Understanding Sampling Distributions: Calculations & Insights

Name:____Tyler Griffith_______________________ Worksheet on Sampling Distributions 1. Let the random variable, X, represent Cholesterol levels of adults in the US. Let’s suppose that we know that X follows a normal distribution with a Mean (µ) of 200 mg/dl and a standard deviation (σ) of 35 mg/dl. a. Begin by selecting one individual from the population. Calculate P( 165 ≤ X ≤ 225). Draw a normal curve to represent your calculation and then find the probability. 0.6038195529 P=0.60 b. Find the 2 cut-of points that represent the middle 90% of these cholesterol values. Draw a normal curve to represent your calculation and then find the cut-off points. Lower cut off 142 Upper cut off 258

c. Next suppose you take a sample of 50 individuals and calculate a sample mean for your sample. This sample mean is one possible sample mean. The set of all possible sample means from samples of size n = 50 form what is called the Sampling Distribution of the Sample Mean. Describe the Sampling Distribution of the Sample Mean. Center: he center of the sampling distribution of the sample mean is equal to the population mean, which is 200 mg/dl in this case. Spread: Spread: The spread of the sampling distribution of the sample mean is given by the standard error of the mean (SE), which is the population standard deviation divided by the square root of the sample size. In this case, the standard error (SE) is 35/Sqrt50 mg/dl. Shape: The sampling distribution of the sample mean is approximately normally distributed, especially when the sample size is reasonably large (as indicated by the Central Limit Theorem). d. What is the probability of obtaining a sample mean of 185 mg/dl or higher from you sample of 50 individuals ? Draw a normal curve to represent your calculation and then find the probability. P=0.9988

2. Data from the CDC states that the average time it takes a 13-year-old girl to run a mile is 11 minutes with a standard deviation of 2 minutes. The data follows a normal distribution. a. Calculate the probability that a randomly selected individual 13-year- old girl could run a mile in 9.5 minutes or less. Would you be surprised if you selected one individual girl and her time to run a mile is 9.5 minutes? Why or why not? Draw and label a normal curve for this situation. 0.2266 No, I would not be surprised if her to time to run is 9.5 minutes because the probability is not unsual. b. Suppose a random sample of 30 13-year-old girls is selected. Calculate the probability that the sample average (x-bar) is 9.5 minutes or less. Would you be surprised if your sample average for this group of 30

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girls is 9.5 minutes? Why or why not? Draw and label a normal curve for this situation. 0.0000 Yes I would be surprised because the probability is very low meaning it is an unusual event. .

6_SamplingDistributions

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