Lesson 9 - 1

Logic in Constructing Confidence Intervals about a Population Mean

where the Population Standard Deviation is Known

Objectives• Compute a point estimate of the population mean

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

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

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

Vocabulary• Point Estimate – value of a statistic that estimates the value of a

population parameter

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

• 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%

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

• Robust – minor departures from normality do not seriously affect results

• 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:PE

_x

MOEMOE

Margin of Error Factors

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

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

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

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

PE

_x

MOEMOE

Margin of Error, E

The margin of error, E, in a (1 – α) * 100% confidence interval in which σ is known is given by

where n is the sample size and zα/2 is the critical z-value.

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

σE = zα/2 --- n

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 

PE

_x

MOEMOE

Level of Confidence (1-α)

Area in each Tail (α/2)

Critical ValueZ α/2

90% 0.05 1.645

95% 0.025 1.96

99% 0.005 2.575

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

• Sample: simple random sample

• 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

• 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

where zα/2 is the critical z-value.

σLower bound = x – zα/2 --- n

σUpper bound = x + zα/2 --- n

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?

MOE = 1

(Z 1-α/2 σ)²n = ------------- MOE²

n = (Z 1-α/2 σ )²

n = (1.96∙ 6 )² = 138.3

n = 139

Summary and Homework

• Summary– 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

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

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

Homework• 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