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Chapter 20: Inference about a Population Mean Two most common types of statistical inference tool. 1. Confidence Interval C.I. for Estimating a Population Mean (with σ Known) x − E <µ< x + E σ where E = zα/2 · √ n (Margin of error) 2. Tests of Significance (Hypothesis Test) Test Statistic Formulas (with σ Known) For mean: z= x − µx σ √ n If we do not know σ....... ( s σ ) Use √ instead of √ . n n (The standard error of x) We must verify two conditions. 1) The data collected “SRS” way. 2) n ≥ 40. 3) If n is not ≥ 40, make sure the data appear close to normally distributed. 3. Confidence Interval C.I. for Estimating a Population Mean (with σ Unknown) x − E <µ< x + E where E = tα/2, df s ·√ n (Margin of error) 4. Tests of Significance (Hypothesis Test) Test Statistic Formulas For mean: (σ unknown) t= x − µx s √ n Student t Distribution t= x−µ √s n with Degree of freedom = n − 1 Degree of freedom for a collection of sample data: the number of sample values that can vary after certain restrictions have been imposed on all data values. Student t Distribution 1. The Student t distribution is diﬀerent for diﬀerent sample sizes. 2. The Student t distribution has the same general symmetric bell shape. 3. The Student t distribution has a mean of t = 0. 4. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution. Ex. A job placement director claims that the average starting salary for nurses is $48, 000. A sample of 40 nurses has a mean of $46, 900 and a sample standard deviation of $1800. Test the claim at α = 0.01. Ex. A rental agent states that the average rent for a studio apartment in Springside is less than $1250. A sample of 50 renters shows that the mean is $1150 and the standard deviation is $75. Is there enough evidence to reject the agent’s claim? Use α = 0.10. Ex. Listed below are predicted high temperatures that were forecast before diﬀerent days Predicted high temperature forecast three days ahead 79 86 79 83 80 Predicted high temperature forecast five days ahead 80 80 79 80 79 Use a 0.05 significant level to test the claim that the paired sample data come from a population for which the mean diﬀerence is µ = 0. Ex. Atkins Weight Loss Program (Population Mean: σ Not Known) 50 individuals participated in a randomized trial with overweight adults. After 12 months, the mean weight loss was found to be 4.6 lb, with a standard deviation of 1.8 lb. a. What is the best point estimate of the mean weight loss of all overweight adults who follow the Atkins program? b. Construct a 99% of confidence interval estimate of the mean weight loss for all such subjects. Ex. Car Pollution (Population Mean: σ Not Known and also n ≤ 30) 95% confidence; n = 7, x = 0.12, s = 0.04. (Assume that the sample is a simple random sample and the population has a normal distribution.) Finding P-value. (t) 1. The claim is that for the nicotine amounts in king-size cigarettes, µ > 1.10 mg. The sample size is n = 25 and the test statistic is t = 3.349. 2. The claim is that for the tar amounts in king-size cigarettes, µ > 20.0 mg. The sample size is n = 25 and the test statistic is t = 1.733. 3. The claim is that for measurements of standard head injury criteria in car crash tests, µ = 475 HIC (standard head injury condition units). The sample size is n = 21 and the test statistic is t = −2.242. 4. The claim is that for pulse rates of women, µ = 73. The sample size is n = 40 and the test statistic is t = 2.463. Which distribution and formula do we use? Is σ known? ⇒ Yes ⇒ Use z α2 values ⇓ Must be normally distributed or n > 30 No ⇓ Is n ≥ 40? ⇒ Yes ⇒ Use t α2 ,n−1 values s in place of σ in the formula. ⇓ No ⇓ Is population normally distributed? ⇒ Yes ⇒ Use t α2 ,n−1 values s in place of σ in the formula. No ⇓ Use a nonparametric or the bootstrapping method