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 (∑x i = $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

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

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

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

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

Example t Pg 389 ex 8.10

Margin of Error Pg 393 Ex 8.14

T-distribution examples Pg 397 #50, #54