This preview shows page 1. Sign up to view the full content.
Unformatted text preview: IV. Inference.
A. Confidence Intervals for Population Mean. (Chapter 8) 1. Point and Interval Estimation. 2. Confidence Intervals – σ known. 3. Margin of Error and Sample Size. 4. Confidence Intervals – σ unknown. Combine point estimation and the point sampling sampling distribution. A. Confidence Intervals for Population Mean.
1. Point Estimate: a single numeric value – our best guess about µx 2. Confidence Interval Estimate • Combine: Point estimate – sample mean, a single numeric value Standard error – summary of variation for the sample mean • Choose a level of confidence: (1 – α) 2. Confidence Intervals – σX known Lower Limit Point Estimate Upper Limit x − zα / 2 ⋅ σ x x x + zα / 2 ⋅ σ x • 95% Confidence Interval: x − 1.96 ⋅ (σ x ) x x + 1.96 ⋅ (σ x ) E = zα / 2 ⋅ σ x = zα / 2 ⋅ σX
n Margin of Error. 1 The Sampling Distribution for Text costs (n=16): Sampling (n=16): We determined previously that 80% of the sample means will fall between the values shown below. $225.89 $320.11 162.75 199.5 236.25 273 309.75 346.5 383.25 The Key to Confidence Intervals
The The methods we use to develop the confidence interval give us random intervals: – Draw random sample – Estimate 3. Margin of Error and Sample Size. Margin of Error: E = zα / 2 ⋅ σx
n = zα / 2 ⋅ σ x Illustrate E: 2 Reducing Margin of Error – what options? E = zα / 2 σx
n Determining Sample Size • Given a desired level of confidence (eg. 90%) • Willing to accept a certain margin of error. • What sample size do you need? What is n? E = zα / 2 σx
n σx is not usually known
• Estimate – draw a small pilot sample just to estimate σx • Use the range (R) 3 4. Confidence interval  σx is not known. • We don’t know µ and we don’t know σ. 4. Confidence interval  σx is not known. tdistribution: a lot like the Zdistribution, but… • The tdistribution  “fatter tails” than the z. • More conservative – wider intervals for (1–α) • Reflects added uncertainty  also estimated σx . • Shape depends on degrees of freedom: df = (n – 1); • Center is zero • As (n – 1) → ∞ , the tdistribution converges to a zdistribution. 4. Confidence interval  σx is not known. The mean and standard deviation for voltages of power packs labeled as 12 volts for a sample of 8 are as follows: x = 11.14 v and s = 0.3 v. . Create a 90% confidence interval estimate for the population mean. 4 ...
View
Full
Document
 Spring '11
 DanielLass

Click to edit the document details