Standard Error and Regression

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I. Standard Error (1 Question) • Given only a small sample of data, we use the sample mean to estimate the population mean. Likewise, we can use sample standard deviation s to estimate the population standard deviation • Standard Error of the Mean. A common application is the Standard error of the mean. This is a standard deviation of the mean, i.e. the variability of the mean from sample to sample. When σ is the standard deviation of the population, represents the standard error of the mean. • Given a sample of data x1, x2,…,xn, the standard error of is o p is the population proportion, with p(1)=p and p(0)=1-p, E(X)=p and SD: • Role within a confidence interval. The standard Error forms part of the construction of the confidence interval, as the range of a confidence interval is defined by the number of standard errors away from the mean. o E.g. the chart below shows a 95% confidence interval consisting of an upper range of 2 standard errors above the mean, to a lower range of 2 standard errors below the mean. II. Interpretation of Confidence Interval (1 Question) • Confidence Intervals (CI) measures how reliable an estimate is. More specifically, it is a range of plausible values for a parameter (such as a mean) based on a sampled population. o Narrow confidence intervals suggest precise answers. A wide confidence intervals suggest we know little about the population. • When is an IID sample from a population t is a predetermined constant that depends on the sample size n and desired confidence level. • “±2 Standard Error” Rule of Thumb: Unless n ≤ 30 and precise confidence is needed, the approximate 95% CI is: • Typically stated at the 95% confidence level. But why 95%? “Ideally we would like a narrow 99.99% interval, since such an interval almost certainly contains the

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