Select three random samples of 5 days Calculate the mean rooms rented for each

# Select three random samples of 5 days calculate the

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the population. Select three random samples of 5 days. Calculate the mean rooms rented for each sample and compare to the population mean. What is the sampling error in each case? Population mean, 𝜇 = Σ𝑥 𝑁 = 0+2+3…+3 30 = 3.13 Example: Sampling Error Random sample #1 : 4, 7, 4, 3 and 1 𝑥 1 = Σ𝑥 𝑁 = 4 + 7 + 4 + 3 + 1 5 = 3.8 Sampling error #1, 𝑥 1 − 𝜇 = 3.8 − 3.13 = 0.67 Random sample #2 : 3, 3, 2, 3 and 6 𝑥 2 = Σ𝑥 𝑁 = 3 + 3 + 2 + 3 + 6 5 = 3.40 Sampling error #2, 𝑥 2 − 𝜇 = 3.4 − 3.13 = 0.2 Example: Sampling Error Random sample #3 : 0, 0, 3, 3 and 3 𝑥 3 = Σ𝑥 𝑁 = 0 + 0 + 3 + 3 + 3 5 = 1.8 Sampling error #3, 𝑥 3 − 𝜇 = 1.8 − 3.13 = −1.33 +ve error indicate that the sample mean overestimate the population mean, and ve error underestimate the population mean Possible combination, 𝑛 𝐶 𝑟 = 30! 5! 25! = 142506 Sampling error for all 142,506 samples = 0 unbiased Sampling Distribution of Sample Mean How to determine how accurate the sample mean is? SAMPLING DISTRIBUTION OF THE SAMPLE MEAN A probability distribution of all possible sample means of a given sample size For a given sample size, the mean of all possible sample means selected from a population is equal to the population mean There is less variation in the distribution of the sample mean than in the population distribution The sampling distribution of the sample mean tends to become bell-shaped Example: Sampling Distribution Tartus Industries has 7 production employees (the population). The hourly earnings of each employee is as below: Employee Hourly Earnings Employee Hourly Earnings Joe \$14 Jan 14 Sam 14 Art 16 Sue 16 Ted 18 Bob 16 1. What is the population mean? 2. What is the sampling distribution of the sample mean for samples of size 2? 3. What is the mean of the sampling distribution? 4. What observations can be made about the population and the sampling distribution? Example: Sampling Distribution 1. What is the population mean? 𝜇 = Σ𝑥 𝑁 = \$14+14+16+16+14+16+18 7 = \$15.43 2. What is the sampling distribution of the sample mean for samples of size 2? 7 𝐶 2 = 7! 2! 5! = 21 The 21 sample means from all possible samples of 2 are shown in the table. These 21 sample means are used to construct a probability distribution Sampling Distribution Example Sample Employees Hourly Earnings Sum Mean Sample Employees Hourly Earnings Sum Mean 1 Joe, Sam \$14,\$14 \$28 \$14 12 Sue, Bob 16,16 32 16 2 Joe, Sue 14,16 30 15 13 Sue, Jan 16,14 30 15 3 Joe, Bob 14,16 30 15 14 Sue, Art 16,16 32 16 4 Joe, Jan 14,14 28 14 15 Sue, Ted 16,18 34 17 5 Joe, Art 14,16 30 15 16 Bob, Jan 16,14 30 15 6 Joe, Ted 14,18 32 16 17 Bob, Art 16,16 32 16 7 Sam, Sue 14,16 30 15 18 Bob, Ted 16,18 34 17 8 Sam, Bob 14,16 30 15 19 Jan, Art 14,16 30 15 9 Sam, Jan 14,14 28 14 20 Jan, Ted 14,18 32 16 10 Sam, Art 14,16 30 15 21 Art, Ted 16,18 34 17 11 Sam, Ted 14,16 32 16 Sampling Distribution Example Sample Mean Number of Means Probability \$14 3 .1429 15 9 .4285 16 6 .2857 17 3 .1429 21 1.0000 3. What is the mean of the sampling distribution? 𝜇 ҧ𝑥 = Sum of all sample means Total # samples = \$14+15+15+⋯+16+17 21 = \$15.43 