Lesson 9 - 1

Logic in Constructing Confidence Intervals about a Population Mean where the Population Standard Deviation is Known

Objectives

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Compute a point estimate of the population mean

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Construct and interpret a confidence interval about the population mean (assuming population σ is known)

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Understand the role of margin of error in constructing a confidence interval

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Determine the sample size necessary for estimating the population mean within a specified margin of error

Vocabulary

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Point Estimate – value of a statistic that estimates the value of a population parameter

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Confidence Interval – for an unknown parameter is an interval of numbers (that the unknown falls between)

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Level of Confidence – represents the expected proportion of intervals that will contain the parameter if a large number of samples is obtained. The level of confidence is denoted by (1 α) * 100%

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α – represents the percentage the parameter falls outside the confidence interval (later known as a Type I error)

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Robust – minor departures from normality do not seriously affect results

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Z-interval – confidence interval

Confidence Interval Estimates

Point estimate (PE) ± margin of error (MOE) Point Estimate Sample Mean for Population Mean Sample Proportion for Population Proportion Expressed numerically as an interval [LB, UB] where LB = PE – MOE and UB = PE + MOE Graphically: MOE PE _ x MOE

Margin of Error Factors

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Level of confidence : as the level of confidence increases the margin of error also increases

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Sample size : as the sample size increases the margin of error decreases (from Law of Large Numbers)

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Population Standard Deviation : the more spread the population data, the wider the margin of error

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MOE is in the form of measure of confidence • standard dev / sample size MOE PE _ x MOE

Margin of Error, E

The margin of error, E, in a (1 – α) * 100% confidence interval in which σ is known is given by E = z α/2 σ -- n where n is the sample size and z α/2 is the critical z-value.

Note: The sample size must be large (n ≥ 30) or the population must be normally distributed.

Using Standard Normal

Measure of Confidence

Z critical value: A value of the Z-statistic that corresponds to α/2 (1/2 because of two tails) for an α level of confidence Level of Confidence (1-α) 90% 95% 99% Area in each Tail (α/2) 0.05

0.025

0.005

Critical Value Z α/2 1.645

1.96

2.575

MOE PE _ x MOE

Interpretation of a Confidence Interval

A (1 α) * 100% confidence interval indicates that , if we obtained many simple random samples of size n from the population whose mean, μ, is unknown, then approximately (1 α) * 100% of the intervals will contain μ. Note that is not a probability, but a level of the statistician’s confidence.

Assumptions for Using Z CI

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Sample: simple random sample

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Sample Population: sample size must be large (n ≥ 30) or the population must be normally distributed. Dot plots, histograms, normality plots and box plots of sample data can be used as evidence if population is not given as normal

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Population σ: known (If this is not true on AP test you must use t-distribution!)

A (1 – α) * 100% Confidence Interval about μ, σ Known

Suppose a simple random sample of size n is taken from a population with an unknown mean μ and known standard deviation σ. A (1 – α) * 100% confidence interval for μ is given by Lower bound = x – z α/2 σ -- n Upper bound = x + z α/2 σ -- n where z α/2 is the critical z-value.

Example 1

We have test 40 new hybrid SUVs that GM is resting its future on. GM told us the standard deviation was 6 and we found that they averaged 27 mpg highway. What would a 95% confidence interval about average miles per gallon be?

PE ± MOE X-bar ± Z 1 α/2 σ / √n 27 ± (1.96) (6) / √40 LB = 25.141 < μ < 28.859 = UB 95% confident that the true average mpg ( μ) lies between LB and UB

Example 2

GM told us the standard deviation for their new hybrid SUV was 6 and we wanted our margin of error in estimating its average mpg highway to be within 1 mpg. How big would our sample size need to be?

(Z 1 α/2 σ)² n = ------------ MOE² MOE = 1 n = (Z 1 α/2 σ )² n = (1.96∙ 6 )² = 138.3

n = 139

Summary and Homework

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Summary

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We can construct a confidence interval around a point estimator if we know the population standard deviation

σ

The margin of error is calculated using σ, the sample size n, and the appropriate Z-value

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We can also calculate the sample size needed to obtain a target margin of error

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Homework

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pg 458 – 465; 1-3, 10, 11, 19, 21, 23, 26, 37, 40, 47

Homework

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1: sample size, confidence level, and standard deviation 2: we widen our interval to become more confident that the true value is in there 3: the greater the sample size the more the law of large numbers helps assure the point estimate is closer to the population parameter 10: No, it does not look normal -- normality plot questionable 11: Yes, normality plot ok and box plot symmetric-like 19: σ= 13, x-bar=108, n=25 21 23 26: PE=103.4 minutes, c)(98.2,108.6) d) (99.0,107.8) e) decreased 37: 40: n=670 47: 4 times