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Statistical Analysis and Estimation Techniques: Confidence Intervals, Point Estimates, and Population Proportions

This assignment provides solutions for estimation techniques using confidence intervals and population proportions.

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    Statistical Analysis and Estimation Techniques: Confidence Intervals, Point Estimates, and Population Proportions 1. A random sample of 10 economists produced the following forecasts for percentage growth in real domestic product in the next year: 2.2 2.8 3.0 2.5 2.4 2.6 2.5 2.4 2.7 2.6 Use unbiased estimation procedures to find point estimates for the following: a. The population mean? 1 1 ˆ 25.7 2.57 10 x x n        b. The population variance?     2 2 2 2 2 1 1 ˆ 66.51 25.7 10 0.0512 1 10 1 s x x n n                      c. The variance of the sample mean 2 2 0.0512 ˆ 0.00512 10 x s n     d. The population proportion of economists predicting growth of at least 2.5% in real gross domestic product .     Here, Number of economists predicting growth of at least 2.5% in the sample 7 Sample size 10 The estimate of population proportion of economists predicting growth of at least 2.5% in real gross domestic X n   product, 7 ˆ 0.7 10 X p n    e. The variance of the sample proportion of economists predicting growth of at least 2.5% in real gross domestic product .

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          2 ˆ Here, Sample size 10 ˆ 0.7 1 0.7 1 0.7 ˆ 0.021 10 p n p p p n         2. A college admissions officer for an MBA program has determined that historically applicants have undergraduate grade point averages that are normally distributed with standard deviation 0.45. From a random sample of 25 applications from the current year, the sample mean grade point average is 2.90. a. Find a 95% confidence interval for the population mean?                2 0.025 Here, Sample size 25 Sample Mean 2.90 Population standard deviation 0.45 The 95% confidence interval on population mean is: 0.45 2.90 25 2.90 1.96 0.09 2.90 0.18 2.7 n x x Z Z n                                          2,3.08 b. Without doing calculations, explain whether a 99% confidence interval for the population mean would be wider than, narrower than, or the same width as found in part ( a ) . The 99% confidence interval for the population mean would be wider than as found in part ( a ), since the critical value increases with the increase in confidence level. c. Suppose that population standard deviation is 0.56 (instead of 0.45), without doing calculations explain whether a 95% confidence interval for the population mean would be wider than, narrower than, or the same width as found in part (a). With a population standard deviation of 0.56 , the 95% confidence interval for the population mean would be wider than as found in part (a), since the width is directly proportional to the standard deviation.

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