18 of 19 3 Sampling Without Replacement If n > 0.05 N σx = σ n N–n N–1 finite population correction factor Agresti/Franklin Statistics, 1e, 19 of 19 4