Note that we are looking at the true mean, μ, not y. The problem for us is that we need to find the endpoints of our interval (a, b).
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1 Confidence Intervals 1) What are confidence intervals? Simply, an interval for which we have a certain confidence. For example, we are 90% certain that an interval contains the true value of something we re interested in. A more specific example: We are 90% certain that the interval 66 to 75 inches contains the true mean (μ) for height for men (numbers made up). Note that we are looking at the true mean, μ, not y. Usually we want confidence intervals for means, but we can also construct them for variances, medians, and other quantities. Some of these can be quite useful. The problem for us is that we need to find the endpoints of our interval (a, b). This is what this section of notes is all about how do we find (a, b)? Let's use a neat example from the text as an analogy: An invisible man walking a dog We might not really be interested in the dog (well, some people might be), but rather in the man. We can see the dog - this is our y, or sample mean. We can t see our man - this is our population mean (μ). But, we know that the dog spends most of his time near the man - in fact, the further away from the man the dog gets, the less time he spends. 2. Some more properties of the normal curve. If we draw a curve for amount of time spent away from the man, it ll be a normal curve, peaking where the man is. We use the dog to estimate where our man is, and we use y to estimate where μ is. If we use the above two facts, we can construct an interval for μ. (We use the normal curve to try and figure out where μ actually is - actually, we'll wind up using the t-distribution, but more on that soon). We already discussed reverse lookup. In this case we're interested in reasonable numbers like 90%, 95%, etc.
2 Since we re interested in numbers like 90%, 95%, 99%, etc., we need to look these up in our normal tables. For instance Pr{Z < z}=.90 => z = 1.28 Pr{Z < z}=.95 => z = 1.65 Pr{Z < z}=.99 => z = 2.33 (Remember, the symbol => means implies ) But this time we re interested in an interval (for instance, the middle 90%), so we really want: Pr{z 1 < Z < z 2 } =.90 => z 1 = and z 2 = 1.65 Note: we look up the value for Pr{Z < z}=.95, and then take advantage of symmetry. If, for example, we want 90% of the area in the middle of our curve, that means each end of our curve has 5% that we don t want. So we look up the value for 95%, and then just add a negative sign for the lower part of our curve.
3 Similarly, we get: Pr{z 1 < Z < z 2 } =.95 => z 1 = and z 2 = 1.96 Pr{z 1 < Z < z 2 } =.99 => z 1 = and z 2 = Constructing confidence intervals. So where are we? (Notice that the values 1.64, 1.96, and 2.58 will come up a lot!) Well, suppose we took a sample and calculated y. What we want to figure out is the probability that a certain interval around y includes μ. Or, putting it another way, we d like something like: Pr{y 1 < Y < y 2 } =.90, where y 1 and y 2 are such that they would have a 90% chance of including the true value for μ. But we know how to do this! (It s just not obvious yet). Pr{-1.65 < Z < 1.65} = 0.90 We just did this above. We also know that the following has a standard normal distribution: Z = Y / n So we can now substitute this for Z in the first equation and then isolate μ (we'll skip most of the algebra, but it's really not difficult): From this we get: Ȳ μ Pr { 1.65 < σ / n } < 1.65 = 0.90 = Pr {Ȳ 1.65 σ < μ < Ȳ σ n n } = 0.90 see note in text below. y±1.96 n Which will give us the endpoints for an interval that will contain μ with 90% confidence. Problem: we DO NOT know σ. Well, let's just substitute s for σ. Unfortunately, this means that our Z is no longer normally distributed; in other words, we can t use our normal tables. Why? Here's a partial explanation.
4 Z has a normal distribution and is equal to the following: Z = Ȳ μ σ/ n Everything in this equation (except Y ) is a constant. Y has a normal distribution, so Z has a normal distribution. But if we substitute s for σ, we have: Z = Ȳ μ s/ n But s is NOT a constant, it's a random variable (like Y ) s has it's own distribution. And it's not normally distributed. This means Z does not have a normal distribution. Or, in other words: Z Ȳ μ s/ n So was all the math above useless? No, or we wouldn't have done it. Without delving deep into the theory, the above quantity has a t-distribution: t = Ȳ μ s/ n So let's finally introduce the t-distribution. It looks a lot like the normal, but generally has fatter tails:
5 [Incidentally, notice that as ν, the t-distribution the normal distribution.] The t-distribution has only one parameter, the greek nu or ν (NOT μ). Nu (ν ) represents the degrees of freedom. To use the t-distribution we need to know what ν is. This can occasionally get complicated, but for the moment it's simply: degrees of freedom = d.f. = ν = n - 1 Historical note: the t-distribution was first derived by W.S. Gosset, while working for the Guiness brewery. Guiness did not allow him to use his own name in publishing, so he published under the pseudonym Student. The Guinness brewing company now has a plaque near its entrance commemorating Gosset. Because it has fatter tails, the t-distribution will require bigger numbers for the same probability. What does this mean?? Suppose you want the middle 90% of a normal distribution. The cut-offs (the z- values) are given above and are and 1.65, or in terms of probability: Pr{z 1 < Z < z 2 } =.90 => z 1 = and z 2 = 1.65 But now if we use a t-distribution (assuming ν = 10), and we want the middle 90%, we have:: Pr{t 1 < T < t 2 } =.90 => t 1 = and t 2 = Let's give some examples: Notice that to get the same probability (.90), we need to use bigger numbers (in terms of absolute value) for t. This also means that our confidence interval will be bigger using the t-distribution than using the normal distribution. But remember, we have to use the t-distribution (at least unless n is really large). See the t-table on our web page. This is arranged backwards from your normal tables. In other words, the numbers for the areas in the main part of the normal table are now along the top, and the numbers from the margins of the normal table are now inside the table.
6 Why? because with the t-distribution we re hardly every interested in anything except the values for 90%, 95%, 99%, etc. 4) Finally, let's actually calculate some confidence intervals. Here's how to calculate, for example, a 95 % CI: y t 95%, ν SE y and y + t 95%, ν SE y or s y ± t 95 %,ν SE which is the same as y ± t 95 %, ν n Where the critical t-value is from table 4, with ν = n-1 = degrees of freedom. Concerning subscripts: t above has a subscript of 95%. This indicates that we want the value of t which puts 2.5 % of the area into each tail. t requires another subscript indicating the degrees of freedom. Remember, there are an infinite number of t-distributions, so knowing the degrees of freedom is important. Many texts do things a little differently here. Some texts always use the one sided probability of the upper tail for the subscript, which might be a bit confusing. Some texts use one or two sided probabilities - which is actually less confusing. For now it's easier to stick with % in the subscript. We'll start using probabilities in the subscript soon enough. So here is a specific example. We take a sample of six ostrich eggs and get an average of 1.4 kg, and a standard deviation of 0.175kg. Let's construct a 90% CI for a sample of 6 ostrich eggs: ȳ±t 5,90 % s n = 1.4±t 5,90 % = 1.4±(2.015) = 1.4±.1275 which gives us: (1.31, 1.57)
7 Two comments: 1) Always finish the calculation. 1.4 ± is NOT an interval. 2) We're calculating actual mathematical intervals. Always put the smaller number first. For example, (1.53, 1.27) is NOT an interval. We can also calculate a 95% confidence interval: 5. Comments on confidence intervals. We follow the same procedure as above, but now our critical value for t becomes In other words, t 95%,5 = 2.571, and we get a confidence interval of: (1.28, 1.60) Confidence interval gets bigger if we want to be more certain. Where would the ends be if we wanted to be 100% certain?? Exactly what does the CI tell us? It tells us how confident we are that the limits of our CI contains the true value of μ. Here's another way of looking at CI's: What does the CI not tell us? If we construct % CI s 95 of our CI s (on average) will contain the true value of μ. Similarly, if we construct % CI's, than 90 of them (on average) will contain the true value of μ. Pr{20.9 < μ < 42.5} = 0.95 This is absolutely WRONG. μ is a constant, and so are obviously the endpoints of our interval. The statement above is then either correct or incorrect, never anything in between. The probability is either 0 or 1. For instance, suppose we know that μ = What happens to the above statement??? Remember: μ is a constant, not a random variable ( Y, on the other hand, is a random variable).
8 Let's do one more example. Let's look at the diameter of American toad eggs. We get a sample of size 97 and want to get a 95% CI. For a sample of size n = 97, we get an average length of 1.5mm and a standard deviation of.25mm. Our SE y = = Problem: how many degrees of freedom do we use? There is no row for d.f. = 97. We always use the row with less d.f. than we have (always round down). In this case we use 90 df (otherwise our CI would be too small (too big is okay, too small is bad)). so t 95%,90 = Constructing a CI, gives us 1.5 ± (.02538) And finally we get: (1.45, 1.55) Again, what does this mean? We are 95% confident that the true average diameter of toad eggs is between 1.45mm and 1.55mm.
Note that we are looking at the true mean, μ, not y. The problem for us is that we need to find the endpoints of our interval (a, b).
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