Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the linear correlation coefficient from a Minitab display. Here's our problem statement: The Minitab output shown below was obtained by using paired data consisting of weights in pounds of 26 cars and their highway fuel consumption amounts in miles per gallon. Along with the paired sample data, Minitab was also given a car weight of 5,000 pounds to be used for predicting the highway fuel consumption amount. Use the information provided in the display to determine the value of the linear correlation coefficient. Be careful to correctly identify the sign of the correlation coefficient. Given that there are 26 pairs of data, is there sufficient evidence to support a claim of linear correlation between the weights of cars and their highway fuel consumption amounts? Part 1 OK, the first part of this problem asks for the linear correlation coefficient. And to find that, we're going to take a look at the Minitab display here. So here's the Minitab display. And notice there's quite a bit of stuff here. But everything we're really going to need is at the top of the display here. The linear correlation coefficient is normally listed in software output with the variable R. Well, R is not shown here, but we do have R squared. So we can take this value for R squared, and if we take the square root of R squared, that leaves us with R. So all I have to do is take the square root of this value here. So 0.636. Notice I'm converting from the percent to a decimal. Take the square root, and there's my R value. But we know that values for the linear correlation coefficient can be positive or negative. So which is it here? If I take a positive number and square it, I get a positive number. If I take a negative number and square it, I also get a positive number. So how do we know whether this is positive or negative? Well, look at the model that they're giving us, the regression equation. If you look at the value for the coefficient in front of your independent variable here, notice it's negative. That means this line, when you graph it, is going to have a negative slope. And aligned with a negative slope is a negative linear correlation coefficient. So the value for R we're looking for is going to be negative. Now I know that I need put in my negative sign and then put in the number we calculated rounded to three decimal places. Excellent! Part 2 And now the last part of the problem asks, "Is there sufficient evidence to support a claim of linear correlation?" Well, let's go back to our Minitab display, and we're going to look at this table here with our predictor values and the actual testing that was run on each of them. Notice here, we want the one for the model itself, which is going to be the independent variable. There's a P-value here on the end. So this value here is the one we want to look at.
And that's a number that's either zero or a number that's so low it's practically zero. And so therefore we're just going to say zero. And when you have a P-value of zero, any reasonable significance level you would test against, you're going to be inside the region of rejection. And when you're inside the reason of rejection, you reject the null hypothesis. And every time you reject the null hypothesis, there's always sufficient evidence. So is there sufficient evidence the answer's going to be yes. Good job! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video.
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Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform hypothesis testing on the variances of skull measurements. Here's our problem statement: Researchers measured skulls from different time periods in an attempt to determine whether interbreeding of cultures occur. Results are given below. Assume that both samples are independent simple random samples from populations having normal distributions. Use a 5% significance level to test the claim that the variation of maximal skull breaths in 4,000 BC is the same as the variation in 150 AD. Part 1 OK, we've got our two samples here. Sample size is n. The sample mean is x-bar, and the sample standard deviation is s. The first part of the problem asks us, "What are the null and alternative hypotheses?" Well, the null hypothesis is by definition a statement of equality, so we're not going to select Answer option C. Of the three answers that remain, we're going to be looking to see what is the alternative hypothesis so we can select the correct answer. Typically, the alternative hypothesis is going to reflect the claim. What's the claim here? Well, here we're testing the claim that the variation of maximal skull breaths is the same for both of those years. So we're looking for something that says they're equal. Well, equality by definition belongs to the null hypothesis. So we have to take the compliment of the claim as our alternative and say that they're not equal to. And if I look at the answer options that are left, Answer option B is going to be the one we want because here it says not equal to. Fantastic! Part 2 The next part of this problem asks for the test statistic. Notice the test statistic is an F-score. The F-score is calculated very easily. We can do that inside or outside of StatCrunch. I'm going to use StatCrunch just because I'm a little lazy. We could do it well enough in our calculator. But like I said, I'm just a little bit lazy, so I'm just going to let the computer do it for me. Hey, I love living in the 21st century. So here we have StatCrunch. I'm going to go to Stat --> Variance Stats (because we're dealing in variance) --> Two Sample (because we have two samples we're comparing) --> With Summary (because we don't have actual data, just summary stats). Here in the options window, we're asked for the variances. But keep in mind that when you're looking for an F-score, there's one little caveat you've got to remember. And that is typically when we're putting in our summary stats here in the option window, the sample that's listed first is normally Sample 1, and the sample that's listed next is Sample number 2. But when you're dealing with an F-score, the sample with the greater variance or the greater standard deviation is always going to be Sample number 1. So notice here the sample listed second has the higher standard deviation. So even though the samples is listed second, it's actually going to be Sample number 1. Notice also we're asked here for the variance, but we're given standard deviation. So that means we've got to take this number and square it to get the variance. So here in my calculator, I'm going to put in the standard deviation that I'm given, I square it, and that gives me the sample variance. Notice I'm putting all of the numbers in. I do the same thing for the other sample. And now I perform a hypothesis test. I need to make sure that these values are correct. This is typically always going to be 1 here for your claimed value. And that's what we have here. When you take any number and divide it by itself, if they're the same, you're going to get 1 out. So we're just gonna leave that alone. And then here we need to make sure that this inequality sign matches, and it does. So now I hit Compute!, and here's my F-score right here in the table. If you ever see an F-score that's less than one, that means you flip-flopped your samples. So go back and reverse. You've got Sample 1 and Sample 2 inappropriately labeled. You've got to flip those around and put them in the right order. That's a little caveat you got to remember when you're dealing with your F-score. Fantastic! Part 3 Now it asks for the P-value, and the P-value is there in my results window, that last value there at the end of the table. I'm asked to round to three decimal places. Fantastic! Part 4 And now I'm asked for a conclusion for the hypothesis test. Well, with a P-value of almost 80%, it doesn't it matter what significance level we would compare that with; anything that would be reasonable to use, we're going to be greater than. So we're definitely outside the region of rejection. Therefore we're going to fail to reject the null hypothesis. Whenever you fail to reject the null hypothesis, there is always insufficient evidence. So we want to select Answer option C. Nice work!
And that's how we it Aspire Mountain Academy. Be sure to leave your comments below, and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform hypothesis testing on two independent sample means of soda can fill volumes. Here's our problem statement: Data on the weights in pounds of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete Parts A and B below. Use a 1% significance level for both parts. Part A1 OK, the first part for Part A says, "Test the claim that the contents of cans of diet soda have weights with the mean that is less than the mean for the regular soda. The null and alternative hypotheses we can get very easily if we understand that the null hypothesis is by definition a statement of equality. So therefore we're not going to select Answer option A because here the null hypothesis is not a statement of equality. The alternative hypothesis will help us to select between the answer options that remain. And we get the alternative hypothesis typically by reflecting the claim. In this case, the claim here is that the diet soda weight has an average that is less than the average from the regular soda. So here in our summary table, we can see that the diet soda is going to be the first sample and the regular soda is going to be the second sample. So diet is less than the regular, so 1 is going to be less than 2. So I go down here and look for the answer option that remains with the alternative hypothesis, that 1 is less than 2. Good job! Part A2 Now I'm asked for the test statistic. And notice here our test statistic is a t-score because it says t right here. So I'm going to open up StatCrunch, and I'm going to resize this window so we can see everything better of what's going on. OK, so in StatCrunch, now I can go to Stat --> T Stats (because we're looking for that t-score) --> Two Sample (because I've got two samples I'm comparing) --> With Summary (because I'm given a summary table and not actual data). Here we put in the values for each of the different samples. So notice here we've got n (which is our sample size), x-bar (which is the sample mean), and then s (which is the sample standard deviation). So I'm just going to put those values in here for the first sample. And I'm going to do the same thing for the second sample. And now this box here for Pooled variances is left unchecked. You want to leave that alone; it needs to be unchecked. Down here for our hypothesis test, we need to make sure that this matches what we selected over here. Notice the difference in the way StatCrunch writes it versus the way it's written in your assignment. That's OK. We'll just recognize that they're algebraic equivalents. Here it's going to be equal to each other. So when you take a number and subtract the same number from itself, that's going to give you zero. So we got to leave that claim value alone. You need to make sure that this inequality sign for the alternative hypothesis matches. And now we've got everything we need. And here's our test statistic right here, second to last value in the table in the results window. So I'm going to put that in here. I'm asked around to two decimal places. Nice work! Part A3 Now, I'm asked for the P-value. The P-value is located here in the results table, last value. Notice it says <0.0001; the number is so small --- it's not zero, but it's so small that it might as well be zero. So that's just what I'm going to put here in my answer field. Well done! Now I'm asked to state a conclusion for the test. The P-value was zero, and with a P-value of zero, you're always going to be inside that critical region, the region of rejection. Therefore we reject the null hypothesis. So we're going to choose Answer options B or C. And every time you reject the null hypothesis, there is always sufficient evidence. So Answer option C is the one we want. Fantastic! Part B1 Now Part B asks for a confidence interval appropriate for the hypothesis test that we just conducted. I could go through the menu options again, but I'm a little lazy. So I'm going to go back to click Options here on my results window, click on Edit, and it takes me right back to the options window where I flip the radio button over to confidence interval. I need to put the right level in. And here it says use a 1% significance level for both parts. So normally we would say 1% significance level means 99% confidence interval because I'm just subtracting that from 100%. But we've got two samples. Therefore, I have to subtract 2α. So what I really want is 98%. And here in my results window, you can see the lower and upper limits that I need to put into my answer fields. So I'm going to do that here. I'm asked to round to three decimal places. Well done! Part B2 And now this final part asks, "Does the confidence interval support the conclusion found with the hypothesis test?" Well, what does the confidence interval say? Where is zero with respect to the confidence interval? Well, zero is not inside the confidence interval. Zero is outside the confidence interval. We look at these numbers as though they're on a number line. Zero is going to be here to the right outside the confidence interval. Therefore, there's going to be a difference between the two main values. They're not the same, but that's what the null hypothesis here says. The null hypothesis says they are the same. We have evidence that they're not the same. Therefore we're going to reject this statement because it's false.
That's the same thing that we got here from the hypothesis test --- rejected the null hypothesis. So we're going to say, yes, the confidence interval does support the conclusion from the hypothesis test. They’re the same conclusion, because the confidence interval contains in this case only negative values. And you can see both our upper and lower limits are negative. So only negative numbers are going to be in between. Fantastic! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below, and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform hypothesis testing on two proportions of common attributes. Here's our problem statement: Two different simple random samples were drawn from two different populations. The first sample consists of 40 people with 19 having a common attribute. The second sample consists of 1800 people with 1,272 of them having the same common attribute. Compare the results from a hypothesis test of p1 = p2 where the 1% significance level and a 99% confidence interval estimate of p1 - p2. Part 1 OK, the first part of this problem asks us for the null and alternative hypotheses for our hypothesis test. The null hypothesis is by definition a statement of equality, so we know we're going to be selecting Answer option B, Answer option C, or Answer option D. To distinguish between these three options to find the correct answer, we need to look at the alternative hypothesis. Notice they're all different in each case. The alternative hypothesis typically reflects the claim unless the claim has any semblance of equality to it, in which case we're going to take the compliment for our alternative hypothesis. If you read the problem statement up here, you'll notice that the typical words testing the claim that don't appear here in the problem statement, so we have to kind of infer what's going on from the words that were given. Here it says, "Compare the results from our hypothesis test of p1 = p2." Right here, what we're saying here is that the claim is that the population proportions are equal to each other. That's the claim. Well, equality by definition belongs to the null hypothesis, so therefore we want to take the compliment of this. And that's going to be not equal to, so it looks like C is going to be our correct answer here. Excellent! Part 2 The next part asks for the test statistic. To get the test statistic, we need to run a hypothesis test and so I'm going to open StatCrunch. And we're going to move this window here so we can see better everything that's going on. OK, here in StatCrunch, I don't need any actual data because I've got summary statistics here in the problem statement. So to run a hypothesis test, I'm first going to go to Stat --> Proportion Stats (because we're dealing with proportions) --> Two Sample (because we have two samples we're comparing) --> With Summary (because we don't have actual data, just summary statistics). Here in the options window, I need to provide the summary stats for each of my samples. In the problem statement, it says the first sample consists of 40 people with 19 having the common attribute. So the success here is having the common attribute. So that's going to be 19. And there's 40 total in that sample. And then I do the same thing for the second sample. The radio button for our hypothesis test is already selected, so I'm good there. Notice how StatCrunch writes the hypothesis test a little differently than what you see in your assignment. So you have to do a little bit of algebraic equivalence here. If p1 is equal to p2 and I subtract p2 from each side, then that's going to leave zero on the right side. Or looked at another way, if I were to say that these two are equal and I subtract them, any number minus itself is going to be zero. So this is the number that we want for our claimed value here in our options window. Make sure that this inequality sign for your alternative hypothesis matches, and it does. So now we've got everything we need. I press Compute!, e voila! Here at the very end of the table, second to last value, I see my test statistic, which is a z-score. Nice work! Part 3 The next part asks us to identify the critical values. The critical values are going to come from the distribution that we're using. And because we're dealing with proportions and we have a z-score for our test statistic, we're going to need to use the standard normal distribution, because z-scores come from the standard normal distribution. So back here in StatCrunch, I go to Stat --> Calculators --> Normal. To know whether we use a one-tail or two-tail test, we need to go back here and look at our alternative hypothesis here. The alternative hypothesis has an inequality sign of not equal to; that means we have a right tail test --- excuse me, a two tail test --- one tail on the right and one tail on the left. And so therefore we want to use the Between option here in our calculator. So we've got two tails that we're looking at for our distribution. For our critical region, the significance level is 1%. That means there's 1% of the area under the distribution curve found the tails of the distribution. That means the area in between, which is what StatCrunch is measuring here, is going to be 99%. So I just stick 99% here, hit Compute!, e voila! Here are my critical values that I need to enter here in my answer field. And I could enter in the negative and then a comma and then the positive one, but I'm a little lazy. So I just like to use this plus or minus sign, and then that way I don't have to type the number twice. I only type the number once. Excellent! Part 4 Now the next part asks for a conclusion based on the hypothesis test. Look here, we're asked to use the test statistic to evaluate the hypothesis test. And that's easy enough to do. So here we have our critical values plus or minus 2.57, so that's going to be the edge here of this red area. The test statistic itself, -3.17. Well, -3.17 is located here on the number line, which is inside the left tail of our critical region. Since we're inside the critical region, we're inside the region of rejection, and therefore we fail to reject --- excuse me, we reject the null hypothesis because we're in the reason of rejection. So here the test statistic is in the critical region, so we reject the null hypothesis. And whenever we reject the null hypothesis, there is always sufficient evidence. Good job! Part 5 Now the next part asks for a 99% confidence interval. We could go through the menu options again, but yeah, I'm a little bit lazy. So I'm just going to go back here to this previous window that we had. Click on the Options button, and in the drop down menu, select Edit. That takes me back to the options window where all I do is select the radio button for confidence interval, make sure my level is at the right level, and hit Compute!, e voila — the lower and upper limits for my confidence interval, which I can stick here in my answer field. I'm asked to round to three decimal places. Nice work! Part 6 And now this next part asks for a conclusion based on the confidence interval. Well, we have to look and see where is zero. Is it inside or outside of the confidence interval? And there's the lower limit. There's the upper limit. Zero is going to be outside. If we look at this as a number line, zero is going to be over here to the right of our interval where zero is outside the interval. So it's not included in the interval. And that means there's going to be a difference between the two parameters that we're evaluating. And so that means we're going to reject the null hypothesis, because obviously the null hypothesis here are saying that they're the same. Well, our confidence interval says that can't be. So therefore we're going to reject this because it's a false statement. Nice work! Part 7 And now the last part of the problem asks, "How do the results from the hypothesis test and the confidence interval compare?" Well, notice here in both cases, we rejected the null hypothesis. So the results are going to be the same. In each case, we suggested that the population parameters are not equal to each other, because we rejected the null hypothesis which says that they are equal to each other. I check my answer. Nice work!
And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform hypothesis testing on standard deviations of drive through service times. Here's our problem statement: The accompanying data are drive through service times (in seconds) recorded at a fast food restaurant during dinner time. Assuming that dinner service times at the restaurant's competitor have a standard deviation of 53.5 seconds, use a 0.025 significance level to test the claim that service times at the restaurant have the same variation as service times at its competitor's restaurant. Use the accompany data to identify the null hypothesis, alternative hypothesis, test statistic, and P-value. Then state a conclusion about the null hypothesis. Part 1 OK, this first part of the problem wants us to identify the null and alternative hypotheses. The null hypothesis is by definition a statement of equality. So we know that the right answer is not going to be Answer option A or Answer option B, because in these instances, the null hypothesis is not a statement of total equality. So it's gotta be Answer option C or D. How do we choose between them? We'll look at the alternative hypotheses. The alternative hypothesis is generally a statement that reflects the claim, unless the claim has any semblance of equality to it, in which case we take the complement. What's the claim here? Well, look back in the problem statement. Here's your key words right here --- we're "testing the claim that" --- and then what follows is the actual claim here, service times at the restaurant have the same variation as service times at the competitor's restaurant. So we're going to say that they're equal. Well, equality by definition belongs to the null hypothesis. So therefore we have to take the complement and say that it's not going to be equal to, and that's going to be Answer option C. Nice work! Part 2 Now we're asked to compute the test statistic. And to do this, we have to actually run the hypothesis test. So we're going to click on this icon here to take a look at our data and then click on this icon to open up our data in StatCrunch. OK, we've got our data here in StatCrunch. I'm going to resize this so we can see more what's going on. And we don't need this window any more. OK, so here in StatCrunch, we're going to run a hypothesis test by going to Stat --> Variance Stats (because we're looking at standard deviation and variance is the only option given here in StatCrunch when we're dealing with standard deviation) --> One Sample (because we've got just the one sample) --> With Data (because we have actual data here in the options window). I'm going to select the column where my data is located, and then I've got to make this area here for Hypothesis Test match the hypothesis test we got in the previous part. But notice we're dealing here with variance. That means we've got to square everything that goes in the options window. So notice here in our original hypothesis we're using sigma as our population parameter, but here in StatCrunch in the options window, we've got sigma squared. So we've got to square everything that comes out. So 53.5 squared is what I want to put there in the options window for my claimed value. And now that I've done that, I can go ahead and hit Compute!, and out comes the chi squared test statistic, which is what we got here, chi squared. I'm asked to round to two decimal places. Excellent! Part 3 The next part of the problem asks for the P-value, which we get by looking here at the last value here in the table. I'm asked to round to three decimal places. Good job! Part 4 Now the last part asks us to state a conclusion about the null hypothesis. Well, our P-value (16.1%) is definitely greater than the 2.5% significance level we're asked to use for comparison. So therefore we're outside the reason of rejection, and there is not sufficient evidence. Whenever you fail to reject, there's not sufficient evidence. And we're doing this because the null hypothesis is not rejected; we fail to reject. Excellent!
And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find a confidence interval given summary sample data. Here's our problem statement: Refer to the accompanying data display of results from a sample of airport data speeds in megabytes per second. Complete Parts A through C below. Part A OK, Part A says, "Express the confidence interval in the format that uses the less than symbol given that the original list of data use one decimal place around the confidence interval limits accordingly." OK. Typically when students encounter a problem like this, by this part of the term you're looking at Chapter 7 here, you've gotten into the habit of running to StatCrunch or running to Excel to solve a problem. And here you just need to take a step back and realize the answer is sitting right in front of you. They're giving you a data display, and look — "t interval" OK, an interval using the, the uh, I guess the Student t distribution, which makes sense because we're looking at means here, and here's our lower and upper limit for a confidence interval. So it's just staring you right in the face. All you gotta do is follow the instructions here, round to two decimal places as needed. So that's what we're going to do. Excellent! Part B Now Part B says, "Identify the best point estimate of mu and the margin of error to get the best point estimate of μ." We need to recognize μ is the population mean, and we learned back in Chapter 6 that the mean is an unbiased estimator. That means the sample mean is going to target the population mean; it makes the best point estimate for your population parameter. So the sample mean, which we represent with x-bar, is listed here in my problem statement. So I'm just going to round that to two decimal places. Fantastic! And now the margin of error. Don't be taking this number here and running with it. That's not your margin of error. Your margin of error is not represented with S(x). This is a standard deviation, so don't be using that. That's not your margin of error. We're going to have to actually calculate margin of error, which we can get pretty easy enough. We've got the point estimate, so we've got the central point for our confidence interval. And we've got the upper and lower bounds here. So if I just take this point estimate and subtract out the lower limit, that distance is going to give me my margin of error. Now alternatively, I could be subtracting the point estimate from the upper limit, but it gives you the same number. The other way to go about it is to just average the two out. But then that just gives you this point estimate here. Then you've got to actually, you know, do one or the other with it. So I just find it easier to just take the point estimate and then subtract out the lower limit. Well done! Part C And now the last part, Part C, asks, "In constructing a confidence interval estimate of μ, why is it not necessary to confirm that the sample data appear to be from a population with a normal distribution?" OK, let's look at each one of these answer options and see which one represents our best option here.
Answer option A says, "Because the sample is a random sample, the distribution of sample means can be treated as a normal distribution." Well, OK, we would hope that the sample is obtained through some random sampling method, but that's just saying how we get the data. The distribution comes from what the data represents, not how we get the data. And so there's no real connection between the two here. So Answer option A isn't going to work for us. Answer option B says, "Because the sample standard deviation is known, the normal distribution can be used to construct the confidence interval." OK, so here we've got a ... sample standard deviation is known. It's listed up here in our summary stats table. But again, in all, this is talking about how the data is spread out. And the distribution comes from what the data represents. So again, there's a faulty logical connection here. Answer option B isn't going to work for us. Answer option C says, "Because the population standard deviation is known, the normal distribution can be used to construct the confidence interval." Well, same thing here. We don't really know what the population standard deviation is, but if we did, that doesn't necessarily tell us what distribution we're supposed to be using to represent our data. Again, the distribution comes from what the data represents, not from how it's sampled and not from what the numbers are --- although, you know, it kind of plays in part to what the numbers are, but it's more representative of what the numbers are represented. That's where the distribution type comes from. So now we're just left with Answer option D: "Because the sample size of 50 is greater than 30, the distribution of sample means can be treated as a normal distribution." Oh, bingo! That's what we're looking for right here. Remember, the Central limit Theorem says that, when your sample size is greater than 30, you can use the normal distribution as an approximation for whatever distribution that data actually represents. So here we've got n = 50, so our sample size is greater than 30, and that means the Central Limit Theorem applies. So we're going to just select Answer option D. Good job! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the best point estimate and confidence interval for proportions. Here's our problem statement: Use the sample data and confidence level given below to complete Parts A through D. In a study of cell phone use and brain hemispheric dominance, an Internet survey was emailed to 2,606 subjects randomly selected from an online group involved with ears. 1064 surveys were returned. Construct a 90% confidence interval for the proportion of returned surveys. Part A OK, here we've got a link to our z-score table, but that's the old school way of doing it, and it's a little calculation involved. So we're going to be lazy (because I'm just feeling lazy today), and we're going to use StatCrunch. So I'm going to go in and open up StatCrunch. And it's not so much that I need any data in StatCrunch, but I just need the functionality. So ultimately we're going to be asked to construct a confidence interval, so let's just go ahead and construct that to begin with. I want a 90% confidence interval for the proportion of returned surveys. So to get my confidence interval, I'm going to go to Stat --> Proportion stats (because we're dealing with proportions) --> One Sample (because I've got just the one sample) --> With Summary (because I don't have any actual data, just summary stats). The number of successes is defined by how many successes we have where success is determined by the confidence interval here. We want a conference interval on the proportion of returned surveys. That means returning a survey is the definition of success. 1064 surveys were returned, so that's the number I'm going to put here in this field. The number of observations is the total number. That's 2606. I want a confidence interval and a 90% confidence limit. So here's my confidence interval. Notice the sample proportion is calculated here for you. You get the same number if you just go ahead and just take these numbers that you put in the options window just divide that out in your calculator. Just take the number of successes divided by total number of observations and, see, I'm going to get the same number here. But I just like doing it in StatCrunch because I gotta go through those steps anyway to get the confidence interval, which it's going to ask me later on the problem. So it's just easier if I just use this. So here's the number I need. Good job! Part B Part B asks us for the margin of error. Now some of you may be tempted to take this next number here and just use this, but that's not the margin of error. That's a value called the standard error, which is calculated a little differently. So this number that you see here is going to be incorrect. How then do we get the margin of error? Well, you can take the upper and lower limits of your confidence interval and you can average them out, but that just gives you the point estimate that's in the middle. We already have the point estimate here. And so if I just subtract the lower limit from the point estimate, that distance will be the margin of error. Alternatively, I can take the upper limit and subtract out the point estimate. That'll give me the same number. We already have the point estimate here in the calculator, so I'm just going to go ahead and subtract out that lower limit, and also putting in all my numbers so that I can avoid rounding errors as much as possible. And there's my margin of error. Whoops, it rounds up. Nice work! Part C And now the confidence interval just comes from the lower and upper limits there at the end of my table. So I'm going to put those in here. Good job! Now before we move on to part D, I just want to mention that some students --- they follow the steps and they get different numbers here. They get, you know, the software from Pearson keeps coming back to them and saying, "Hey, you got the wrong answer here." And typically that's a result of using the wrong confidence level. So if you're getting the wrong answers here, even though you're going through these steps, go back to your options window and make sure you've got the right confidence level in. And more than half the time, that's the problem the students are encountering when they get the wrong answer there. Part D OK, Part D says, "Write a statement that correctly interprets a confidence interval. Choose the correct answer below." A confidence interval is simply a range of values, and what we're trying to do is figure out what is the true value of the population parameter. So when we make a confidence interval, we're saying that we have a certain level of confidence that the true value of the population parameter is between the upper and lower limits. Here our confidence level is 90%, so we're 90% confident that the true value of the population proportion is between these two values here. So let's look and see which answer option most, you know, best fits that statement.
Answer option A says, "One is 90% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion." That sounds a lot like what we just went over, so let's mark that. But before we hit Check Answer, let's go ahead and check the other answer options just to make sure. "There's a 90% chance that the true value of the population proportion will fall between the lower bound and the upper bound." Close, but it's not quite the same sentiment. "90% chance of falling in." Yeah, I mean I --- it sounds like the same as 90% confidence, but you know, the way statisticians think, it's not the same thing. So we don't want to pick that answer. "One has 90% confidence that the sample proportion is equal to the population proportion." No, no, that's not it either. I mean, yes, proportions are unbiased estimators, and so the sample statistic will tend to target the population parameter. But yeah, that's ... that's not really the same thing as saying we're 90% confident that it's lying between an upper and a lower bound. Our final answer option says, "90% of the sample proportions will fall between the lower bound and the upper bound." Oh yeah --- well, sort of. I mean, if you were to take an infinite number of samples and then look at the proportion and do a distribution on the proportions for those samples, then yeah, 90% of them will fall between the upper and the lower bounds. But that's not what the confidence interval is suggesting. The confidence interval is not suggesting infinite anything. It's just saying we're 90% confident that the true value was between the bounds of the confidence interval. So we're not going to select any of these other answer options. We want this first answer option here. Well done! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below and let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the area under a normal distribution curve using StatCrunch. Here's our problem statement: Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. Solution OK, to find the area for the shaded region, we could use the tables that we have links to here in the problem. But I find it's much easier to just use StatCrunch instead of the tables for solving a problem like this. We can use the tables, but it requires us to find the area under the curve to the left of each value and then subtract the one from the other to get the area in between. With StatCrunch, all you do is just put the numbers in the calculator and it does all that hard heavy lifting for you. So that's what we're going to do to solve this problem.
So I'm going to open up StatCrunch. And I don't necessarily need anything in the data table. I just need the functionality that StatCrunch provides so I can get into that calculator for my distribution. It says the scores are normally distributed, so I want the normal calculator. To get that, I go to Stat --> Calculators --> Normal. Now here it says that the mean of the distribution is 100 and the standard deviation is 15. So I'm going to put those values in here in my calculator. And I'm looking for the area in between two values. So I want to come up here and select the Between option. And then I just come down here and put those two values that I want to get the area in between. And when I press Compute!, out comes the area in between those two values. That is the area in between the shaded region, which is same as the probability of selecting something between those two values. So I'm going to put my answer in my answer field. Good job! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below. Let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find the z-score for a standard normal distribution using a table. Here's our problem statement: Find the indicated z-score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. Solution OK. We're looking for a z-score, and notice the z-value or z-score that we're looking at over here is on the right side of the distribution. That means the z-score is going to have a positive value. If the z-score were directly in the middle, the z-value or z-score would be zero. If it were to the left, on the left side of the distribution, then it would be negative. But here we find it on the right side, so therefore it's going to be positive.
That means we want the second page of our table, because the first page of the table lists the negative z-score values. We want a positive z-score because it's here on the right. So here's my table. Notice it's the areas that are listed under here in the table, which are also the same as the this which correlate with the z-scores. So that area is to the left of the z-score. That's exactly what we have here in our problem statement. This area under the curve, which is 0.9817, is to the left of the z-score we're looking for. So here we're going to look in the table for this value (0.9817) and then get the corresponding z-score. So 0.9817 --- if I look here in my table, you gotta scroll down --- 0.98 ... 0.9817. OK, so here's the 0.9817. So remember the z-scores are listed in two parts. The first part is over here on the left. So I go over here to the left. Here's the first part: 2.0. And then the last part is going to be up here at the label for the column, 0.09. So we add those two together, and we get 2.09. So there's the first part ... 2.09. That's my z-score. So I put that here in my answer field. Good job! And that's how we do it at Aspire Mountain Academy. Be sure to leave your comments below. Let us know how good a job we did or how we can improve. And if your stats teacher is boring or just doesn't want to help you learn stats, go to aspiremountainacademy.com, where you can learn more about accessing our lecture videos or provide feedback on what you'd like to see. Thanks for watching! We'll see you in the next video. |
AuthorFrustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help. Archives
July 2020
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