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A study by Wolfinger in 2015 showed that the probability of divorce decreases with marriage age for people who were married between 18 and 32 years old. Would it be...

Question

A study by Wolfinger in 2015 showed that the probability of divorce decreases with marriage age for people who were married between 18 and 32 years old. Would it be appropriate to use this decreasing trend to predict the probability of divorce for a person who was married at 65 years old? Why or why not? What poor practice in regression does this illustrate?Note: His study found that the probability of divorcing when married after 32 actually has an increasing trend!

A study by Wolfinger in 2015 showed that the probability of divorce decreases with marriage age for people who were married between 18 and 32 years old. Would it be appropriate to use this decreasing trend to predict the probability of divorce for a person who was married at 65 years old? Why or why not? What poor practice in regression does this illustrate?Note: His study found that the probability of divorcing when married after 32 actually has an increasing trend!



Answers

Ages of couples again Has the trend of decreasing difference in age at first marriage seen in Exercise 28 gotten stronger recently? The scatterplot and residual plot for the data from 1980 through 2010, along with a regression for just those years, are below.
a) Is this linear model appropriate for the post-1980 data? Explain.
b) What does the slope say about marriage ages?
c) Explain why it’s not reasonable to interpret the y-intercept.

So our scatter plot looks roughly like this. It goes down a little bit at the start and then goes up fairly subtly afterwards. So for and were asked if there is a clear pattern. So our answer is no. There's not really a clear pattern, because at the start there's kind of a pattern that goes down. And then at the end there's a pattern that goes up. But overall across the entire data is that there's not really a shape that we could use to describe this reliably. So no and for B um, were asked if the association is strong. So since there since we said there's no clear trend, there isn't really an association to look at. So no, the association is not strong. Forsee were asked if the correlation is high. So for the correlation to be high, we would need to have all the points fairly close together and along roughly along a straight line. So if it were just this part of the data, you wouldn't see that the correlation would be fairly high. But we've got a large number of outliers here. So in reality, if we're doing at least squares regression line, we might get something that looks a little bit like this, and for that we would not have a high correlation and for D were asked if a linear model is appropriate. So given our previous answers that there's no clear trend, there's not really a strong association of any kind on The correlation is very weak. Um, we do not have evidence to say that a linear model would be appropriate in this case.

Her first stuff is going to be to identify the relevant information that were given in the question. So we've got our r squared Are slope and our intercept. So first were asked about the correlation between age difference and year. So we recall that the relationship between the correlation coefficient R and R squared is quite literally that if you were to square the value of are, then you will get R squared. So in order to figure out the correlation coefficient from R squared, you need to take thes square root off r squared. But one thing that we need to be careful about is that there is always a positive and negative square root. And we need to use the scatter plot to identify whether we're gonna be taking the positive route for the negative group here. So if we look at the Scott of what we see, that there is a, uh, negative linear relationship between the year and the difference in age and so that tells us that we're going to be using the negative square root. So the negative square root of 75.5% or 0.755 is going to be our correlation coefficient. So now we just need to take that into our calculator. So the spirit of Sarah, 0.755 is given by roughly 860.869 And so when we take the negatives covered of that, we get a correlation coefficient of negative 0.869 were where should be were asked to interpret the slope of this line. So since we've got year along the X axis and we've got the age diff on the Y axis are, Slope tells us that for every one year increase, the age difference decreases by roughly 0.1571 And finally for C were asked to predict the average age difference in the year 2015. So we're going to need to identify our least squares regression equation. So that's given by, uh, negative 0.1571 times x less 33.396 So we need to figure out which value of X we're going to plug into our equation. So the years start at 1980 we're fast about the year 2015. So that's going to be 35 years. So we're gonna plug in value 35 for X, and we're going to take this into our calculator. So we've got negative zero point 01571 times 35 plus 33.396 gives us 32.846 roughly. Okay, sorry. I just made a mistake. I thought that the X variable was being measured in the number of years from 1980 but that is not correct. So instead, we just get to you, plug the year directly into our equation. So if we plug in 2015 and we put that into our calculator, what we get is roughly one point 74035 and for d were asked reasons why we might not place much faith in this prediction. And the reason why is because we're extract light, extrapolating outside the scope of the data set that were given. So the data that were given on Lee goes until the year 2000 and we're trying to make predictions about the year 2015

Okay, So here we are taking a look at the data set which is describing the divorce rate during the 19 seventies, the rate being the number of couples to get divorced per 1000 couples. What we are trying to do here is developed the line of best fit. That is the linear regression that best describes the data doing that. We'll be using our graphing calculators. I am using 80 I 84 at the bottom right of your screen. You can see the steps that I've taken that's helpful to you at all. Um, but yeah, let's begin to start. I've simply plotted these points on the graph you see here just a basic scatter plot just to get a feel for what the data looks like when it's graft. But to graph to actually produce this linear regression, I've got into stat edit and put in my data into my l one l two going to Stack Falcon, calculating that linear regression our calculator finds for us that why is equal to 0.2 oh seven x plus 3.44 where X is our year and why being the rate the divorce rate. Um, from this, if we wanted to graph this, we can see that our intercept is 3.44 So that will sit somewhere right in here. And we have a slope of about point 207 so we could sketch a graph. That probably fits the data pretty well. Maybe. Look, something like that. That would be our why function or are linear function that we just found. Now, if we wanted to determine how well this actually fits the data, we want to look at our coefficient of correlation, and that's going to show up directly under your regression results on your calculator, you should have an r squared value and in our value r squared being r coefficient of determination and ours are coefficient of correlation. With that, we can see that are coefficient of determination is 0.95 and R coefficient of correlation 0.97 can add one more decimal place make more accurate than real. Um, this is telling us that around 95% of our of the variation in our y so in the divorce rate is explained by the information included in the status that so that's pretty good. Um, here, if we wanted to say we wanted to predict the divorce rate in year, uh, let's see, say, 1987. So if we wanted to predict that, we need to determine 1987 which year that would fall in our data set. So we started at 1970 one. Oops. Starting at 1971. That was our first year. We had, um, a total of 10 years. Sorry. Uh, so from 1970 to 1979 that was our data set, which was 10 years. So we wanted 1987 minus 1970. It was a 17, but we'll add one. We're at 18. So 18 years later. So this is gonna be the X value that we include in our function We just found. So let's go ahead and plug that in to determine what the divorce rate is predicted to be during the year 1987. Doing that we have y is equal to 0.2 oh, seven times 18 plus 3, +44 Calculating that we can get two Oh, seven times 18 us. Okay, here we go. So we get the predicted divorce rate is equal to 7.166 We can round that we'd like to about 7.2.

In question number 43. Given the verdicts one and three, we write to the function in verdicts form, which is? It affects equal to aim boy X minus one squared plus three. Where a is constant to be found. We use the fact that f of zero equal to five to find a so five equal to a by zero minus one squared plus three. So five equal to eight for three. So a will equal to two and that the equation for the problem would be ffx equal to to buy X minus one square pus. Three. That the final answer that question. Okay?


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