Elementary Statistics

Transcription

1 Chapter 7 Estimation Goal: To become familiar with how to use Excel 2010 for Estimation of Means. There is one Stat Tool in Excel that is used with estimation of means, T.INV.2T. Open Excel and click on the Stat button in the Quick Access Bar. Scroll down until you see T.INV.2T. Select that tool. Here is what you should see: The input to the Probability box depends on the confidence level. If you want to construct, say a 95% confidence interval, the input would be: If you want a 99% confidence interval the input would be: Deg_freedom stands for Degrees of Freedom. No, it s not about the Tea Party. It s a statistic thing, and it equals one less than the sample size. If for example, the sample size is 20, the input for Deg_freedom would be: 28

2 The value returned from the tool is called the t critical value. We will learn how to use this value later, but for now, try using the tool to find the t critical value for a 95% confidence interval and a sample size of 30. Your input should look like the following: The value returned is rounded off to four decimal places. This value is the t critical value which will be explained later. 29

3 Goal: To become familiar with how to use Excel 2010 for Estimation of the Standard Deviaion. There are two Stat Tools in Excel that are used for estimating standard deviayions, CHISQ.INV which is the left tail tool and CHISQ.INV.RT which is the right tail tool. In Excel 2007 there is just one tool, CHIINV. Select CHISQ.INV from the Stat menu and here is what you should see: The input to the Probability box depends on the confidence level. If you want to construct, say a 95% confidence interval, the input would be: Deg_freedom is similar to the t-statistic. It equasl one less than the sample size. For example, if the sample size is 20, the input for Deg_freedom would be: The value returned from the tool is called the critical value. We will learn how to use this value later, but for now, try using the tool to find the critical value for a 95% confidence interval and a sample size of 30. Your input should look like the following: 30

4 The value returned is rounded off to four decimal places. Unlike the z-statistic and the t-statistic, when working with chi square critical values, we also need to calculate the right hand critical value. Select CHISQ.INV.RT from the Stat menu and here is what you should see: The inputs are similar to those for the left-hand tool: 31

5 The value returned from the tool is called the off to four decimal places. critical value. The value returned is rounded 32

6 Elementary Statistics Goal: To understand the concept of a Confidence Interval. Imagine a normally distributed population from which we draw a sample. Assuming that the mean of the population is µ and the standard deviation is σ, we know from the Central Limit Theorem, that the mean of the sample is a member of the set of all means which form a sampling distribution. Furthermore, we know that the mean of this sampling distribution,, is equal to µ and the standard deviation of the sampling distribution,, is equal to where n is the sample size. Since we have selected a relatively small sample from a much larger population, we would not expect to equal µ. The following figure shows this relationship. For example, µ might be equal to 100 and might equal We would like to find the number, E, such that for any sample with mean, such that we are 95% confident that the interval, actually contains the true value of µ. Of course, since Statistics is a branch of mathematics, we have to be precise about what we mean when we say, 95% confident. Let s go back to the concept of a sampling distribution. Imagine that we take hundreds of samples, each with its own mean,. For each such mean, we construct the interval, using the same value of E for each interval. E is such that 95% of these intervals will actually contain the true value of µ. See the figure below. 33

7 Elementary Statistics The horizontal line represents the population s true mean, µ, and the intervals shown are centered on various sample means, As you can see, all but one of the intervals contain µ. We call E the margin of error. Now, we know from the Central Limit Theorem that the sampling distribution of the sample means is normal in shape, with mean, and standard deviation,. If we wanted to convert a particular to a z-score we would use the formula: If we choose z to be then the only that would satisfy the above equation would be those that fall within units from in the sampling distribution: Well, as you can see from the figure above, those are the 95% of that we re looking for. By the way, use NORM.S.DIST to verify that z = will give you an area of 95% under the curve centered on zero, and remember that NORM.S.DIST is a one-tail left tool. If we rearrange the terms in the equations above, and remember that z can be positive or negative, we get the following: From this we see that the E we are looking for is given by: In other words, if is within E units of then would be within E units of. The z in the equation above is often written as. Alpha,, is called the significance level and is equal to. Another way to look at alpha, is that it s the area under the tail or tails. 34

8 Elementary Statistics Confidence intervals are always centered under the curve, so alpha will be the sum of the area under the two tails. That s why the notation is used. is also sometimes referred to as a critical value. It s that value along the z-axis where the tail begins. Take another look at the figure above. We can use NORM.S.INV to find but remember that this Excel tool is a left one-tail tool. As an example, suppose that z, we would use the following,, the value corresponding to the 99% confidence interval. To find NORM.S.INV(.995) because the area under both tails would be 0.01, hence the area under the right tail would be and the area to the left of z would Precisely What We Mean By 95% Confident Exactly what do we mean when we say that we are 95% confident that will contain µ? Consider the sampling distribution of all. Define to be the critical values corresponding to the left and right tails for Therefore, 95% of the lie between. Now, we know that when translating from the x-axis to the z-axis, we use the following transformation: If we apply this formula to the sampling distribution, we get: and by the Central Limit Theorem, we get: Furthermore, from the way we defined, (C.I. = 95%) we get: and therefore, we see that: Therefore, if 95% of the fall between, then 95% of the intervals, will contain µ. 35

9 Finally, we can precisely define what we mean when we say that we are 95% confident that µ lies somewhere in. What we are really saying is that 95% of the intervals, would contain µ if we were to take very many samples and calculate the mean of each sample. These concepts are not easy to digest. Before going on, please reread the above text several times until you grasp them. 36

10 Finding the Confidence Interval of a Mean the Student t-distribution Now that you completely understand the theory behind confidence intervals, let me point out a serious problem. We still can t find E. Take a close look at the formula for E We can easily find and n is just the sample size, but if we don t know µ, we most likely will not know the value of σ. So after all this work, what do we do? Well, there s a loophole. If the size of the sample is large, say we can substitute s, the standard deviation of our sample, for σ, and we ll still get pretty good results, at least to three or four decimal places of accuracy. However, what if the size of the sample was less than 100? In that case, there wasn t anything that could be done until the early 1900 s when William Gosset, working for the Guinness Brewery Company solved the problem. Instead of using the Standard Normal Distribution to find z, he came up with a whole new distribution called the Student t-distribution. This distribution is quite similar to the Standard Normal Distribution, except that the intervals based on the following formula, 37

11 will be wider than the ones using z, because t will always be a larger number then z for any given value of. For example for n = 20, we have the following: Alpha z t Notice that I said above, for n = 20. The exact shape of the t-distribution depends on the size of the sample. The larger the sample, the more the distribution will look like a perfect normal curve, and the closer the critical value of t will be to the corresponding critical value of z. Shown below are t-distributions corresponding to various values of n, superimposed on a normal curve. Other than the fact that we use T.INV.2T to calculate t instead of using NORM.S.INV, there really is no difference between how we calculate E and how we use it, EXCEPT that we use s instead of σ, and that is a very important difference, because we can always calculate s from our sample regardless of its size. Keep in mind that T.INV.2T is a two-tail tool, unlike NORM.S.INV which is a one-tail tool. The input to NORM.S.INV is whereas the input to T.INV.2T is simply. 38

12 Elementary Statistics Example Suppose. The 99% confidence interval is be 99% confident that lies between and In other words we can Finding the Confidence Interval of a Proportion Finding the confidence interval for a proportion is very similar to that of a mean. First, when taking a poll with only one of two responses possible, the percentage of affirmative answers is given by: where x is the number of affirmative answers, n is the total number in the sample and is usually expressed as a decimal. When you take a poll like this, you are essentially dealing with a binomial distribution, except that the population tends to be very large. Mathematically, a binomial distribution tends toward a normal curve as n increases, so we can base our confidence interval on the normal curve, i.e. we use. The formula for finding E, the margin of error is given by: where Example Let That plus or minus 3.04 percent, also stated as points. 39

13 Elementary Statistics The confidence interval is then We can also, beforehand decide how big we want our margin of error to be and choose the sample size appropriately. We simply solve for n: Of course, we don t know and until after we have taken the sample. However, by using and we will get a value of n that will be valid for all other values of and. Example Let 40

14 Finding the Confidence Interval of a Standard Deviation Calculating a confidence interval for a standard deviation is very different than that of a mean. In the first place, we don t really have a concept of a margin of error. Also, the confidence interval is not centered on s. Finally, s is not a good estimator for σ, but is good for. The confidence interval is given by: We then take the square root of all sides to get the confidence interval for σ. are Chi Square statistics. Just as we use the Normal Distribution to find z and the Student t Distribution to find t, we use the Chi Square distribution to find the two chi square statistics. Unlike the former two probability distributions, Chi Square is not symmetrical: Like the t-distribution, the exact shape of the Chi Square curve depends on the sample size. If n is the sample size, then n 1 is the degrees of freedom. Example Let 41