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Unformatted text preview: Chapter 6 – Introduction to Formal Statistical Inference Section 6.1: LargeSample Confidence Intervals for a Mean In most realworld situations values of interest based on an entire population are unknown. What we can do for a desired parameter is use a sample to identify an interval of values likely to contain the unknown parameter and quantify how likely the interval is to cover the correct value. Definition : A confidence interval (CI) for a parameter (or function of one or more parameters) is a data based interval of numbers thought likely to contain the parameter and possessing a stated probability based confidence or reliability. Notation : The Greek letter α will be used to represent a significance level, while 1 α stands for the confidence level…the higher the confidence the more likely that a constructed CI will contain the unknown parameter value of interest. The most common choices for α are .10, .05, and .01. These imply confidence levels of .90, .95, and .99, respectively. Confidence intervals for the population mean μ , for known σ (the population std. dev.) • Recall from Chapter 5 (by the Central Limit Theorem) that X ~N( μ , σ 2 n ). • For a given choice of α , we will need to find the values of the N(0,1) distribution that put 1 α probability in the middle of the distribution and α /2 probability in each of the two tails of the distribution (giving a total probability of 1 α + α /2+ α /2 = 1). We refer to these two values as  z 1 2 α and z 1 2 α . 1 α α /2 α /2 z 1 2 α z 1 2 α • We can use the normal table as we did in Chapter 5 to find these values. The values for the common α ’s of .10, .05, and .01 are given in the following table. 1 α z 1 2 α .01 2.575 (for a 99% CI) .05 1.96 (for a 95% CI) .10 1.645 (for a 90% CI) • We now derive a (1 α ) × 100% confidence interval for μ . If X ~N( μ , σ 2 n ), then under the transformation Z n n = = X X μ σ μ σ 2 , we have that Z~N(0,1) (this follows from what we learned in Chapter 5). Now, Z n n Z = ⇒ = X X μ σ σ μ . By definition, P( z 1 2 α < Z < z 1 2 α ) = 1 α . 2 Example 6.1: A study compares lead levels of children whose parents worked in a factory where lead was used to a control group of children of the same age and from the same neighborhood. There are 33 children from the “exposed” group and 33 children from the control group. It is known that σ =14.41 for the exposed group and that σ =4.54 for the control group. Based on the data it is found that x =31.85 for the exposed group, and x =15.88 for the control group. Find a 95% CI for μ (for each of the groups).Note that there is a relationship between the significance level, α , and the width of the constructed CI. As we decrease α , we have more confidence that our interval covers the true value of interest (i.e. 1 α is higher). However, decreasing α will give us a wider interval (this can be seen by looking at the z 1 2 α values from the table). values from the table)....
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This note was uploaded on 03/26/2008 for the course STAT 105 taught by Professor Bingham during the Spring '08 term at Iowa State.
 Spring '08
 Bingham

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