Estimating a Population Mean. Section 7-3
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1 Estimating a Population Mean Section 7-3
2 The Mean Age Suppose that we wish to estimate the mean age of residents of Metropolis. Furthermore, suppose we know that the ages of Metropolis residents are normally distributed but we do not know the mean (μ) or standard deviation (σ). One option that we have is to take a sample and find the sample mean (x-bar).
3 The Sample When 25 Metropolis residents were surveyed, the sample had a mean age of 42.0 and a standard deviation of 10.6 If we wish to be 95% confident that our interval contains the true mean of the age of Metropolis residents, we can use the sample mean (x-bar) as a point estimate. We now know the center of the interval, but we still need to find the margin of error.
4 Finding E Because we are constructing a confidence interval for a mean rather than a proportion, we will be using a slightly different formula to find the margin of error. This formula is s E t /2 n
5 Deciphering the Symbols The s and the n are familiar to us since they are the sample standard deviation and sample size respectively. But what is t α/2? The reason that a new critical value is used is because the population standard deviation (σ) is unknown so we use the sample standard deviation instead.
6 Buy Why T? The reason that a different critical value is used is because the sample standard deviation (s) is a biased estimator of the population standard deviation (σ). The value of t α/2 actually varies with the value of α and the sample size. An important concept with respect to the t distribution is the idea of degrees of freedom.
7 Degrees of Freedom The degrees of freedom for a sample is the number of values that can be freely assigned and still have the sample meet some restriction. If, for example, a sample consists of 25 values and the sum must be equal to 1826, then 24 of the values can be freely assigned and the 25 th will be determined to make the sum equal to the target of 1826.
8 Degrees of Freedom The sample mean is the sum divided by the total number of observations. This means that a sample of size n will have n-1 degrees of freedom when a target mean is given. The degrees of freedom is used to determine the value of t α/2 when using Table A-3. When using Table A-3, the degrees of freedom is used to determine the row and the value of α is used to determine the column.
9 Back to Metropolis Recall that for the sample of metropolis residents, n = 25, x-bar = 42.0, s = 10.6 Since we wish to be 95% confident that our interval contains the mean age of the population, α = = The sample has 25 1 = 24 degrees of freedom. α represents the area in two tails, which is 0.05 in our case.
10 Constructing the Interval Using table A-3, t α/2 = E E s t /2 n
11 Constructing the Interval Lower Limit = x-bar E = = 37.6 Upper Limit = x-bar + E = =46.4 So the 95% confidence interval for the mean age of Metropolis residents is: (37.6, 46.4) The good news regarding confidence intervals is that they can be constructed using the TI-83 and TI-84 calculators.
12 Interpreting the Confidence Interval The correct interpretation for the confidence interval is: "We are 95% confident that the interval from 37.6 to 46.4 contains the true mean age of Metropolis residents (μ) Note that this is similar to the correct interpretation of a confidence interval for a proportion.
13 Homework Section 7-3 (page 357) 9 to 17 (Odd)
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