Stats Ch 8

Confidence Intervals for One Population Mean

Estimating a population mean

• Point estimate: Use a statistic to estimate the corresponding parameter. (Often we use a point estimate…that is we use the mean of a sample to estimate the mean of the population, or the st. dev of a sample to estimate the st. dev of a population, etc)

• Most of the time there is sampling error. So we often need confidence intervals

Confidence Intervals

• Confidence-Interval Estimate – Confidence interval (CI): an interval of numbers

obtained from a point estimate (the range: from 100 to 200)

– Confidence level: The confidence we have that the parameters lies with in the CI (the percent: 90% confident)

– Confidence-interval estimate: a combination of the confidence interval and the confidence level (we are 90% confident that mu is between 100 and 200)

Example

• A random sample of the costs of 45 weddings was taken (∑xi = $608,580). The population standard deviation is $5205. Obtain a point estimate for the population mean. Determine a 93% confidence interval for the population mean. Interpret your results.

Ex

• Pg 383 problem 49 (also interpret the results)

CI when sigma is known

• Confidence level vs α: CL = 1 – α…..so using algebra α = 1 – CL

• is the z-score with an area of α to the RIGHT under standard normal curve

z

One-Mean z-Interval Procedure

• Assumptions: Simple random sample, normal distribution or large sample, σ is known

• 1. Sketch picture• 2. Find α and use Table C to find

• 3. Find CI--use and • 4. Interpret the CI

2z

n

zx

2

n

zx

2

CI

• Pg 375 ex 8.3

• How does the range of the CI (precision) effect the CL

Margin of Error

• The margin of error is how far away (above OR below) your CI critical points are from

• Formula:

• How can we decrease the margin of error (which will increase the precision)?

• Pg 379 ex 8.8

x

CI when sigma is NOT known

• Studentized version of which is• There is a different t-distribution for each sample size.

We identify the particular t-distribution by the degrees of freedom which is df = n – 1

• Basic properties of t-curve:– Total area under the curve is 1– Extends indefinitely in both directions as it approaches the

horizontal axis– Symmetric about 0– As df increases, the t-curve looks more like the standard

normal curve

xns

xt

/

The t-table (Table D)

• Uses notation. The t-value with an area of α to the RIGHT

• Pg 388 ex 8.9• Find a CI for one-mean t-interval is almost the exact

same way as with one-mean z-interval• Assumptions: Simple random sample, normal

distribution or large sample, σ is unknown• 1. Sketch picture• 2. Find α and use Table D to find with df = n – 1 • 3. Find CI--use and • 4. Interpret the CI

t

2t

n

stx2

n

stx2

Example t

• Pg 389 ex 8.10

Margin of Error

Pg 393 Ex 8.14

T-distribution examples

• Pg 397 #50, #54