Statistical Inference Confidence Intervals - Tests 3 Notes - Statistics | STAT 2000, Study notes of Statistics

Download Statistical Inference Confidence Intervals - Tests 3 Notes - Statistics | STAT 2000 and more Study notes Statistics in PDF only on Docsity! Page ① of 89 Chapter 8: Statistical Inference: Confidence Intervals 8.1 What are Point and Interval Estimates of Population Parameters? When we first began our discussion of statistics, we mentioned that there were two branches of statistics: descriptive and inferential. The inferential branch uses sample information to draw conclusions about the population. One of the most common uses of the inferential branch is to use sample statistics, such as⎯x, to estimate population parameters, such as μ. It makes sense that if we take a large enough sample,⎯x should be pretty close to the actual value of μ. But the chances are pretty small that⎯x turns out to be exactly μ. This is why we call⎯x a point estimate of μ. The key here is that sample statistics estimate population parameters. For example,⎯x is a point estimate of μ and is a point estimate of p. So we know the value of⎯x is probably pretty close to μ, but we want to get even closer. So rather than just say⎯x is close to μ, we are going to build an interval around⎯x, and then say that μ probably lies somewhere on this interval. We call this an interval estimate. Here’s an example: p̂ Page ② of 89 Suppose you were asked to estimate the average age of all the students in our class. You might survey 10 students and find their average age to be 20. This sample mean of 20 would be a point estimate of μ. BUT you could also express your guess by giving a range of ages centered around your sample mean. So your guess could be 20 give or take 2 years. This “give or take 2 years” part is what we call the margin of error which we will talk about more later. So mathematically, your guess would be 20 +/- 2 which would be the interval estimate. Suppose you were then asked how confident you were that μ, the mean age of all students, was within your interval estimate of 18 to 22 years old. You might say “I am 95% confident that the mean age of all students is within 18 to 22 years old.” In statistics, we construct intervals for the population mean that are centered around an estimate. This estimate is⎯x, the sample mean. Since we can’t get the full population mean, we go for the next best thing. We take a sample and calculate a sample mean. And what we add and subtract from the sample mean to get the interval estimate is the margin of error. When we construct these interval estimates, we call them confidence intervals. Page ③ of 89 So a confidence interval of a parameter consists of an interval of numbers, and this interval is our point estimate +/- our margin of error. Just as in our example above where our interval was 20 +/- 2, or in other words, 18 to 22. 20 is our point estimate and 2 is our margin of error. We call the value we obtain when we take the point estimate minus the margin of error, in our example 20 – 2 or 18, the lower limit or lower bound. And we call the value we obtain when we take the point estimate plus the margin of error, in our example 20 + 2 or 22, the upper limit or upper bound. You will also see the notation of the lower and upper limit in parentheses for confidence intervals. In our above example, the confidence interval may be written as (18,22). Page ④ of 89 It is also important for us to note the level of confidence of a confidence interval. In our example before, our level of confidence would have been that we were 95% confident that the mean age of all students in the class was somewhere on our interval. So the level of confidence is the probability that the interval contains the population parameter, in this case, μ. We will see in examples that as we increase our level of confidence, we will get wider and wider intervals. We will be constructing two different types of confidence intervals: 1. In Section 8.2, we will be calculating the confidence interval for the population proportion, p. 2. In Section 8.3, we will be calculating the confidence interval for the population mean, μ (like our classroom age example). Before we get to these sections, let’s make sure we understand the terms in the example on the next page. In this example, the confidence interval will already be constructed for us. In Sections 8.2 and 8.3, we will actually learn how to construct these confidence intervals. Page ⑤ of 89 Example: Suppose a farmer is trying to estimate the average number of peaches per tree in his orchard. He does not want to count every peach on every tree, so he takes a random sample of a few trees and calculates a 95% confidence interval based on the sample. That 95% confidence interval for the mean number of peaches per tree in the orchard is 112 to 148 peaches per tree. This means that we are 95% confident that the population mean, μ, for the number of peaches per tree is somewhere between 112 and 148 peaches per tree. What is the lower limit? What is the upper limit? What is the level of confidence? What is the width of the confidence interval? What is the sample mean,⎯x? *Remember, the sample mean is always the middle of the confidence interval. The sample mean,⎯x, will always be on the confidence interval, but the population mean, μ, may or may not be on the confidence interval.* What is the margin of error? Page ⑥ of 89 Example of a Confidence Interval for a proportion: Suppose there is an election coming up, and we want to predict what proportion of the votes that candidate A will receive. Suppose we took a random sample of 200 voters and found that 112 of these voters said they would vote for candidate A. What proportion of the voters in our sample said they would vote for candidate A? In other words, what is the sample proportion for this sample? We are trying to predict what proportion of all voters will vote for candidate A by using this sample. What if I told you that the margin of error for a 95% confidence interval to be used to predict the population proportion is equal to 0.07. Construct and interpret this interval. Candidate A will win if he/she gets more than 50% of the votes. Using the interval, are we 95% confident that more than half of the voters will vote for candidate A? Page ⑦ of 89 8.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion? Recall from Section 8.1 that confidence intervals can be written in the general format: point estimate +/- margin of error. The point estimate and margin of error change depending on what parameter is being estimated. For example, we looked at an example of a Confidence Interval for μ, so our point estimate was⎯x. Now we will consider the format of the Confidence Interval for the population proportion, p. The point estimate for this type of Confidence Interval is the sample proportion, = x/n, where x is the number of individuals in the sample with the desired characteristic and n is the sample size. So we know what goes before the +/-, the point estimate, and we can calculate that easily. Now we need to know how to calculate what goes after the +/-, the margin of error. p̂ Page ⑧ of 89 The margin of error will always be a multiple of the standard error. In Section 8.2, we discuss confidence intervals for population proportions, so the standard error will be: Why is it now in the formula and not p? So the margin of error will always be some number times the standard error we see above. The number we multiply the standard error by to get the margin of error TOTALLY depends on the level of confidence. The general formula for a confidence interval for the population proportion is: You can see that the margin of error is this “Z” value times the standard error. Later on in this chapter, we will see how to get this Z value, because this Z value TOTALLY depends on our level of confidence, how confident we want to be that the population proportion is on our interval. For now, we will just focus on 95% confidence intervals, where this Z value equals 1.96. p̂ Page ⑰ of 89 In this example, we saw that: As the level of confidence increases, the margin of error increases and the confidence interval gets wider. ALSO, as the level of confidence decreases, the margin of error decreases and the confidence interval gets narrower. This applies to all confidence intervals, like in the picture below: Why is this true? With a 95% confidence interval, we want to be 95% confident that the population parameter is on the interval. But with a 99% confidence interval, we want to be even more confident (99% confident) that the population parameter is on the interval. So to be that much more sure the proportion is on the interval, we need a wider interval. Page ⑱ of 89 We have seen what happens when we change the confidence level, what about if we change the sample size? As the sample size increases, the margin of error decreases and the confidence interval gets narrower. As the sample size decreases, the margin of error increases and the confidence interval gets wider. So the opposite happens when we increase the sample size. The confidence interval gets narrower. Why is this true? As we increase our sample size, the sample statistic we obtain (whether we are looking for a mean or a proportion) is a better representation of the population. So as we increase our sample size, our point estimate is a better and better estimate, and we don’t need such a wide confidence interval. Page ⑲ of 89 RECAP: The following symbols go along with the following terms when calculating the confidence interval for the population proportion: Term Symbol Point Estimate Margin of Error Standard Error Confidence Interval Page ⑳ of 89 HOW CAN STATCRUNCH CALCULATE THESE CONFIDENCE INTERVALS FOR US? Look back at our example where we wanted to get a 90% confidence interval for the population proportion of ALL Atlantans that own dogs on page 30 of our notes. We got the 90% confidence interval which has a lower limit of .50369 and an upper limit of .69631. On page 31, we got the 99% confidence interval which has a lower limit of .44917 and an upper limit of .75083. Guess what, STATCRUNCH can get these values for us. Go to Stat Proportions One Sample With Summary Here we can type in how many Atlantans owned dogs in our sample. In our sample, 42 of 70 Atlantans owned dogs. Put those numbers in just like this and hit Next: On the next screen choose “Confidence Interval”, and we want a 90% confidence interval, so change the 0.95 to 0.90: Page 21 of 89 Hit Calculate, and here is what we get: It tells us the Sample Proportion, which is the point estimate, which is .6, the same thing we got in part (a). It also tells us the lower limit (.50369) and the upper limit (.69631), the same values we calculated! Notice, it also gives us the standard error. The only values it does not give us are the margin of error, and the Z-score used in the formula, so we still would need to know how to get those by hand. Now get the 99% Confidence Interval and check it against our answers of (0.44917, 0.75083). Page 22 of 89 Section 8.3 How Can we Construct a Confidence Interval to Estimate a Population Mean? Recall from Section 8.1 that confidence intervals can be written in the general format: point estimate +/- margin of error. Remember the point estimate is a single value that is our “best guess” for the parameter. What single number is the “best guess” for a population mean if we only have a sample from the population? The sample mean. So the sample mean is the point estimate part of the confidence interval formula. It is the center of the confidence interval, so now we need to know the margin of error. We need to know what to add and subtract from the point estimate to get the lower and upper limits of our confidence interval. Just like in Section 8.2, the margin of error will be some number times the standard error. But the formula for the standard error when we are talking about means is: Standard error = s / √n where s = standard deviation from our sample We saw the formula for the standard error back in Chapter 7 was σ / √n, but we don’t know anything about the population so we don’t know σ, so we have to use the standard deviation from our sample, s. Page 23 of 89 So we are this far into our formula for the confidence interval for the population mean: ⎯x +/- (some number)*s / √n All we have left to find is the “some number”. We saw in confidence intervals for the population proportion, that this “some number” ended up being a Z-score that corresponded with the level of confidence. For confidence intervals for the population mean, the “some number” still corresponds with the level of confidence, but it is from a new distribution that we call the T-distribution. So if you look on Stat Calculators you will see a calculator just called “T”. Before we see how to get these “T values” let’s talk about the properties of this T-distribution, and how the T- distribution or T-curve is different from the normal distribution or normal curve. Page 24 of 89 Properties of the T-Distribution 1. The T-distribution is centered at 0 and is symmetric about 0, like the standard normal distribution. 2. The total area under the curve is 1. The area to the right of 0 is 0.5 and the area to the left of 0 is 0.5, like the standard normal distribution. 3. The T-distribution is different for different values of n, our sample size. 4. The area in the tails of the T-distribution is a little greater than the area in the tails of the normal distribution. 5. As the sample size n increases, the T curve looks more and more like the normal curve. Since the T-distribution looks different for different values of n, we always have to type in what we call the “degrees of freedom” on the T calculator. The degrees of freedom we have to put in the T calculator = n – 1. The degrees of freedom on the T-calculator is abbreviated as “DF”. So DF in StatCrunch = n – 1 and we always have to put that into the T Calculator. Try some different DF values in StatCrunch and see how the T-distribution changes for different sample sizes. Again it is Stat Calculators T Try DF = 5. Then try DF = 500, this one looks more like our normal curve. Page 25 of 89 So our confidence interval formula for the population mean is: Lower limit:⎯x − T · sn Upper limit:⎯x + T · sn These intervals are valid when we : 1. use a random sample AND 2. either use a sample size > 30 OR when we are sampling from a normal population. So we can get the sample mean, sample standard deviation and n value, but we haven’t yet talked about what the T value is that we want from the T Calculator. To get the T value is just the same as getting the Z value when we were doing confidence intervals for the population proportion in Section 8.2. The only difference is that the T value depends on BOTH the confidence level and the sample size. Page 26 of 89 Let’s find the T value for a 95% confidence interval if the sample size we used is n = 32. First, draw a curve with .95 in the middle and find the area of both tails: Next put in the right tail area = .025 in the T Calculator AND put DF = 32 – 1 = 31. Hit Compute and you get T = 2.03951 Page 27 of 89 Let’s do a few more: These are the same thing they are asking you to get on Homework 8.3-8.4. a) Find the t-score for a 99% confidence interval for a population mean with 5 observations in our sample. First, draw a curve with .99 in the middle and find the area of both tails: Next put in the right tail area = .005 in the T Calculator AND put DF = 5 - 1 = 4. Hit Compute and you get T = So now we can construct confidence intervals for the population means. We can get all the symbols in these formulas: Lower limit:⎯x − T · sn Upper limit:⎯x + T · sn Finally, let’s do some examples. Page 28 of 89 Example 7 in Section 8.3: Ipods are sold all the time on eBay. We have the prices for a random sample of seven Ipods that recently sold on eBay: 235, 225, 225, 240, 250, 250, 210 We will assume these prices are normally distributed. We want to find the 95% confidence interval for the population mean price of Ipods sold. In other words, we want to construct an interval of numbers, and be 95% confident that, if we averaged the price of ALL Ipods sold on eBay, the average price would be on our interval. We need to find: Lower limit:⎯x − T · sn Upper limit:⎯x + T · sn Let’s break it down. n = 7 because we are using a sample of 7 Ipods sold on eBay. How do we get ⎯x and s? Easy, we can list our seven prices in StatCrunch, go to Stat Summary Stats Columns Choose our column with the data and get: so ⎯x = 233.57143 and s = 14.6385 Now finally we need to get the T score: Page 37 of 89 What if we are not dealing with a population proportion example, but a population mean example? That is, we want to know what sample size we need so that the sample mean we get is close enough to the true population mean. For example, maybe we want to estimate the average income for an entire company. We want to take a sample of their employees, and get a sample mean of their income. And we want this sample mean income to be within $5000 of the entire company’s mean income with 95% confidence. We can determine what sample size is needed so that whatever sample mean income we get, it will be within $5000 of the population mean income, and we can be 95% confident of that. Here is the formula we use to determine sample size for estimating the population mean: 2 2 2 Zn m σ = where σ is the provided standard deviation, m is the margin of error, and Z is obtained just like before. Page 38 of 89 Example: An estimate is needed of the mean height of women in Ontario, Canada. A 95% confidence interval should have a margin of error of 3 inches. A study ten years ago in this province had a standard deviation of 10 inches. (a) About how large a sample of women is needed? (b) About how large a sample of women is needed for a 99% confidence interval to have a margin of error of 3 inches? Page 39 of 89 Chapter 9 Statistical Inference: Significance Tests about Hypotheses (Hypothesis Testing) 9.1: What are the steps for performing a Significance Test? In this section we will introduce the language and steps of significance testing. The procedures will be addressed in later sections of Chapter 9. Basics of Significance Testing 1. A statement is made about a population parameter. 2. A claim is made that this statement is incorrect. 3. Evidence (sample data) is collected in order to test the claim. 4. The data are analyzed in order to support or refute the claim. Example: A car manufacturer advertises a mean gas mileage of 26 mpg. A consumer group claims that the mean gas mileage is less than 26 mpg. A sample of 33 cars is taken and the sample mean for these 33 cars is 25.2 mpg. Significance testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of one or more populations. We use sample data to test hypotheses. Page 40 of 89 The Five Steps of a Significance Test: 1. Assumptions 2. Hypotheses 3. Test Statistic 4. P-value 5. Conclusion Page 41 of 89 1. Assumptions – each type of test will have certain assumptions that we need to check (ex. is the sample size large enough?) 2. Hypotheses Each significance test has two hypotheses about a population parameter: the null and alternative hypotheses. The null hypothesis, denoted H0 (read “H-naught”) is a statement to be tested. The null hypothesis is assumed true until evidence indicates otherwise. In this chapter, it will be a statement regarding the value of a population parameter. In our car example, the null hypothesis is H0 : μ = 26 mpg This is the statement made by the car manufacturer that we have to accept as true before we test the claim. The alternative hypothesis, denoted HA, is a claim to be tested. Generally, this is a statement that says the population parameter has a value different, in some way, from the value given in the null hypothesis. In experiments, we are usually trying to find evidence for the alternative hypothesis. In our car example, the alternative hypothesis is HA : μ < 26 mpg This is the claim made by the consumer group, that the mileage is less than what the car manufacturer stated. Page 42 of 89 There are three ways to set up the null and alternative hypotheses. 1. Less than test (left-tailed test) H0: parameter = some value HA: parameter < some value Example: A car manufacturer advertises a mean gas mileage of 26 mpg. A consumer group claims that the mean gas mileage is less than 26 mpg. 2. Greater than test (right-tailed test) H0: parameter = some value HA: parameter > some value Example: A newspaper states that a candidate will receive 46% of the votes in an upcoming election. An analyst believes the percentage will be higher than 46%. 3. Not equal to test (two-tailed test) H0: parameter = some value HA: parameter ≠ some value Example: Five years ago, the average daily rainfall in a jungle was 2 inches. A scientist thinks it is different now. We always test about population parameters, like μ and p. We never test about sample statistics, like⎯x and p-hat, because they change with every sample. Page 43 of 89 Example: Determine whether the significance test is left- tailed, right-tailed or two-tailed. a) H0: μ = 26 HA: μ < 26 b) H0: p = 0.46 HA: p > 0.46 c) H0: μ = 2 HA: μ ≠ 2 Page 44 of 89 3. Test Statistic In all of these tests, we will be testing the population parameter based on what we get in a sample. The test statistic tells how far away the sample statistic is from the assumed population parameter. It tells us this information in terms of how many standard errors away the sample statistic we get is from the assumed population parameter. Think of the car example. The manufacturer states that their cars get 26 mpg. We take a sample of their cars, and in our sample, their cars get 25.2 mpg on average. We want to see how far away our sample mean, 25.2, is from the assumed population parameter, 26, in terms of the standard error. Example: If the test statistic = -1, then the sample mean of 25.2 was only one standard error below the population mean of 26 mpg. Draw a curve to represent this. Example: If the test statistic = -2.5, then the sample mean of 25.2 was 2.5 standard errors below the population mean of 26 mpg. Draw a curve to represent this. We will see the formula for how to calculate this test statistic for different tests in Sections 9.2 and 9.3. Page 45 of 89 4. P-value The p-value is the probability of getting a sample statistic as far away (or further) from the parameter as we did if the null hypothesis is true. In our car example, the p-value is the probability that we would get a sample mean of 25.2 or lower if the true population mean equals 26. Example: If the test statistic = -1, then the sample mean of 25.2 was only one standard error below the population mean of 26 mpg. Draw a curve and SHADE IN THE AREA REPRESENTING THE P-VALUE. Example: If the test statistic = -2.5, then the sample mean of 25.2 was 2.5 standard errors below the population mean of 26 mpg. Draw a curve and SHADE IN THE AREA REPRESENTING THE P-VALUE. Page 46 of 89 If this p-value is NOT very small, then we got a sample statistic pretty close to the population parameter we were testing, and we will NOT reject that population parameter, like in our first example where the sample mean was only one standard error below the population mean. BUT if this p-value is very small, then we got a sample statistic pretty far away from the population parameter we were testing, and we will reject that population parameter, like in our second example where the sample mean was 2.5 standard errors below the population mean. We will see how to use our calculators in StatCrunch to calculate this p-value in Sections 9.2 and 9.3. Page 47 of 89 5. Conclusion At the end of these significance tests, we state our conclusion. Our conclusion will be one of these two: 1. If our sample statistic is “significantly” different from the stated population parameter, like if our sample mean is way off from the stated population mean, then we reject H0 and accept HA. When the null hypothesis is rejected, we say that there is enough evidence to reject the null hypothesis and accept the alternative hypothesis. 2. If our sample statistic is NOT “significantly” different from the stated population parameter, like if our sample mean is pretty close to the stated population mean, then we DO NOT reject H0 and we CANNOT accept HA. When the null hypothesis is NOT rejected, we say there is NOT enough evidence to reject the null hypothesis and accept the alternative hypothesis. NOTE: We NEVER accept H0, we just don’t reject H0. Page 48 of 89 Example: According to the American Hotel Association, the average price of a room was $78.62 per night in 1998. An analyst believes that this value has increased since then. a) Determine the null and alternative hypotheses. What type of test is this? b) Suppose sample data indicate that we should not reject H0. State the conclusion of the researcher. Page 57 of 89 5. Conclusion - .19324 > .05 The P-Value > the given significance level, so we will not reject H0 and state that there is not sufficient evidence to reject H0 and accept HA at the .05 level of significance. Interpretation: So we are not rejecting the magazine’s statement that the population proportion of people voting for candidate A is 40%. Our sample proportion was not far enough away from .4 or 40% to reject it. This was done at the 5% level of significance. The level of significance is saying that if we wanted to reject the null hypothesis, there needed to be less than a 5% chance of us getting the sample proportion that we got, and there wasn’t. There was a 19.324% chance of us getting our sample proportion or higher. Page 58 of 89 Now, that we have done all this work and see the steps in a hypothesis test, let’s see how StatCrunch can easily do it for us. Go to Stat Proportions One Sample with summary In our sample, we had that 86 out of the 200 people in our sample said they would vote for candidate A. So the number of yeses or “successes” = 86 And the number of observations = 200 Hit Next and choose “Hypothesis Test” The magazine states that 40% or 0.4 of the entire population will vote for candidate A. So enter the null hypothesis that the proportion = 0.4. Our alternative hypothesis is our claim, we think that proportion is really higher than 40% or 0.4, so choose “>” Hit Calculate and what we get is on the next page: Page 59 of 89 Same test statistic = .86603 and same p-value = .1932 So much easier! Now use StatCrunch to check and see how the conclusion would have changed if in our sample 96 out of 200 said they were voting for candidate A, instead of only 86 out of 200. Page 60 of 89 Example: In a 1998 article, the credit card industry asserted that 50% of college students carry a credit card balance from month to month. In a random sample of 300 college students, 174 carried a balance each month. Is this sufficient evidence to claim that the proportion is different from 50% at the .01 significance level? 1. Assumptions - Is this a random sample and is the data categorical? Is n * p0 ≥ 15 and n * (1-p0) ≥ 15? 2. Hypotheses – Set up the null and alternative hypotheses Before we get to step 3, what is p-hat, the sample proportion? Page 61 of 89 3. Test Statistic - We want to see how far away our sample proportion is from the assumed population proportion in terms of the standard error. So our sample proportion is 2.77128 standard errors above the assumed population proportion. 4. P-value - Because HA has ≠, this is a two tailed test. Draw a graph with the population proportion in the middle and shade the area to the right of our sample proportion. Also, shade the same symmetrical area on the left. This shaded area of both tails is the p-value. Draw a standard normal curve and shade the area to the right of our positive Z-score, and to the left our our negative Z-score. This shaded area is also the p-value. Page 62 of 89 We need to find the area in both tails because this is a two- tailed test. We can do this on StatCrunch by putting in the positive Z- score and finding the area to the right of it. Our area in the left tail (area to the left of the negative Z- score) will be the same because the graph is symmetrical. So our p-value = .00279*2 = .00558 This means that there is a .00558 or only a 0.558% chance that we would get a sample proportion this far off from the population proportion, if the assumed population proportion is correct. Page 63 of 89 5. Conclusion - .00558 ≤ .01 The P-Value ≤ the given significance level, so we will reject H0 and state that there is sufficient evidence to reject H0 and accept HA at the .01 level of significance. Interpretation: So we are rejecting the credit card company’s statement that the population proportion of college students that carry credit card debt is 50%, and accepting our claim that the population proportion of college students that carry credit card debt is different from 50%. We got a sample proportion far enough away from 50% that we could make this claim. This was done at the 1% level of significance. The level of significance is saying that if we wanted to reject the null hypothesis, there needed to be less than a 1% chance of us getting the sample proportion that we got, and there was. There was only a 0.558% chance of us getting a sample proportion this far from the population proportion. Page 64 of 89 Again, let’s see how StatCrunch can easily do it for us. Go to Stat Proportions One Sample with summary In our sample, we had that 174 out of the 300 college students in our sample had credit card debt. So the number of yeses or “successes” = 174 And the number of observations = 300 Hit Next and choose “Hypothesis Test” The credit card company states that 50% or .5 of the college students carry credit card debt. So enter the null hypothesis that the proportion = .5 Our alternative hypothesis is our claim, that we think that proportion is really different from 50% or .5, so choose “≠” Hit Calculate and what we get is on the next page: Page 65 of 89 Same test statistic = 2.77128 and same p-value = .0056 Page 66 of 89 Page 67 of 89 Section 9.3: Significance Tests about Means In this section, we run significance tests to test values of population means that have been given, like our car manufacturer example: Example: A car manufacturer advertises a mean gas mileage of 26 mpg. A consumer group claims that the gas mileage is less. A sample of 33 cars is taken and the sample mean for these 33 cars is 25.2 mpg. Before we work through examples, we need to see how the 5 steps in the test have changed. We will see the big change is the new formula for the test statistic, and the fact that we treat the test statistic as a t-value rather than a Z-score like we used in Section 9.2. Page 68 of 89 Steps in a Significance Test about a Population Mean: 1. Assumptions – When performing significance tests about a population mean, we need the following assumptions to be true: a) The variable is quantitative b) The data are obtained using randomization (like a random sample) c) The population distribution is approximately normal or we are using a sample size ≥ 30. 2. Hypotheses – We set up the hypotheses just like we have seen before. The null hypothesis will be μ = value (like μ = 26) If it is a two-tailed test, the alternative hypothesis will be μ ≠ value (like μ ≠ 26) If it is a one-tailed test, the alternative hypothesis will be either μ < value or μ > value (like μ < 26) Page 77 of 89 Problem on Homework Chapter 9 An industrial plant claims to discharge no more than 1000 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental action group decides to monitor the plant, in case this limit is being exceeded. Doing so is expensive, and only a small sample is possible. A random sample of four hours is selected over a period of a week. Test at the 0.05 significance level. Assume the distribution of wastewater is approximately normal. The observations are below. 2000, 1500, 3000, 2500 1. Assumptions - Is this a random sample and is the variable quantitative? Is the population normally distributed? 2. Hypotheses – Set up the null and alternative hypotheses We aren’t given the sample mean and standard deviation but we can put these numbers into StatCrunch (Stat Summary Stats Columns) and get that the sample mean = 2250 and the sample standard deviation = 645.49725. Page 78 of 89 3. Test Statistic – We want to see how far away our sample mean is from the assumed population mean in terms of the standard error. So our sample mean is 3.87298 standard errors above the assumed population mean. 4. P-value - Because HA has >, this is a right tailed test. Draw a graph with the population mean in the middle and shade the area to the right of our sample mean. This shaded area is the p-value. Draw a t curve and shade the area to the right of our t-value. This shaded area is also the p-value. Page 79 of 89 We need to find that area to the right of our t-value. We can do this in StatCrunch by putting in the t-value and finding the area to the right of it. So our p-value = .01523 This means that there is only a .01523 or a 1.523% chance that we would get a sample mean this far off from the population mean, if the assumed population mean is correct. Page 80 of 89 5. Conclusion - .01523 ≤ .05 The P-Value ≤ the given significance level, so we will reject H0 and state that there is sufficient evidence to reject H0 and accept HA at the .05 level of significance. So we are rejecting the industrial plant’s statement that the population mean of the wastewater per hour that the plant puts out is 1000 gallons, and accepting our claim that the population mean is greater than 1000 gallons. Page 81 of 89 Now let’s see how StatCrunch can do it for us. Put in the four values from our data. Go to Stat T Statistics One Sample with data and select the column you put the data in. Choose “Hypothesis Test”. Our null hypothesis is that the mean = 1000 gallons so input 1000 in the first blank. Our claim, the alternative hypothesis, is that it is greater than 1000 gallons so put in “>” in the second blank and hit Calculate. Here is our output: Look, the same test statistic = 3.87298 and the same p-value = .0152. Page 82 of 89 Page 83 of 89 Section 9.4: Decisions and Types of Errors in Significance Tests Type I and Type II errors As we have stated earlier, we use sample data to determine whether or not we will reject the null hypothesis. BUT because we are using only sample information, we could reach an incorrect conclusion. Four possible outcomes from Significance Testing 1. We reject H0 when in fact H0 is false and HA is true. This decision is correct. 2. We do not reject H0 when in fact H0 is true. This decision is correct. 3. We reject H0 when in fact H0 is true. This is incorrect. This is called a Type I error. 4. We do not reject H0 when in fact H0 is false and HA is true. This is incorrect. This is called a Type II error. Below is a chart that breaks down these outcomes Reality Decision H0 is true H0 is false Do not reject H0 Correct Type II Error Reject H0 Type I Error Correct Page 84 of 89 Example: According to the U.S. Department of Justice, the mean age of a death row inmate in 1980 was 36.7 years. A district attorney believes that the mean age of a death row inmate is different today. a) Determine the null and alternative hypotheses. b) If a Type I Error was made, which of the following statements would apply? i. The null hypothesis was not rejected. ii. The null hypothesis was rejected. iii. The researcher decided there was enough evidence to indicate a change in the mean age. iv. The researcher decided there was not enough evidence to indicate a change in the mean age. v. Actually the mean age has changed significantly. vi. Actually the mean age has not changed significantly. Page 85 of 89 c) If a Type II Error was made, which of the following statements would apply? i. The null hypothesis was not rejected. ii. The null hypothesis was rejected. iii. The researcher decided there was enough evidence to indicate a change in the mean age. iv. The researcher decided there was not enough evidence to indicate a change in the mean age. v. Actually the mean age has changed significantly. vi. Actually the mean age has not changed significantly. Page 86 of 89 Interpretation of Error: H0 : the defendant is innocent HA : the defendant is guilty A Type I Error means we reject the null hypothesis (so we don’t think the defendant is innocent), when the null hypothesis is really true (the defendant really is innocent). So in this type of error, we have put an innocent person in jail. A Type II Error means we do not reject the null hypothesis (so we think the defendant is innocent), when the null hypothesis is really false (the defendant really is guilty). So in this type of error, we have let a guilty person go free. Page 87 of 89 The Level of Significance We never really know whether the results of a significance test result in an error or not. However, just as we place a level of confidence in the construction of a confidence interval, we can limit the probability of making errors. The symbol we use is α: α = level of significance We will always be given the level of significance when we are performing significance tests. For example, they might say “Run this test at the 5% level of significance.” which would mean α = .05. What they are really saying is “If you are going to reject the null hypothesis, make sure there is less than a 5% chance that you would get this extreme a sample statistic.” We then take the calculated p-value and compare it to this level of significance, α. If our calculated p-value is less than the stated significance level, we can reject the null hypothesis. Page 88 of 89 Section 9.5: Limitations of Significance Tests We can also perform significance tests using a confidence interval, and we have actually already done problems like this. If the confidence interval contains the value we are testing, then we do not reject the null hypothesis. If the confidence interval does not contain the value we are testing, then we do reject the null hypothesis. So which is better: a significance test or a confidence interval? The consensus is that confidence intervals are actually better. Why? The reason is confidence intervals actually give us a range of possible values for the population parameter. In a significance test, we can only test if the population parameter is above, below or not equal to a stated value.