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Lecture6_print

# Lecture6_print - Chapter 7 Inference on Population Mean(s...

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Chapter 7: Inference on Population Mean(s) Using t Procedures Readings: Chapter 7 September 26, 2009 1 One-sample t Procedures When the population standard deviation σ is KNOWN , use z procedures: Confidence interval for the population mean μ : ¯ x ± z * σ n Hypothesis testing for the population mean μ ( H 0 : μ = μ 0 ): Test statistic z = ¯ x - μ 0 σ/ n Both inference procedures are based on the sampling distribution of ¯ X : ¯ X N ( μ ¯ X = μ, σ ¯ X = σ n ) , or equivalently, Z = ¯ X - μ σ/ n N (0 , 1) . When the population standard deviation σ is NOT KNOWN (lecture 6) : Use the sample standard deviation s to estimate σ – Standard error of the sample mean: SE ¯ X = s n The sampling distribution of ¯ X - μ s/ n is: T = ¯ X - μ s/ n t ( n - 1) , where t ( n - 1) denotes the t distribution with n - 1 degrees of freedom . Confidence interval for the population mean μ : ¯ x ± t * s n Hypothesis testing for the population mean μ ( H 0 : μ = μ 0 ): Test statistic t = ¯ x - μ 0 s/ n 1

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The t -Distribution When the population standard deviation σ is unknown, T = ¯ X - μ s/ n has a t -distribution. There is a different t -distribution for each sample size, so t ( k ) stands for the t -distribution with k degrees of freedom. Degrees of freedom for T = ¯ X - μ s/ n is k = n - 1. As the degrees of freedom increase, the t -distribution looks more like the normal distribu- tion (because as n increases, s σ ). t -distributions are symmetric about 0 and are bell shaped. They are just a bit wider than the standard normal distribution. To use the t-table (Table D): The t-table is set up differently than the normal table. The upper-tail probability on the top of the table refers to the area to the right of the t * values, called the critical value . We start with the desired area and read off the critical values (in the normal table, we started with a z and we read probabilities). Note the “confidence level C ” on the bottom. A 90% confidence level gives the same critical value as the upper tail probability of 0.05 does. What happens if your degrees of freedom are not on the table? Always round DOWN to the next lowest degrees of freedom to be conservative. 2
Example 1 : How accurate are radon detectors of a type sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows: 91.9 97.8 111.4 122.3 105.4 95.0 103.8 99.6 119.3 104.8 101.7 96.6 The sample mean is ¯ x = 104 . 13 and the sample standard deviation is s = 9 . 40. a. Find the 90% confidence interval for the population mean. Now re-do the above example using SPSS completely. To do just a confidence interval: enter data, then Analyze Descriptive Statistics Explore. Click on ”Statistics” and change the CI to 90%. Then hit “OK”.

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Lecture6_print - Chapter 7 Inference on Population Mean(s...

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