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Statistics

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Feb 20, 2024

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docx

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5.1 Confidence Interval for Population Mean (Population Standard Deviation σ is Known) Suppose we want to know the mean (average) for a large population but have difficulty obtaining all the population data. We may conduct a survey to obtain sample data to estimate the population mean. Confidence level For the standard normal distribution, 95% confidence level corresponds the middle 95% area as follows: The total area is 100% or 1. Then the sum of the remaining area (two tails) is: α = 1 – 0.95 = 0.05. Hence Each tail area is α/2 = 0.025. In general, the middle area can be denoted as CL (confidence level). The two tail areas are denoted α/2:

Critical Values For CL = 95%, we can use the z-table to find the corresponding z-scores (or critical values) -1.96 and 1.96 as shown in the following: z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 : -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 The often-used CL are 90%, 95% and 99%. Their critical values are as follows: z 0.05 (or 0.04) z 0.06 z 0.08 (or 0.07) -1.6 0.05 -1.9 0.025 -2.5 0.005 In general, we use -z α/2 and z α/2 to denote the critical values:

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