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Stat 226 SI: Sampling Distribution Supplemental Instruction Iowa State University Leader: Carly Course: Stat 226 Date: 3/4/13 Agenda: Opening Activity: Fill-in-the-blank Main Activity: Sampling Distribution Problem Closing Activity: T/F Questions Fill-In-The-Blank: 1.) Population parameters μ and σ are typically unknown values. 2.) The sample mean 𝑥̅ will following an approximately normal distribution if the sample size is large enough. 3.) The symbol for the mean of the sampling distribution of 𝑋̅ is 𝜇𝑋̅ , which is equal to μ. 4.) The standard error of the sampling distribution of 𝑋̅ is 𝜎𝑋̅ , which is equal to 5.) If 𝑋̅follows a normal distribution, it is notated by 𝑋̅~𝑁 (μ, σ2 n 𝜎 √𝑛 . ). 6.) The notation for the standard error of the sample mean distribution is SE(𝑋̅), which is equal 𝜎 to 𝑛. √ 7.) The z-score of a sampling distribution of a sample mea is calculated by 𝑍 = 𝑥̅ −𝜇 𝜎 √𝑛 . Sampling Distribution Problem: A farmer claims that, on his apple orchard, the number of apples per tree follows a normal distribution with a mean of 230 apples and a standard deviation of 20 apples. (a.) Describe the shape of the sampling distribution with a sample size of 4 trees. Normal – Because the population distribution is known to be normal, we assume the sampling distribution is also normal, regardless of the small sampling size. (b.) 𝜇 = 230 𝑎𝑝𝑝𝑙𝑒𝑠 𝜇𝑋̅ = 230 𝑎𝑝𝑝𝑙𝑒𝑠 (c.) SE(𝑋̅) = 20 √4 = 20 2 = 10 𝑎𝑝𝑝𝑙𝑒𝑠 (d.) Does the mean of the sampling distribution change if the population distribution is no longer normal? If the population distribution is no longer normal, we must apply the CLT, which says the sampling distribution will be approximately normal if the sample size is sufficiently large. If 4 trees is a sufficiently large sample size, then the mean of the sampling distribution will still be equal to the mean of the population distribution. (e.) Does the standard error of the sampling distribution change if the population distribution is no longer normal? See answer D. This also applies for the standard error. It will still be equal to the standard deviation of the population distribution divided by the sample size if a sample size of 4 trees is sufficiently large. (The benchmark number for a sufficiently large sample size is 30.) (f.) How does the standard error change if the sample size increase to 16? If the standard error quadruples in size here, the standard error decreases by 50%. 𝜎 20 20 = = = 5 𝑎𝑝𝑝𝑙𝑒𝑠 𝑛 4 √ √16 (g.) If the number of apples per tree did not follow a normal distribution, could we find the probability that a single tree has fewer than 200 apples? No. True/False Questions: If X does not follow a normal distribution, and it has mean equal to 15 and standard deviation 2 2 equal to 2 and sample size of 2, 𝑋~𝑁 (15, ( ) ). √2 FALSE 2 4 If 𝑋~𝑁(10, 42 ), we need to use the central limit theorem to assume that 𝑋̅~𝑁 (10, ( 𝑛) ). √ FALSE If X does not follow a perfectly normal distribution, but it is symmetric and bell-shaped, 2 𝜎 𝑋̅~𝑎𝑝𝑝𝑟𝑜𝑥. 𝑁 (𝜇, ( 𝑛) ). √ TRUE