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College administrator is curious to know if ACT scores can be used as predictor of performance for first vear undergraduate The administrator collects the following...

Question

College administrator is curious to know if ACT scores can be used as predictor of performance for first vear undergraduate The administrator collects the following data for randomly selected students_ACT Scores First Year GPA25 3.18 3.9 2,91,85 2.2 2.8 3.5Build scatter plot for this data_Use the special calculator features to find the correlation coefficient for this data [Round the final answer to three decimal places.Use the special calculator features to find the equation of the regression l

college administrator is curious to know if ACT scores can be used as predictor of performance for first vear undergraduate The administrator collects the following data for randomly selected students_ ACT Scores First Year GPA 25 3.18 3.9 2,9 1,85 2.2 2.8 3.5 Build scatter plot for this data_ Use the special calculator features to find the correlation coefficient for this data [Round the final answer to three decimal places. Use the special calculator features to find the equation of the regression line. [Round the final answer to three decimal places | Find the residual for the student that scored a 27 on his/her ACT,



Answers

Here is a sample of ACT scores (average of the Math, English, Social Science, and Natural
Science scores) for students taking college freshman calculus:
$\begin{array}{ll}{24.00} & {28.00} \\ {24.00} & {25.00} \\ {28.00} & {24.50}\end{array}$
$\begin{array}{ll}{27.75} & {27.00} \\ {30.00} & {23.25} \\ {22.50} & {28.25}\end{array}$
$\begin{array}{ll}{24.25} & {23.50} \\ {26.25} & {21.50} \\ {21.25} & {19.75}\end{array}$
26.25
26.00
(a) Using an appropriate graph, see if it is plausible that the observations were selected from a
normal distribution.
(b) Calculate a 95$\%$ confidence interval for the population mean.
(c) The university ACT average for entering freshmen that year was about $21 .$ Are the calculus
students better than average, as measured by the ACT?

In question. That imprint fight. We have data from a sample off 40 college students on. We're going to be comparing there. It's cause for the city and the GP averages after the first time in college. So the fast part of the question because us to construct a scatter diagram and we need to ensure that the A city's caused around the horizontal axis on the GP on the vertical axis. Then we also need to identify the male and female students. So this case, we just have to selected deter a city and GP here. Then we go to the in sacked tub and select this cutter dagger. As you can see on the horizontal axis, we have the GP and sorry. On the horizontal axis we have the yes, it is cause on on the vertical axis, you have the GP. Now we just need to identify the male and female students. All of them have the same color, but you need to identify the mean and female students. So what we need to do here is to select the detail and right select deter. Then we need to add the Siri's just add the series. We can call the Siri's there. Do you want to identify meals? So heels? We have X valleys, this ridge, and, uh, we have y values these fringe. So now, as you can see, these are color. The the blue color there that you cannot there represents the meal. So when you compare, you notice that in clinical feather patterns for males and females, you notice that that difference for mills the points distribution indicate a positive dependency. That is, the points appear to be, we think are rising light. But for females, the points distribution indicates little or no dependency, as you can see the orange dot and that points are between horizontal line. So that's the comparison between the patterns from meals and females. Then we need to see based on this scatter diagram, whether we can predict the students GPS based on the arts course. So if you have a student has an act score off 25 what would the GP for that students be? So we need for that information to remove the series view, and then we need to select the deter and odd a trendline. So when you are the trend line and show the equation tonight is going to be possible for you to drew that prediction. So in our kiss, the question of the line East why equals zero point 0 to 51 x plus two point 5 to 7. And remember the X stands for the city's cause and the wife stands for the GP. So if you want to predict the value, why, then you just need to take in the formula zero point zero to times. Yes, exit is called 25 and at that, too, to be five. That gives us a GP off 23.1 Pfeifle. So remember this is the GPS exit thescore off identifies, able to predict using. So the question, Oh Crassus to tell whether they appears to be any relationship between a city's cause on the fast on GP. And as you can see from this cutting diagram, the slope is very small, so we can conclude that these a slight relationship between yes, it is cause on the first term GOP. The relationship is much stronger for males than females, as you have seen before, I and so we can conclude that there seems to be a slight relationship under the relationship is also positive, Which means that if you have ah, high a CT school, you're most likely going to have Ah hi Gp in your faster

This exercise is once again looking at S 80 schools. But this time looking at how we can use an S A T to predict a GPS and a late gentile people could perform a college level. So part a asked us to write the equation of the regression light. Now, everything we need for this is given in the question. So we are predicting our g p a. They want g p A. Remember the hot to say it's a prediction predicted g p A is now are a value is given by the intercept Kerr fish and the question so minus 1.26 to remember the minus side and then our s a T coefficient is again given to us for the positive. No point no to no, not 21 for that city. Uh, that is old neat moving on pop be asked us to explain what the Y intercept of this Russian line indicates. Now Warren stepped is given to us hit now Usually in this imminent in any context, the won't set would represent predicted G p a. When s a T equals zero. Um, but it's hard for that toe kind of make any sense in this context because we've got a negative number as the intercept. But it does indicate a predicted debate when a city is zero and then looking at C l sister interpret this slope, which is this done? So what this is saying is for inch one point set in groups, we would predict E g p a. Increase off that value of the slopes off. No point, no, no to want full quints. That's what that means. So we will probably miss asking us to predict the GPS. A freshman who scored a combined 21 hundreds on This is just substituting into the equation we worked out in part A. So I predicted GDP is one point minus 1.262 plus no point, no North 214 as a very substrate in that 2100 these two moles plan together, come to four point for no for So if we have negative one point 262 that's 4.494 that is gonna come to three point to three to So the answer point for Part D and finally looking at Part E on it not finally looking at high, it says, Based on the statistics, How effective do we think SAT schools would be in predicting academic except success during the first most of the freshman year up to college? So the best indicator of this a one of the best and kids to this is our r squared value, which we given as 22.1%. Now it's showing that 22.1% of the variability in GP A schools in the first Master can be explained with this model so it could be explained with the S A T schools. Why was no high? It says it can give some indication. So I'd say Molterer indication. So while it shouldn't be used to fully gauge what people are gonna get, it can give you some kind of idea. But yeah, 80 well, 78.9% of very billeting do free schools could be explained with other things, so it definitely shouldn't be taken as everything that goes towards GP a school now in the last five, it says. As a student, if one it says as a student, would you rather have a positive or negative residual in this context. So? So that s manager about the or residual is Ah, actual or observe school minus are predicted school. Now, I did a We'd want the actual school three o'clock as possible. So for if our actual scores higher than I predicted, in other words, we did better than they thought. That would be groups and indigent that Gregson zero. So we'll be looking a positive residual. And I was a student. You definitely will, Max, if you get a negative residual.

So the scatter plot looks roughly like this. We see that there is a positive linear relationship between the combined s A T score and the freshman G p A. So for the first question, we're asked how strong the evidence is that the S a T score is a good predictor of G p A. And given that we just said that there's a strong, positive linear relationship between the combined s A T and the G p A. We can say that there is some strong evidence that s a T score is a predictor of freshman G p A. And for the next question were asked what concerns us about the statistical methodology for the conclusions that were reached. So what concerns us is the fact that we're using this combined US 18 which is a little bit problematic, because if we were to look at individual level data, maybe it looks something more like this, and that relationship could be a lot less strong, and perhaps they would still have the same least squares regression line in either case. But the correlation would be much more week when we're looking at the individual level data rather than the aggregate data

In this question, we revisit the grades of the statistics course from Question 47. This time we want to look for a linear relationship between homework marks in midterm marks. So what we've done is we've added the two midterm scores for each student together, and now we're looking for a linear relationship between homework and midterm total. So I've performed a linear regression in our and this is my output. So the linear relationship is defined by this intercept coefficient and this slope coefficient and for Part B were asked to comment on the conditions for a regression and whether the slope is statistically significant. So let's start off by looking at some of the scatter plots I made in our. So this is just a raw scatter plot of the midterm sums versus the homework, and it seems to show a linear relationship from here to the top, right? It's a pretty good in your relationship. You would say it's linear enough to meet the condition, and here I have a scatter plot of the residuals. It looks like a nice random scatter plot, which speaks to the independence assumption, and it also looks like what it has equal variance as you move from left to right, so it's not fanning in its not fanning out, and that speaks to the equal variance assumption. And third, I have a normal probability plot of the residuals, and it makes a pretty good straight line. This is actually quite good from the bottom left to the top right, and that suggests that the residuals where the errors are normally distributed. So we have normal data, so we may conclude that the conditions for inference are met. We would also note that it's not a random sample. It's an exhaustive sample from one class. So we would caution against using these results, extrapolating these results outside of that particular class and also whether the slope is statistically significantly different from zero. The P value is very small, so based on that very small, P value would say that there is a significant, um, difference from zero for this low and one other thing to take note of this, the R squared value is just a little over 50% which suggests that only about 50% of the variability in the midterm scores can be accounted for by this model and also the residual standard deviation is over 18 which is quite a significantly large standard deviation. So we're talking about grades that are in the seventies, eighties and nineties. So if you combine the midterm grades into two and add them, we're talking about numbers that are between 142 100. So 18.3 is pretty significant relative to that. Which is to suggest that the relationship or the association between homework raid and midterm grades is not strong. So we would be careful predicting midterm grades from the homework raid, and that's really what's asked of us in part C. Do you think you can accurately judge a student's performance without giving the midterms?


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