Intro Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to use a calculator display to perform hypothesis testing on proportions. Here's our problem statement: A certain drug is used to treat asthma. In a clinical trial of the drug, 20 of 287 treated subjects experienced headaches, based on data from the manufacturer. The accompanying calculator display shows results from a test of the claim that less than 10% of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution, and assume a 5% significance level to complete Parts A through E below. Part A OK, Part A asks, “Is the test two-tailed, left-tailed, or right-tailed?” To determine this, we need to know what our alternative hypothesis is. And the alternative hypothesis comes from the claim that's being tested. Here the test is that were claiming that less than 10% of treated subjects experience headaches. You can see that right here in the calculator display as well that the proportion that we're looking for is less than 10%. This has no equality with it, and so we can adopt this claim as the alternative hypothesis. Since this is the alternative hypothesis, we know that this is a one-tailed test to the left because “less than” is like an arrow pointing to the left. Excellent! Part B Part B asks for the test statistic. You can see that here in the calculator display. Notice they've asked for a z-score — so “z equals” then there's your answer field. In here you've got your z-value, so you just take that right off the calculator display. There's no need to put any data into StatCrunch. There's no need to do any number crunching in StatCunch. The display for the results of the test is right here, so you can just take that from the display that they have right there. Well done! Part C Part C asks for the P-value, which is right below the test statistic in the calculator display. I'm asked around to four decimal places. Well done! Part D Part D asks, “What is the null hypothesis, and what do you conclude about it?” Well, the null hypothesis is by definition a statement of equality, so there's only one correct answer. And it's the one with a statement of equality. Excellent! Now we're asked to decide whether to reject the null hypothesis. As we go back to our P-value, which is a little more than 4%, we're asked to test our claim at the 5% significance level. Our P-value is less than our significance level. Therefore, we’re inside the region of rejection, and we're going to reject the null hypothesis, because the P-value is less than or equal to the significance level alpha. Well done! Part E And now Part E asks, “What is the final conclusion?” Well, we rejected the null hypothesis, and so therefore there is sufficient evidence. But do we want to support the claim or do we want rejection of the claim? Well, to determine that, we go back to our alternative hypothesis. Remember, our alternative hypothesis matches the claim that was being made. Because it matches, we want to select “support the claim.” If it didn't match, then we would select “warrant rejection.” But it does match, so we're going to select “support the claim.” And we rejected the null hypothesis, so there is sufficient evidence. 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.
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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 construct a stem-and-leaf plot. Here's our problem statement: Construct a stem-and-leaf plot of the test scores 68, 73, 85, 75, 89, 89, 87, 90, 98, and 100. How does the stem-and-leaf plot show the distribution of these data? Part 1 OK, we could go old school and construct the stem-and-leaf plot from our data set by hand, but I'm gonna use technology because I'm living in the 21st century where we have this wonderful technology that can do a lot of the heavy lifting for us. So that's what I'm gonna do. So here in StatCrunch I have the actual data set that I'm given here in the problem statement. And now that I've put this in, I'm ready to construct my stem-and-leaf plot. To do that, I go up to Graph –> Stem and Leaf. In the options window, I'm gonna select the column where my data is located and then hit Compute! and here's my stem-and-leaf plot already made for me. Now all I need to do is figure out which of these answer options matches the stem-and-leaf plot given to me in StatCrunch. And that's going to be — looks like this one. I check my answer. Nice work! Part 2 And now the second part of our problem asks, “How does the stem-and-leaf plot show the distribution of this data?” Well, if you look at the way this is formatted it looks kind of like a histogram on its side. And that's essentially what we're looking at, where the stems correspond to the categories or classes of a histogram and these leaves here correspond to the heights of the bars.
So the more numbers you have within a given category or class, then that means you've got a greater frequency count for that category. That's really what we're looking at here. So what are our answer options? The lengths of rows are gonna be similar to the heights of the bars in a histogram because this is on its side. So we're looking at answer options C or D. And then longer rows of data correspond to higher frequencies. I check my answer. 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 use StatCrunch to perform hypothesis testing on means of tornado lengths. Here's our problem statement: A dataset includes data from 500 random tornadoes. The display from technology available below results from using the tornado lengths (in miles) to test the claim that the mean tornado length is greater than 2.7 miles. Use a 5% significance level. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim. Part 1 OK, the first part of our problem asks us to identify the null alternative hypotheses. The null hypothesis is by definition a statement of equality, so looking at our answer options, we can see answer option D is not going to be correct. Now, to select from the remaining answer options, we need to determine the correct alternative hypothesis. And to do that, we look at the claim. What is the claim that we're being asked to evaluate? Well, here in the problem statement, it says we're testing the claim that the mean tornado length is greater than 2.7 miles. So we want that mean to be greater than 2.7, and that's this answer option here. I check my answer. Good job! Part 2 Now we're asked identify the test statistic. Normally in a problem like this, we would have data and we dump it into StatCrunch, and we use StatCrunch to give us the results. In this case, we don't need to do that, because we click on this icon, we have this display from technology which already gives us what we need. So here's our test statistic, located here, the T stat. We're asked to round to two decimal places. Nice work! Part 3 Now we’re asked for the P-value. That's also there in that technology display, right next to the test statistic. I'm asked to round to three decimal places. Good job! Part 4 And now we're asked to state the final conclusion that addresses the original claim. We do that by comparing our P-value with our significance level. So here's our P-value, 3.9%. We're asked to compare that with the alpha level of 5%. So our p-value is definitely less than our significance level. That means we're inside the region of rejection, and therefore we reject H0. Because we reject H0, we can say that there is sufficient evidence. 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 use StatCrunch to perform hypothesis testing on standard deviations of piston diameters. Here's our problem statement: The piston diameter of a certain hand pump is 0.4 inch. The manager determines that the diameters are normally distributed with a mean of 0.4 inch and a standard deviation of 0.005 inch. After recalibrating the production machine, the manager randomly selects 25 pistons and determines that the standard deviation is 0.045 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 10% level of significance? Part 1 OK, the first part of our problem asks us to determine the correct hypotheses. For this test, we're looking at a test on standard deviation, so we want to select population standard deviation as our parameter. The null hypothesis is always a statement of equality. And then the question becomes “What value do we put here?” Typically this is the claimed value. And the claim that we're making is do we have enough evidence to conclude that there's a decrease in the diameter? So if we're looking for a decrease in the diameter, the value that we want to put here is the standard diameter that we're measuring against — not the sample that we're looking at, but for the population. The standard belongs with the population. So that's the one that we want to select. Then of course, here the claim is that we're looking at a decrease from that value. Good job! Part 2 Now I'm asked to calculate the test statistic. This is the part where I would love to have StatCrunch do the heavy lifting for me. However, I'm not aware of any feature inside StatCrunch which will actually calculate this for us. We can do hypothesis testing on the variance, but the test statistic that we get is for the variance, and it cannot be manipulated or somehow used to get the test statistic for the standard deviation. So we're gonna have to go old school, unfortunately, and calculate this out by hand. Our test statistic is computed by using this equation right here. So we've got 1 minus the sample size (so this is essentially our degrees of freedom) times our sample variance divided by our population variance. If we put the numbers from the problem statement here under this equation, this is what we get. There's 25 in our sample size, so that number goes in for n. And then our sample standard deviation is 0.0045, and then the population standard deviation, 0.005. If I go ahead and compute this out, I end up with 19.44, so there's my test statistic — 19.44. Fantastic! Part 3 Now we're asked to find the P-value. And here you can stay old-school and use a chi-square table, or you can actually use the calculator inside StatCrunch and calculate your answer that way. And that's the route I'm going to be taking. So here in StatCrunch I'm going to click on Stat –> Calculators –> Chi-square. Now here's my chi-square distribution calculator. The degrees of freedom is one less than the sample size, so sample size of 25 gives us 24 degrees of freedom. Here we want “less than” because that matches the alternative that we're looking at; we want that left tail on our hypothesis test. And then here we're going to put in our test statistic 19.44, and then Hit Compute! And then this is the area under the curve that's less than the test statistic. And that is the definition of the P-value. So the P-value is the area under the curve for your distribution that's bounded by the test statistic. In this case, we’re asked to round to three decimal places. Good job! Part 4 Now we're asked to evaluate the hypothesis test, make a conclusion. So the P-value we have, 27%, we're comparing that with a significance level of 10%. Definitely we're outside of that region of rejection, so 10% is this — or [rather] the significance level. We're outside the region of rejection. So the P-value is going to be greater than the level of significance, which puts us outside the region of rejection. So we fail to reject (or do not reject) the null hypothesis. And because we fail to reject the null hypothesis, there is not sufficient evidence. 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 use StatCrunch to perform hypothesis testing on two proportions. Here's our problem statement: A simple random sample of front seat occupants involved in car crashes is obtained. Among 2763 occupants not wearing seatbelts, 31 were killed. Among 7830 occupants wearing seatbelts, 19 were killed. Use a 5% significance level to test the claim that seatbelts are effective in reducing fatalities. Complete Parts A through C below. Part A OK, Part A asks us to test the claim using a hypothesis test, and the first part of the hypothesis test is forming our null and alternative hypotheses. We're doing a hypothesis test on proportions, and so our parameter in our hypothesis is going to be proportions, which we see here. But which of these answer options is the right one? Well, let's figure that out. By definition, the null hypothesis is always a statement of equality. So right off the bat, we know that answer option A, C, and F are incorrect. So we're having to choose between answer option B, D, and E, because all of these answer options have the null hypothesis as a statement of equality. To determine the correct alternative hypothesis, we need to go and look at the claim. What is the claim that's being made? Back here in the problem statement, we can see we're testing the claim that seatbelts are effective in reducing fatalities. OK, so we have two groups: One group who was not wearing seatbelts, the other group who was wearing a seatbelts. And it says here that we're supposed to consider the group not wearing seatbelts as the first sample and the group wearing seatbelts as the second sample. This is the same order in which they're listed here in the problem statement. That's great, because now we see that there's no semblance of equality that's being made in the claim; it's just one is greater than the other. So we can adopt the claim as our alternative hypothesis. The group wearing seatbelts — if the claim is that the seat belts are going to be effective in reducing fatalities, and that means the group wearing the seatbelts is going to have a lower proportion of deaths than the group not wearing a seat belt. So p2 (the group wearing the seatbelts) will be less than p1. And that's what we see right here. So I'm going to check that answer. Excellent! Now we're asked to identify the test statistic. And to do that I'm going to pull up StatCrunch. Notice there's no icon or any data that you have to dump into StatCrunch. And because there's no icon to click on, it's often a good idea for you to keep a copy of StatCrunch open, just in case you need to access it like you do here but you don't have any data. We don't need any data for the data table. We just need the functionality of StatCrunch. So for that, I'm going to go to Stat –> Proportion Stats –> Two Sample –> With Summary. Here in the options window, I’m going to list my summary stats. And they're listed here in the problem statement. Again, we're asked to make the first sample the group that's not wearing seatbelts and the second sample the group that is wearing seatbelts. That's the same order that they're listed here. The number of successes is the part of the whole that we're looking at. So for that first sample, it's going to be 31. I know it's really weird to consider that people dying is considered success, but try not to think of it that way. Try to think of it as you're looking for the part of the whole that you're trying to examine. And since we're looking at fatality rate, we want to look at the number of deaths. I put the total number from that group in here. And I'm going to do the same thing for the second sample. Now down here I want to make sure this radio button for hypothesis test is selected. This is the default selection, so we're already there. And now I want to make sure that these fields match the hypotheses that we established over here. Notice how the formatting is a little different. Here in StatCrunch you've got p1 minus p2. Over here you've got p1 equals p2. If I just subtract p2 from both sides here, I get p1 minus p2 equals 0, so that's okay. I leave that alone, and then when I make sure that this inequality sign matches. And now I'm all ready to go. I hit Compute! and here in my results window, the second number to the end of my table is my test statistic. I'm asked to round to two decimal places. Well done! And now I'm asked for the P-value. The P-value is right next door, the last number listed in the column. Notice when we have this “< 0.0001" listed here, that's practically zero. So I can just put zero in here. Excellent! And now I'm asked to make a conclusion based on the hypothesis test. We do that by comparing the P-value with a significance level. We’re asked to test at 5%. The P-value is zero, so that's going to be lower than any significance level that we would use for testing. So we're definitely less than the significance level. That means we're inside the region of rejection, so we're going to reject the null hypothesis. And because we reject the null hypothesis, there is sufficient evidence. Good job! Part B Well, so much for Part A. Now Part B asks us to test the claim by constructing an appropriate confidence interval. I could go back here into StatCrunch and go back to the menu options, but it's much quicker if I go back to my results window, click on Options, and then Edit, go right back to the options window. Then all I have to do is flip this button, and now I'm going to be testing for the confidence interval. The question is “What's the appropriate confidence level for an appropriate confidence interval?” To do this, we need to go back and look at our alpha (significance level), which is 5%. Normally in constructing a confidence interval, we would take 1 minus alpha, but here in this case, because we're looking at two samples, we need to select 1 minus 2 alpha. So I need to take twice alpha. So twice 5% is 10%, subtract that from 1, I get 90%. This is the appropriate confidence level for my appropriate confidence interval. I press Compute! and then here at the end of my table I see the lower and upper limits that I need to put into my answer field. I'm asked to round to three decimal places. Well done! Now we're asked to make a conclusion from the confidence interval. And to do this, we always look for where is zero with respect to our confidence interval. Zero is outside the confidence interval. It's not inside the confidence interval, so our confidence interval limits do not include zero. And so therefore, because zero is not inside the confidence interval, there's going to be a significant difference between the two proportions. So they're not equal. And which side of zero is my confidence interval on? Well, zero is over here to the left, so all of these values here are positive. And because all these values are positive, that means this difference is always going to be positive. Well, what does that mean? If this difference is always positive, that means p1 is always going to be greater than p2. And so, go back and look at what are these correspond to. p1, remember, was the proportion of deaths from people who did not wear the seatbelts. p2 is the proportion of people who died and they were wearing the seatbelts. So wearing a seatbelts leads to lower death, lower numbers of death, or fatality rate. So the fatality rate is higher for those not wearing the seatbelts. Nice work! Part C And now Part C asks, “What did this suggest? What did the results suggest about the effectiveness of seatbelts?” If we go back and we look at what we've actually concluded from the hypothesis test and the confidence interval — remember that for proportions, they don't necessarily match up and when they don't match up you want to go with the hypothesis test — in this case they actually are matching up. Both the hypothesis test and the confidence interval lead to the conclusion that the fatality rate is higher for those not wearing the seatbelts. And so we have a pretty good statistical case for suggesting that the use of seatbelts is associated with a lower fatality rate than not using the seatbelt. So I'm going to select that answer. 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 construct a relative frequency distribution from frequency count tables in Excel. Here's our problem statement: Construct one table that includes relative frequencies based on the frequency distribution shown below. Then compare the amounts of tar in non-filtered and filtered cigarettes. Do the cigarette filters appear to be effective? (Hint: the filters reduce the amount of tar ingested by the smoker.) If we click on this icon here, we can see the frequency counts that were given for this problem. Notice that some of the classes appear for the filtered but not the non-filtered and vice-versa. All that is saying is that, for the classes that don't appear, there are no counts for that category for either the non-filtered or the filtered. Typically in solving these problems, I'm dumping data into StatCrunch and working inside StatCrunch. StatCrunch is actually more clunky for this type of operation. You can make it work, but, as I say, it's more clunky, so I prefer to dump my data into Excel. If enough of you comment back and say that you want to see this worked in StatCrunch, I'll make another video working it in StatCrunch. But as I say, the data (as is) is a little clunky to work with so — wow! OK. So this is funky. Let's try this. I’d like to just copy you. Let's just paste you in here. OK, that works. Part 1 So now I have my data here in Excel. I don't need this any more. Before I start moving things around, I'm going to do a little formatting here so we can see a little bit more of what's actually happening and going on. The first thing I'm going to do is format the headers. And let's expand you out, and let's expand you out, and you out, and you out. All right, so now our headers look a little bit better. And the next thing I want to do is make another column here because what I'm going to do is actually calculate the percentages that we need for our table here in MyMathLab (or MyStatLab, whichever way you want to call it). We're actually going to calculate those answers here in Excel and then just transfer them over to help us get the right numbers in the right spot. OK, I'm gonna line up the categories here. So see how this category here — 18 to 23 — is at the top for the non-filtered, but 18 to 23 doesn't appear until you get down here for the filtered. So what I'm going to do is just take those cells and move them down. So now we're all “even Steven” with that. And then I'm gonna go and fill in the missing categories here. I'm gonna do the same thing over here. OK. Got ya. All right. Now I'm ready to start calculating out. If I want, I could put zeros in here, because there are no counts for these particular classes. But that's not really needful. First thing we need to do is relative frequency is based on the total number accounts for a particular item. So I'm gonna sum up the frequency counts for the non-filtered cigarettes — 25. So here I'm actually going to divide each one of these numbers in the frequency column by the total, which is 25. Here I actually put 25, or if I wanted to I could just select the cell with the sum and then, so I keep the same cell when I copy the formula down across multiple rows or columns, I press F4, and I get those dollar signs. And that tells Excel we want to keep the cell reference constant when we copy the cell down. Before I copy this down, I want to make sure I have it in a percentile. That way, I don't have to do any conversions; I can just flip the number straight from here over into my answer field. So now this is what I need to look at. I'm going to copy this down, and there are my frequency counts. I can do the same thing over here. Take the sum all of these numbers here, and I can put the same formula in that I did over here. Percent, we want that to appear as a percent. Now I could just take you and copy you down. So now I got the numbers that I need. Because there's so many numbers here, I want to actually make this a little bit more identifiable. So I'm gonna put you in there, I'm gonna shade you a little bit something over here, I'm gonna shade you a little bit, center you, and, if I want to, I can actually put a border around it that actually helps me see it a little bit better. So now all I need to do is just transfer the numbers over. So 6 to 11 for the non-filtered is this number here (zero), so that's what I'm going to place here (zero). And then I just place the numbers in one at a time. Excellent! Part 2 Now the next part of the problem asks, “Do cigarette filters appear to be effective?” Well, let's look at our distribution table here. So we have for the non-filtered cigarettes, the bulk of the frequency counts are here in the higher tar classes. For the filtered cigarettes, the bulk of the data is in the lower tar classes. So it appears that the filtered cigarettes have lower tar than the non-filtered cigarettes, generally speaking. And that's what we want; that's the purpose of the filters — to reduce the tar. So it does appear that they're effective. So yes, because the relative frequency of the higher tar classes is greater for non-filtered cigarettes, this seems to be our answer, and it is. 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 perform correlation hypothesis testing. Here's our problem statement: Use the given data set to complete Parts A through C below. Use alpha equals 5%. Part A OK, Part A asks us to construct a scatterplot. To do this, we're going to use StatCrunch. And the first step to do that is putting the data inside StatCrunch. I’m going to resize this window so we can see better what's gonna go on. OK, now inside StatCrunch, I could go to Graph and then select Scatter Plot. But I know from the problem I'm given here, I'm gonna have to do a regression analysis anyway, and the scatterplot comes as part of the regression analysis. So let's just do that. It's less buttons to push. To get there, I'm going to go to Stat –> Regression –> Simple Linear. In the options window, I'm going to select my x-variable and my y-variable. And all the other default selections are good, so we're going to hit Compute! and in the results window, notice up here at the top it says 1 of 2. That means you're looking at page 1 of 2 pages total. The scatterplot is on the second page, which we can get to by hitting this arrow key down here at the bottom of the window. And voila! So now we just match the one that looks the same. Notice how the axes go from 4 to 14 and 4 to about 13, so we're pretty close to the axes that are here. If I wanted to, I could select the little three bar icon in the lower left corner and then change my axes as appropriate. But we're close enough here that we get the general pattern. And we can see clearly that answer option A is the one we want. Well done! Part B Part B asks us to find the linear correlation coefficient. That was done here in our results window. If we go back to page 1, we can see our correlation coefficient here, in this case 0.81. We're asked to round to three decimal places, so I will do that! Good job! Now this next part of Part B asks us to use the linear correlation coefficient to determine whether there is sufficient evidence to support the claim of linear correlation between the two variables. To do this, we need to compare the R-value we found from the problem statement and compare it with the critical R-values. So I'm going to click on this icon to get my table of critical values. Notice here — if I can get rid of this — we have 11 pairs in our sample. So we have 11 pairs, and we were asked earlier to use 5% for our alpha level. So 5% alpha matched with 11 pairs of data gives me a critical R-value of 0.602. The R-value that we obtained is 0.816. That's greater than the critical R-value. Therefore we have sufficient evidence to support a claim of linear correlation. I select the appropriate answer. Well done! Part C Now Part C asks us to identify the feature of the data that would be missed if Part B was completed without constructing the scatterplot. OK, let's go back and look at our scatterplot. And it looks like our scatterplot went away. Oh yeah, because I deleted it. OK, no matter! I'll just make it again. I want to do this.
So here's our scatterplot. What feature would be missed? Well, we've got a really good correlation coefficient, so it's not surprising that our data fit this line reasonably well. But clearly the scatterplot is showing us something that the correlation coefficient can't show us, and that is the presence of an outlier in our data set. If we didn't have this outlier, then this line would slope downward and fit the data that we have remaining much, much better. As it is this outlier is actually pulling that line up a little bit so that we're losing some of the fit. It's not that bad because we still have our value that's greater than the critical R-value. But it would be better if we didn't have that outlier in there, and this is what the scatterplot is showing us. So I'm going to select the answer option that best reflects that reasoning. Good job! And that's how we do it as 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 percentile for a standard normal distribution. Here's our problem statement: Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find P15, the 15th percentile. This is the bone density score separating the bottom 15% from the top 85%. Part 1 OK, the first part of this problem is asking us which of the graphs represents P15. To do this, we need to remember that percentiles are the area of the curve that are below the given percentile value. So here, the given percentile value is 15. Therefore, we want the area of the curve that is below P15. And that area will correspond to 15% of the total. Well, answer option A is not going to work for us, because this is not the area below P15. Answer option B looks like the one that we want to select, but let's look at the other answer options just to make sure. Answer option C has the area shaded to the left, which is good, but this area is no way 15% of the total because the area shaded here is clearly more than half and 15 is not more than 50. So there's no way this is right. And answer option D has the area under the curve shaded to the right, which is not what a percentile is, so that's also incorrect answer. Option B is the one we want. Good job! Part 2 Now the second part of this problem asks us to identify the bone density score. This is actually the z-score that corresponds with P15. To do this, you can actually use a standard normal distribution table — or what I'm going to do is use StatCrunch to do this. So here I have StatCrunch open. And to do this, I'm going to go to Stat –> Calculators –> Normal. The reason why I'm using the Normal calculator is because it says in the promise statement that our scores are normally distributed.
So here's my standard normal distribution; the mean is zero, the standard deviation is one. Those are the default values here in the calculator, so I don't need to make any adjustments there. I want the area to the left, and that's the area that's shaded here, so I don't need to do anything with the inequality sign here. I want to make sure that's to the left and it is. This field here will show me my z-score once I put in the corresponding percentile here, which is in this case 15%. So I put in 15% in decimal form and I hit Compute! And there's my z-score. I'm asked to round to two decimal places. 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 use StatCrunch to perform hypothesis testing on matched pair means. Here's our problem statement: The accompanying table lists the attribute ratings made by a random sample of participants in a speed-dating session. Each attribute rating is the sum of the ratings of five attributes: sincerity, intelligence, fun, ambition, shared interests. Use a 0.05 significance level to test the claim that there is a difference between female attribute ratings and male attribute ratings. Part 1 OK, this first part of a problem is asking us to identify the null and alternative hypotheses. For matched pair means, the parameter that we look at is the difference from the population mean, so we identify that with use of d (subscript). This is our population parameter for our hypotheses. The null hypotheses is by definition a statement of equality, so it will always be that the parameter is equal to some value. Here we're actually looking at matched pair mean differences. And when you're looking at differences, typically the value that you're looking at is zero. The alternative hypothesis — let's see if we can match the claim. The claim here is that there's a difference between female attribute ratings and male attribute ratings, so there's some difference. So it could be that the female is greater than the male, or the female is less than the male. It doesn't matter which side you're on; it's just one or the other. And if you're on either side, there's some difference there. That means we have a two-sided test, and this is going to be, therefore, not equal to (again, the same value) zero. Nice work! Part 2 The next part asks us to identify the test statistic. To do this, I'm gonna let StatCrunch do the heavy lifting for me. So here's my data. I click on this icon so I can dump the data into StatCrunch. And now my data is in StatCrunch. I’m going to resize this window so we can see more what's going on. And now in StatCrunch, I'm gonna go to Stat –> T Stats –> Paired. In the options window, I’m going to tell StatCrunch where to find my data. Notice with the paired sample option, I don't have the ability to do the test with anything but actual data. So I can't provide summary statistics in order to do the test; I have to have actual data. That's just part of the way that the software was coded. We actually do have actual data, so it's not a big problem. You need to select the samples and put them or tell StatCrunch where the data is actually going to be located. So typically you're going to just take the column listed first as the first sample and the one that's listed next as the second sample. Then down here under Perform, we're going to select the radio button for hypothesis test. This is the default selection, so it's already done for us. And the default selections here already match the null and alternative hypothesis for our particular situation, so we don't need to make any changes to the default values here. If there were changes needed, then we could make them. We want to make sure that these fields here match the null internal hypothesis for our particular situation. Once you get that matched up, there's nothing else to do. So just press Compute! and in the results window here on the end of the table is your test statistic. I'm asked to round to two decimal places. Fantastic! Part 3 The next part asks us to identify the P-value. The P-value is right here in the same table in the results window next door to the test statistic. It's the last value in the table. I'm asked to round to three decimal places. Fantastic! Part 4 And now the last part of the problem asks us to make a conclusion on the hypothesis test. First, one thing we need to do is compare the P-value with a significance level. We were given a significance level of 5% here in the problem statement. Our P-value is way more than 5%; we're almost at 88%, so we're definitely greater than 5%. So the P-value is greater than the significance level. Because the p-value is greater than the significance level, we are outside the region of rejection, and therefore we're going to fail to reject the null hypothesis. And because we fail to reject the null hypothesis, there is not sufficient evidence. I check my answer. 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. |
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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