Estimating the Population Mean
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- Holly Stafford
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1 GOALS: 1. Understand that the sample mean is not epected to be eactly the same as the population mean. 2. Understand what a Point Estimate is. 3. Understand the differences between a statistic and a parameter. 4. Understand how a Confidence Interval improves the estimate of a population mean. Study 8.1, # 1, 17(3), 19(5), 21(7), 23(9) If we don't know the value of the Population Mean, μ, what can we use to estimate it? So far, best estimate for μ is the sample mean Can we epect the sample mean to equal μ?
2 7.1 Sampling Error μ = σ = Distributions using DOTPLOTS Samples of 2 C C J R CG CS JR J GS J RG RS GS σ = Samples of 3 CJR CJG CJS CRG CRS CGS JRG JRS JGS RGS σ = 7.10 Samples of 4 CJRG CJRS CJGS CRGS JRGS μ =279.4 σ = 4.35 n=4 n=3 n=2 n=1 σ =4.35 σ =7.10 σ =10.65 σ= Sampling Error μ = σ = n=4 n=3 n=2 n=1 Distributions using DOTPLOTS Samples of 2 C C J R CG CS JR J GS J RG RS GS σ = Samples of 3 CJR CJG CJS CRG CRS CGS JRG JRS JGS RGS σ = 7.10 σ =4.35 σ =7.10 σ =10.65 σ=17.39 Samples of 4 CJRG CJRS CJGS CRGS JRGS μ =279.4 σ =
3 The sample mean is not epected to equal the population mean. Epect SAMPLING ERROR (Difference between and μ) Goal: Reduce amount of SAMPLING ERROR Determine a measure of reliability associated with the estimate Use sample mean,, to estimate the population mean μ We call the sample mean a Point Estimate A Point Estimate of a parameter is the value of the statistic used to estimate the parameter. μ statistic parameter variable fied Point Parameter Estimate μ s σ med η
4 If μ then it lies on either side So, we look for an interval that contains μ, using, known σ, and snc distribution of CONFIDENCE INTERVAL An interval of numbers about a Point Estimate ( ) associated with a percent of confidence that the parameter lies within the interval. Area under SNC between z and +z
5 7.1 Sampling Error Samples of 2 When sampling an unknown C C J CJR R distribution, can use the sample C C mean CJG G CJS to estimate the population Smean CRG JR CRS J GS CGS J JRG RG JRS μ = RS JGS σ = GS RGS σ =10.65 Samples of σ = 7.10 Samples of 4 CJRG CJRS CJGS CRGS JRGS μ =279.4 σ = 4.35 For 2 σ to right or left, use to If took a random sample of n=3 and found = 266.0, then could consider an interval from to Does this interval include μ, the mean of the population that we are trying to estimate? 95.44% Confidence Intervals ± 2 σ 95.44% because = area under SNC between 2σ and +2σ Gertrude Battaly,
6 Estimating the population mean with: 1. sample mean and 2. known population std. dev. 3. sample size, n 95.44% Confidence Intervals ± 2 σ Distribution of sample mean algebra represents the z score for a particular significance level, α For demonstration we used ± 2σ. Most often we use α that results in 95% or 99% confidence SNC: z = μ Gertrude Battaly, 2014 popul σ 95.44% Confidence Intervals ± 2 σ Estimating the population mean with: 1. sample mean and 2. known population std. dev. 3. sample size, n α = = α / 2 = /2 = Gertrude Battaly,
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