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Statistical Sciences
Statistical Sciences 2035
Steve Kopp

Chapter 7 – Estimation of Population Means Confidence Intervals for μ We now move into a statistical topic called “statistical inference” • the goal of inference is to draw conclusions (make inferences) about a population from a sample • often, we are interested in obtaining information about μ (and sometimes σ) For example, we might want to know: 1. The mean tar content of a certain brand of cigarette 2. The mean lifetime of a newly developed steel-belted radial tire 3. The mean gas mileage of a new model car 4. The mean annual income of liberal-arts graduates 5. The mean height of male UWO undergraduate students If the population is small, then we can calculate μ directly from the small number of observations in the population • however, if the population if large, as it often is in practice, it is not practical to calculate μ (sometimes it is impossible) What Do We Do? • we gather data • take a random sample from the population that we wish to draw conclusions about • calculate x (and s) • it seems logical to usex to estimate μ (and s to estimate σ) We will start with the simplest case: Inference about μ from a normal population, where σ is known (section 7.1) Example 7.1 A large hospital wishes to estimate the average length of time patients remain in the hospital. The hospital’s administrators randomly select the records of 49 previous patients and calculate x to be 4.53 days. Previous research on this subject has shown σ = 3.68 days and that the lengths of stay are normally distributed. What can you say about μ? Note For now, we will assume that σ is known (either from a previous study or from historical records) • we will cover the more usual case, where σ is unknown, later Solution to 7.1 1. x = 4.53 is called a point estimate of μ 2. Why is x not equal to μ? Suppose the administrators took several samples: Sample x 1 4.53 2 4.82 3 5.45 4 3.98 All four of these numbers are point estimates of μ • but they vary, they are not all the same However, in real life, you only get to take one sample • since you know that x is not equal to μ, it may be better to calculate a range, or interval, of values, centred around • this interval is called a confidence interval (CI) for μ • it allows us to take into consideration the variability of • it will also indicate how accurate is the point estimate,, that we have calculated We hope that the CI contains the true, but unknown, value of μ • we will state how confident we are that μ lies in the interval • this is given by the confidence level (typical levels are 90%, 95%, 99%) Let’s look for a 95% CI for μ We have: x = length of stay for a patient x ~ N(μ, 3.68) x ~ N(μ, 3.68 = 0.5257) 49 You might think the appropri
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