Lecture Slides on Confidence Intervals About a Population Mean | MATH 130, Assignments of Statistics

Download Lecture Slides on Confidence Intervals About a Population Mean | MATH 130 and more Assignments Statistics in PDF only on Docsity! Confidence Intervals about a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Spring 2008 J. Robert Buchanan Confidence Intervals about a Population Mean Motivation Goal: to estimate a population mean µ based on data collected in a sample. Assumption: the population standard deviation σ is known. This is not strictly required, but simplifies the steps involved J. Robert Buchanan Confidence Intervals about a Population Mean Example Example The mean length of machine parts manufactured by a certain factory is to be estimated by taking a sample of ten machine parts. The lengths in millimeters are as follows. 75.3 76.0 75.0 77.0 75.4 76.3 77.0 74.9 76.5 75.8 The sample mean is x = 75.92. Thus the point estimate of µ is 75.92. J. Robert Buchanan Confidence Intervals about a Population Mean Relationship Between the Point Estimate and µ If the point estimate is close to µ, we are interested in understanding how close. This brings up the notions of margin of error, confidence interval, and level of confidence. J. Robert Buchanan Confidence Intervals about a Population Mean Experiment Suppose the mean age of a sample of college students is 20.1 years. 1 Into what interval would you be willing to place the population mean age of college students with 90% confidence? 20.1±? 2 Into what interval would you be willing to place the population mean age of college students with 95% confidence? 20.1±? 3 Into what interval would you be willing to place the population mean age of college students with 99% confidence? 20.1±? J. Robert Buchanan Confidence Intervals about a Population Mean Confidence Intervals and Level of Confidence Definition A confidence interval for an unknown parameter consists of an interval of numbers. The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted (1 − α) · 100%. J. Robert Buchanan Confidence Intervals about a Population Mean Margin of Error We will express confidence intervals in the form: point estimate ± margin of error where the margin of error depends on 1 the level of confidence (as the level of confidence increases so does the margin of error), 2 the sample size (as the sample size increases the margin of error decreases), 3 the population standard deviation (the greater the spread in the population characteristic, the larger the margin of error). margin of error = E = zα/2 σ√ n J. Robert Buchanan Confidence Intervals about a Population Mean Margin of Error We will express confidence intervals in the form: point estimate ± margin of error where the margin of error depends on 1 the level of confidence (as the level of confidence increases so does the margin of error), 2 the sample size (as the sample size increases the margin of error decreases), 3 the population standard deviation (the greater the spread in the population characteristic, the larger the margin of error). margin of error = E = zα/2 σ√ n J. Robert Buchanan Confidence Intervals about a Population Mean Simulation 100 samples of size 35. 20 40 60 80 100 sample 18 19 20 21 22 23 CI J. Robert Buchanan Confidence Intervals about a Population Mean Example Example The Third International Mathematics and Science Study (TIMSS) in 1999 examined eighth-graders’ proficiency in math and science. The mean geometry score for a sample of 25 eighth-grade students was 47.3 with a standard deviation of 4.4. Construct the 95% confidence interval for the mean geometry score for all eighth-grade students in the United States. J. Robert Buchanan Confidence Intervals about a Population Mean Determining Sample Size If the confidence level, margin of error, and population standard deviation are known, we can estimate the sample size necessary to create the confidence interval. E = zα/2 σ√ n n = ( zα/2 · σ E )2 J. Robert Buchanan Confidence Intervals about a Population Mean