Point Estimate and Confidence Intervals for Population Proportion and Mean, Study notes of Mathematical Statistics

Download Point Estimate and Confidence Intervals for Population Proportion and Mean and more Study notes Mathematical Statistics in PDF only on Docsity! Point Estimate ?̂? = 𝑥 𝑛 ?̂? is the point estimate for the population proportion, 𝑥 is the number of successes, n is the sample size ?̂? = 𝑈𝑝𝑝𝑒𝑟 𝐵𝑜𝑢𝑛𝑑+𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 2 for the population proportion ?̅? = 𝑈𝑝𝑝𝑒𝑟 𝐵𝑜𝑢𝑛𝑑+𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 2 for the population mean 𝐸 = 𝑈𝑝𝑝𝑒𝑟 𝐵𝑜𝑢𝑛𝑑−𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 2 -> 𝐸 is the margin of error (can be used for either) A survey was given to 1500 students in a school to find the proportion of students who are interested in taking a musical course. Determine the point estimate, margin of error, and number of students who responded they would be interested in a musical course, based on the given upper and lower bounds. Lower bound = .682 Upper bound = .746 ?̂? = 𝑈𝑝𝑝𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 + 𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 2 = . 746 + .682 2 = .714 𝐸 = 𝑈𝑝𝑝𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 − 𝐿𝑜𝑤𝑒𝑟 𝐵𝑜𝑢𝑛𝑑 2 = . 746 − .682 2 = 0.032 ?̂? = 𝑥 𝑛 -> 𝑥 = ?̂? ∙ 𝑛 = .714 ∙ 1500 = 1071 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑒𝑑 Confidence Intervals Population proportion: ?̂? ± 𝑧𝑎 2⁄ ∙ √ 𝑝(1−?̂?) 𝑛 -> Use z-table Population mean: ?̅? ± 𝑡𝑎 2⁄ ∙ 𝑠 √𝑛 -> Use t-table Population variance: (𝑛−1)𝑠2 𝜒𝑎 2⁄ 2 , (𝑛−1)𝑠2 𝜒1−𝑎 2⁄ 2 -> Use chi-square table Population standard deviation: √ (𝑛−1)𝑠2 𝜒𝑎 2⁄ 2 , √ (𝑛−1)𝑠2 𝜒1−𝑎 2⁄ 2 -> Use chi-square table 𝑎 2 = 1 − 𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝐿𝑒𝑣𝑒𝑙 2 Common critical values for the POPULATION PROPORTION ONLY: For the population mean, variance, or standard deviation, the degrees of freedom (df), will also be needed to find the critical value. 𝑑𝑓 = 𝑛 − 1 Sample Size **Always round up to the next integer** Population proportion: 𝑛 = ?̂?(1 − ?̂?)( 𝑧𝑎 2⁄ 𝐸 )2 **If there are no prior estimates, then use ?̂? = .5 Population mean: 𝑛 = ( 𝑧𝑎 2⁄ ∙𝑠 𝐸 )2 Examples: A survey of 500 airline passengers found that 338 were satisfied with the service they received from the flight attendants. Calculate and interpret a 95% confidence interval for the proportion of passengers who are satisfied with the service from flight attendants.  Because we are looking for a population proportion, first we need to find the point estimate, and then we will use the z-table in our confidence interval for the critical value. ?̂? = 𝑥 𝑛 = 338 500 = .676 Now, we use ?̂? ± 𝑧𝑎 2⁄ ∙ √ 𝑝(1−𝑝) 𝑛 , and use the z-table. For a 95% confidence interval, the critical value is 1.96 Confidence Level Critical Value (Z-Score) 90% 1.645 95% 1.96 98% 2.33 99% 2.575