9.1 The Logic In Constructing Confidence Intervals For a Population Mean When the Population Standard Deviation is Known Objectives- 1. Compute a point estimate of the population mean 2. Construct and interpret a confidence interval for a population mean, assuming that the population standard deviation is known. 3. Explain the role of margin of error in constructing a confidence interval. 4. Determine the sample size necessary for estimating the population mean within a specified margin of error. Point Estimate: The value of a statistic that estimates the value of a parameter. For example, the sample mean, INCLUDEPICTURE "http://upload.wikimedia.org/math/6/0/6/6061e2e960b7cd1f2381c7a3cc6a1131.png" \* MERGEFORMATINET , is a point estimate of the population mean, µ. Confidence Interval: A confidence interval for an unknown parameter consists of an interval of numbers. In other words, it’s a range of numbers, such as 22-30. 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%. Confidence interval estimates for the population mean are of the form Point estimate ± margin of error Margin of Error: The margin of error of a confidence interval estimate of a parameter is a measure of how accurate the point estimate is and depends on three factors: Level of confidence: As the level of confidence increases, the margin of error also increases. Sample Size: As the size of the random sample increases, the margin of error decreases. This is a consequence of the Law of Large Numbers, which states that as the sample size increases the difference between the static and parameter decreases. Standard deviation of the population: The more spread there is in the population, the wider our interval will be for a given sample size and level of confidence. Critical Value: The value z α/2 Interpretation of a Confidence Interval A (1- α)*100% confidence interval indicates that (1- α)*100% of all simple random samples of size n from the population whose parameter is unknown will contain the parameter. In other words, the interpretation of a confidence interval is this: We are (insert level of confidence) confident that the population mean is between (lower bound) and (upper bound). This is an abbreviated way of saying that the method is correct (1-α)*100% of the time. Constructing a (1-α)*100% Confidence Interval for µ, σ Known Suppose that a simple random sample of size n is taken from a population with unknown mean, µ, and known standard deviation, σ. A (1-α)*100% confidence interval for µ is given by Lower bound: INCLUDEPICTURE "http://upload.wikimedia.org/math/6/0/6/6061e2e960b7cd1f2381c7a3cc6a1131.png" \* MERGEFORMATINET - z α/2 * σ/sqrt(n) Upper Bound: INCLUDEPICTURE "http://upload.wikimedia.org/math/6/0/6/6061e2e960b7cd1f2381c7a3cc6a1131.png" \* MERGEFORMATINET + z α/2 * σ/sqrt(n) Where z α/2 is the critical z-value. Note: The sample size must be large (n INCLUDEPICTURE "http://upload.wikimedia.org/math/a/9/d/a9d9e4092ee6253aac14ec6b8f7959dd.png" \* MERGEFORMATINET 30) or the population must be normally distributed. Robust: Minor departures from normality will not seriously affect the results. -The margin of error, E, in a (1-α)*100% confidence interval in which σ is known is given by E= z α/2 * σ/ sqrt(n) Where n is the sample size. In other words, the margin of error can be thought of as the “give or take” portion of the statement “The mean age of the class is 24, give or take 2 years.” Note: We require that the population from which the sample was drawn be normally distributed or the sample size n be greater than or equal to 30. Determining the Sample Size n The sample size required to estimate the population mean, µ, with a level of confidence (1-α)*100% with a specified margin of error, E, is given by n= ((z α/2 * σ)/E)2 where n is rounded up to the nearest whole number. Caution: Rounding up is different from rounding off. We round 5.32 up to 6 and off to 5. Requirements for Constructing a Confidence Interval about µ if σ Is Known The data obtained come from a single random sample. In Chapter 1, we introduced other sampling techniques, such as stratified, cluster, and systematic samples. The techniques introduced in this section apply only to samples obtained through simple random sampling. Although methods exist for constructing confidence intervals when using other sampling methods, they are beyond the scope of this text. If the data obtained from a suspect sampling method, such as voluntary response or convenience sampling, no methods exist for constructing confidence intervals. If data are collected in a flawed manner, any statistical inference performed on the data is useless! The data are obtained from a population that is normally distributed, or the sample size, n, is greater than or equal to 30. When the sample size is small, we can use normal probability plots to help us judge whether the requirement of normality is satisfied. Remember, the techniques introduced in this section are robust. This means that minor departures from requirements will not have a severe effect on the results. However, we do have to be aware that the sample mean is not resistant. Any outlier(s) in the data will affect the value of the sample mean and therefore affect the confidence interval. If the data contain outliers, we should proceed with caution when using the methods introduced in this section. The population standard deviation, σ, is assumed to be known. It is unlikely that the population standard deviation is known when the population mean is not. We will drop this assumption in the next section.