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College Prep Stats Chapter 7 Important Info Sheet 7.2 A point estimator is a statistic that provides an estimate of a population parameter. Point estimators that we will be using are x and pˆ . A point estimate is the value of that statistic from a sample. Ideally, a point estimate is our “best guess” at the value of an unknown parameter. 𝑥 The sample proportion pˆ is the best point estimate of the population proportion p. 𝑝̂ = 𝑛 A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI. Interpreting a Confidence Interval “We are _____% confident that the interval from _____ to _____ actually does contain the true value of the population proportion p.” The underlined portion should be written in the context of the problem. Critical Values for a Population Proportion p When finding a critical value, use the table below, or the calculator command: invNorm(area to the left, , ) Example: If the given confidence level is 86%, α = 1 – 0.86 = 0.14. Therefore, α/2 = 0.07. To find the correct critical value, find the area to the left (confidence level + α/2) = 0.86 + 0.07 = 0.93. Use the following computation in your technology: invNorn(0.93, 0, 1) Margin of Error for Proportions E z 2 ˆˆ pq n E margin of error z /2 z* critical value pˆ proportion of successes qˆ proportion of failures n sample size Confidence Interval for Estimating a Population Proportion p p = population proportion pˆ = sample proportion n = number of sample values E = margin of error z/2 (z*) = z score separating an area of /2 in the right tail of the standard normal distribution 3 Different Ways to Write a Confidence Interval for the Estimate of a Population Proportion p pˆ E p pˆ E pˆ E pˆ E, pˆ E Sample Size for Estimating Proportion p When an estimate of p̂ is known: When an estimate of p̂ is unknown: n z / 2 E 2 ˆˆ pq n 2 z / 2 2 E 0.25 2 Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. Finding the Point Estimate and E from a Confidence Interval point estimate of p : pˆ upper confidence limit lower confidence limit 2 Margin of Error: E upper confidence limit lower confidence limit 2 7.3: The sample mean x is the best point estimate of the population mean µ. Critical Values for a Population Mean µ when σ is Known See the section for Critical Values for a Population Proportion, p Margin of Error for Means (with Known) E z / 2 n 3 Different Ways to Write a Confidence Interval for Estimating a Population Mean µ (with Known) xE xE xE x E, x E Confidence Interval for Estimating a Population Mean (with Known) = population mean = population standard deviation x = sample mean n = number of sample values E = margin of error z/2 = (z*) = z score separating an area of a/2 in the right tail of the standard normal distribution Finding a Sample Size for Estimating a Population Mean = population mean σ = population standard deviation x = sample mean E = desired margin of error z/2 = (z*) = z score separating an area of a/2 in the right tail of the standard normal distribution z * n E 2 7.4 degrees of freedom = n – 1, for the student t distribution Choosing the Appropriate Distribution How do we know when to use zα/2 or tα/2 (z* or t*)? *If you are working with a categorical variable (estimating a population proportion, p) always use zα/2 (z*). *If you are working with a quantitative variable (estimating a population mean, µ) and you DO know σ, use zα/2 (z*). *If you are working with a quantitative variable (estimating a population mean, µ) and you DO NOT know σ, use tα/2 (t*). **Remember that the population distribution must be normal or n must be large for quantitative variables.** Critical Values for a Population Mean µ when σ is Not Known When finding a critical value, use the following calculator command: invT(area to the left, df) Example: If the given confidence level is 86%, with a sample size of 28, the degrees of freedom will be n – 1, so df = 27. α = 1 – 0.86 = 0.14. Therefore, α/2 = 0.07. To find the correct critical value, find the area to the left (confidence level + α/2) = 0.86 + 0.07 = 0.93. Use the following computation in your technology: invT(0.93, 27) Margin of Error E for Estimate of µ (With σ Not Known) s , where tα/2 has n – 1 degrees of freedom. NOTE: tα/2 = t* E t / 2 n Notation = population mean x = sample mean s = sample standard deviation n = number of sample values E = margin of error t/2 = t* = critical t value separating an area of /2 in the right tail of the t distribution 3 Different Ways to Write a Confidence Interval for the Estimate of a Population Mean μ (With σ Not Known) xE xE xE x E, x E Finding the Point Estimate and E from a Confidence Interval point estimate of : x upper confidence limit lower confidence limit 2 Margin of Error: E upper confidence limit lower confidence limit 2