Confidence Intervals for Population Means: Known and Unknown Standard Deviations, Study notes of Statistics

Download Confidence Intervals for Population Means: Known and Unknown Standard Deviations and more Study notes Statistics in PDF only on Docsity! Chapter 8 Confidence Intervals About a Single Parameter Section 8.1 Confidence Intervals About a Population Mean,  Known Assume we have a population about which we want to know . We discussed how most often we cannot sample the entire population, but instead we use a sample of size n. If we could only look at the sample, what would you use as your best guess for ? A point estimate of a parameter is the value of the statistic that estimates the value of the parameter. The sample mean, x , is the best point estimate of the population mean, .  The sample mean, x , is an unbiased estimator of . E( X ) =   The sample mean provides more consistent estimates of the population mean. The larger your sample, the closer the sample mean gets to the population mean.  In repeated samples, a majority of the sample means will be “close” to the value of the population mean. This characteristic of the sample mean is called efficiency. However, if we only take one sample, is x close to ? 119 A confidence interval estimate of a parameter consists of an interval of numbers, along with a probability that the interval contains the unknown parameter. The level of confidence in a confidence interval is a probability that represents the percentage of intervals that will contain  if a large number of repeated samples are obtained. The level of confidence is denoted (1 - )*100%. We know that P[-1.96 <        n X   < 1.96] = 0.95 We are looking for a bound on . So after a little algebra, we have P                    n X n X    96.196.1 = 0.95 Hence, a 95% Confidence Interval for  is        n X  96.1 Notice that 1.96 is specific to the 95% Confidence Interval. This value is called a critical value. It was obtained from the Z-table. A more general form for a Confidence Interval is        n zX   2/ Common Critical Values Level of Confidence Area in Each Tail Critical Value 90% 0.05 1.645 95% 0.025 1.96 99% 0.005 2.575 Interpretation of a Confidence Interval 120 3) What happened to the size of the confidence interval as we increased our confidence? 4) What other factors affect the size of the confidence interval? The Margin of Error The margin of error, ME, in a (1 - )*100% confidence interval in which  is known is given by n zME   2/ where n is the sample size. We assume that the standard deviation is a fixed quantity. Therefore, the width of the confidence interval is affected by the sample size and the level of confidence.  As the level of confidence increases, the margin of error also increase.  As the sample size increases, the margin of error decreases. Note: ME = half the confidence interval width. 123 Determining Sample Size n The sample size required to estimate the population mean, , with a level of confidence (1 - )*100% with a specified margin of error, ME, is given by 2 2/ *        ME z n  where n is rounded up to the nearest whole number. Example: For the student height example, assume we want a 90% confidence interval of width one-half inch. How big of a sample must be taken? Section 8.2 124 Confidence Intervals About a Population Mean,  Unknown We have looked at building confidence intervals for  when  is known, but what about building a confidence interval for  when  is not known. What would be a sensible point estimate for ? If  is unknown, it seems reasonable to replace  with s and proceed with the analysis. However,        n s X  will no longer follow a standard normal distribution. Instead, it will follow a Student’s t-distribution. Student’s t-Distribution Suppose a simple random sample of size n is taken from a population. If the population from which the sample is drawn follows a normal distribution, then the distribution of         n s X t  follows Student’s t-distribution with n – 1 degrees of freedom. 125 Example: A Gallup poll conducted December 20-21, 1999, asked 1031 Americans, “How much TV do you watch each week?” Results indicate a sample mean of 3.4 hours and a sample standard deviation of 1.8 hours. Construct a 95% confidence interval for the mean number of hours of TV Americans watched each week in 1999. Interpret the interval. 128 Confidence Bound Suppose you are interested in only bounding the value of  above or below with a desired confidence. The procedure is very similar to constructing a confidence interval, only now, to obtain a critical value, you place all the  in one tail. Example: Using the above example, find a 90% upper confidence bound for . 129 Section 8.3 Confidence Intervals About a Population Proportion Point Estimate of a Population Proportion Suppose a simple random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The best point estimate of p, denoted p̂ , the proportion of the population with a certain characteristic, is given by n x p ˆ where x is the number of individuals in the sample with the specified characteristic. Sampling Distribution of p̂ For a simple random sample of size n such that N n  0.05, the sampling distribution of p̂ is approximately normal with mean = p and standard deviation n pq  , provided that np ≥ 10 and nq ≥ 10. 130