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Several years ag0, two companies merged_ One of the concerns after the merger was the increasing burden of retirement expenditures_ An effort was made to encourage ...

Question

Several years ag0, two companies merged_ One of the concerns after the merger was the increasing burden of retirement expenditures_ An effort was made to encourage employees to participate in the 401(k) accounts_ Nationwide 56% of eligible workers participated in these accounts: The accompanying data table contains responses of 30 employees of the company when asked they were currently participating in _ 401(k) account: Complete parts through d.Click the icon to view the data table_Determine the

Several years ag0, two companies merged_ One of the concerns after the merger was the increasing burden of retirement expenditures_ An effort was made to encourage employees to participate in the 401(k) accounts_ Nationwide 56% of eligible workers participated in these accounts: The accompanying data table contains responses of 30 employees of the company when asked they were currently participating in _ 401(k) account: Complete parts through d. Click the icon to view the data table_ Determine the sample proportion of company workers who participate in 401(k) accounts The sample proportion is (Round t0 three decimal places as needed:) Determine the sampling error if in reality the company workers have the same proportion of participants in 401(k) accounts as does the rest of the nation. The error is (Round t0 three decimal places as needed,) Determine the probability that sample proportion at least as arge as that oblained in the sample would be oblained the company's workers have the same proportion of participants in the 401(k) accounts as does the rest of the nation. The probability (Round to four decimal places as needed ) Does appear that = larger proportion of company workers participate in 401(k) accounts than do the workers of the nation as whole? Support your response. because there % chance of obtaining sample proportion greater than the one calculated in part if the population proportion is 56% (Round to the nearest percent as needed )



Answers

The data in 401 $\mathrm{K}$ are a subset of data analyzed by Papke $(1995)$ to study the relationship between
participation in a 401$(\mathrm{k})$ pension plan and the generosity of the plan. The variable prate is the per-
centage of eligible workers with an active account; this is the variable we would like to explain. The
measure of generosity is the plan match rate, mrate. This variable gives the average amount the firm
contributes to each worker's plan for each $\$ 1$ contribution by the worker. For example, if mrate $=0.50$ .
then a $\$ 1$ contribution by the worker is matched by a 50 $\mathrm{c}$ contribution by the firm.
$$\begin{array}{l}{\text { (i) Find the average participation rate and the average match rate in the sample of plans. }} \\ {\text { (ii) Now, estimate the simple regression equation }}\end{array}$$
$$\widehat{\text {prate}}=\hat{\boldsymbol{\beta}}_{0}+\hat{\boldsymbol{\beta}}_{1} mrate,$$
and report the results along with the sample size and $R$ -squared.
$$\begin{array}{l}{\text { (iii) Interpret the intercept in your equation. Interpret the coefficient on mrate. }} \\ {\text { (iv) Find the predicted prate when mrate = } 3.5 . \text { Is this a reasonable prediction? Explain what is }} \\ {\text { happening here. }} \\ {\text { (v) How much of the variation in prate is explained by mrate? Is this a lot in your opinion? }}\end{array}$$

Hard run in this part, you are going to find out what fraction of the families in the sample are eligible for participation in a 401 K plan. What you need to do is to you count the number of cases where the variable E for a one cape equal one you can easily find out there are 3600 37 cases. The total number of observations in the simple is 9000 275. You can easily calculate the percentage, so there's about 39.2% of the sample. U. N estimate a linear probability model explaining for a one K eligibility. In terms of income, age and gender. You will also include income and gender in quadratic form. This is the result table in part three. Based on the results of the regression, you can say that the eligibility depends on age and income. You can easily tell that because there p value of the variables of income and age is highly significant. However, the P value of the variable gender male in particular is not significant. So you could say controlling for age and income eligibility does not depend on gender Part three in this part, we're going to get the fitted value of the eligibility for 401 k from the previous model, and we will see if these fitted values greater than one or less than zero. In some of cases, you will see that there is no such case, are sited. Value remain in the range of zero and one. This is interesting because one limitation of the linear probability model is that it always it could predict fitted values outside the range of zero and one. But we didn't see that happened here in part four. Yes, we will define a new variable. We will use the fitted value and we define a new variable like this, which take a value of one is the fitted value from the bottle is greater than or equal 2.5, and the new variable takes a value of zero. If the fitted value is less than 0.5, we find out that there are 2460 cases. We're the new variable equal one, which means that among over 9000 families in the simple over 2000 are predicted to be eligible for the 401 K plan. In this part, we will construct a confusion matrix. We went tabulate the number of cases of actually ill eligibility to predict it in Eliza Bility. In other words, we tabulate the values of E for a one K versus the values of E for a one k tilda. The numbers in color are the cases where the module predicts correctly. You can see that of the over 5000 families actually ineligible for a 401 K plan, about 82% are correctly predicted not to be eligible. Of over 3000 families actually eligible, only 39% are correctly predicted to be illegible or part seven. We can calculate the overall percent correctly predicted by taking by something the number of cases where the model is correct. Divided by the total number of cases the model does find. Because it has a correct rate of 65% it does better for predicting there in eligibility rather than their eligibility. In this part, we will add a variable PIRA into the regression equation of the previous parts. Pyrite indicates whether a family member has an individual retirement account. This is the regression result. As you can see, the coefficient of PIRA is 0.2 It means that when a family member has I r a, then the chance of the family to be eligible for 401 k increases by 0.2 The P value of this estimation is point 105 It means that this variable is not statistically significant at the 10% level.

Let us go to part in what is the research objective. It is to determine the true population. Proportion off those who participate in an employer sponsored automatic payroll deduction for a 401 K plan to save for a time. So we have to determine the true population proportion to determine the true population proportion. Okay, the true population proportion off what off? Those who participate in an employer sponsored automatic payroll deduction for a 401 K plan to save for their retirement? What is the population? It is an entire set of people who work in the United States. What is the population, all working people, all working people and what is going to be my sample? The sample is a set of people surveyed, the people who are surveyed who are surveyed So fe, you know, go to the questions Question. I have 1172 employees. I have 1172 employees. This is my sample, but deep. What is a descriptive statistic? 27% of the sample is a descriptive statistics because it represents a population the proportion of the samples of 27%. The sentence that is given in the question you get read that this is going to be a descriptive statistics and what exactly can be inferred? Well, we're 95% confident that the two population proportion is in the range from 23 to 31 from 23 to 31. This is the range that we are saying that the two proportion will lie in at 95% confidence level. This will be my answers.

Hello, everybody. And welcome to a econometrics tutorial using our studio. So in this video, we're going to be looking into some endogenous variables and how you can try to use instrumental variables to, uh, fix that problem. So let's begin. First of all, you need to download the data set that we're gonna be using, which is called Old Bridge. So go ahead and open up that data set. Old bridge, open up the data for K, for one case subs That is the data we're gonna be using. And it is data about, uh, retirement funds, age, income and that kind of thing. If you open it up in help, you can see what all the D notations mean in the four. Oh, one k subs data file. So for this example, we're going to want to, um, compare what effect a bunch of these variables have on Pyra, which is whether or not you have an individual retirement arrangement. Uh, we'll be comparing it to income age and then whether or not you have a 41 K, which is a retirement savings plan some firms offer their employees were first going to start with an ordinary least squares model to come up with some in some inferences about the data. So you're going to want to, uh, do a linear regression on this formula here. So we're gonna be comparing Pyra to P 41 K. And what effect that has on it would affect income has on it income squared, age and H squared. And then, of course, an error term. So let's get to that, shall we? Actually, before we get to that, of course, it is also common practice. As you see in here, quite a few of these are factors. So you have. You know, marriage is a factor. Zero or one e 41 K is a factor. Male or female is a factor. Uh, that's not a factor. Uh, P 41 k is a factor and also appears a factor. So it is. It is pretty good practice to change the data, the data for these factors into actual factors in art. Because right now, as you can see down here, Pierre is not being recognized as a factor. Neither is male marriage. Um e 41 k or p 41 k. So it is good practice to do that with your data sets. If you have factors, you have binary variables. Go ahead and switch them again. You don't technically need to do this. The conclusions you can draw from the model are the same. More or less, Uh, enough here, what we do. So it's just good practice to do this. All right? So now is a good time to run the regression. So let's go ahead and name it. Model one. A very original. Mm hmm. All right, so let's run the model and see what happens. Boom. Creating the factors didn't seem to work, So yeah, it is good to create factors, but it's not imperative. So we will avoid it here. All right, so here we have our model. And, uh, for this example, we're going to be mainly focusing on the effect of P 41 k on Pyra. So that's right over here. We have the estimation. It is what appears to be 5% p. Value is very small, So this is really significant. This looks really good. Um, also because both Pierre and P 41 care factors. Um, the number here the B one is actually represented by this equation here. So you have Pura hat pure one hat and peers here at. So this would be the probability that you, um the probability that you have an i r a given. Um, you also have a 41 k account, and then this is the probability that you have an i. R a. Given that you don't have a 41 k account, and that's what you're the one is. So in practical terms, that means that according to this model, a person who has a four oh, one K account is 5% more likely to also have an IRA account given everything else is equal. Uh, but, you know, you might be asking yourself. All right, well, is that it? Is that it can be stopped there? Is that the effect that having a P 401 K has on pere? No. Because if you give it a quick if you think about it, just for a couple seconds might realize that there's some potential problems here, Um, and that both these variables, but mainly this one, since it's supposed to be independent, is actually endogenous. That might mean there's you're missing some kind of variable. That could explain the change in period that isn't explained by that. So say somebody doesn't like saving for their retirement through the government or through their company. So instead they go and save money in like houses. They buy a house and then they'll rent it or sell it later for retirement money. They still have a pension towards saving, but it's not captured. Um, in this checking whether the variable isn't darkness is always good. So let's do that. You're gonna have to have a thing called a TR, So just open that. And then once you have this, you could make your Ivy models. So let's do that. We'll call it Model B. So it's essentially the same thing. Except you are also including your instrument in here, which, in our case, they already have one. For us. It is E 41 case, so rather than whether they participated, participate in it or not, it's whether they're eligible for it or not. So you're getting the same sort of benefit, theoretically, that it's related to your endogenous variable. Uh, but it kinda ditches the baggage, so to speak. All right, and then once you get to this point what you're gonna want to do to, uh, your, uh your formula, your I V formula is just put a straight water and then through that down here. So our knows that you're trying to do a instrument very instrumental variable formula. So give that man a X squared and you basically replacing your P 41 k here with, uh, e 41 k. an important thing here because we're gonna wanna do a couple tests to see whether this instrument is actually, um, strong and required. So we're going to go ahead and have our give us the diagnostics. So remember, from our first our first formula where RB one was 5%. So with our ivy formula, um, it was very it was very significant. So far, I ve Formula RB one is 2% and it's it's not significant at all, really. So we can't really draw any conclusions from what, Like the relationship between p 41 K and, um Pyra. So so is our instrument e four oh, one k actually valuable. And that's where these tests come into play. So, first off, the weak instruments test is to you know, tell you whether your instrument is actually good or not. And the formula that this that are is using for the week instruments test is actually this. So it tests your what you think is an exogenous variable. Um, and then it tests it in this formula, with Z being your instrument and then sigma here being the effect it has. So the week instruments test is it's basically testing whether sigma is zero or different from zero if sigma is zero, that means that this is this whole terrible zero and your instrument is meaningless. So it's a week instrument. But if it's different from zero, then you're rejecting the null hypothesis and you actually have a worthwhile instrument. So if we go back to our and we check the weak instruments test, well, how do we reject the null hypothesis? So the the instrument e 41 K is actually worthwhile. Uh, and then the one husband test is actually just a generic test to check if you're, uh, if your variable is endogenous at all. So the formula that are is using for the horseman test is actually this one down here. So it's testing the null hypothesis that the co variance between your endogenous or what you think is your endogenous variable and your error term is zero. So that, in all hypothesis, actually means that it's exogenous. So you're testing whether it's exogenous or not. If you can refute this null hypothesis, then you have an endogenous variable. So let's see how we did. And it's significant so we can refute the null hypothesis here. Which means that P 401 k is indeed endogenous. So now that you know it's endogenous, you look at your results here instead and yeah, we went over, and already it's 2% but it's not very significant. So based on what we've done here, the we cannot actually come too much of a conclusion on whether you're not having a 41 K account with your employer and having an individual retirement account, uh, arrangement. Rather, having an individual retirement arrangement are correlated at all. Yeah, that was a quick look at endogenous variables and how you can clear up that problem with instrumental variables. Thank you, and I'll see you on the next one. Farewell

Okay. In this question, what we have to do is we have to find confidence in troubles. So what is given over here is that there were 1500 individuals with network of dollar one million or more. Okay, so our n a sample size is 1500. Okay, now, let's simply wanted the questions. What's part A. It says the Soviet reported that 53% of the respondents have lost 25% or more of their portfolio in the past three years. So what is P bar fever? This is the proportion, right? Proportion off a sample. In this case, it is 53%. Or I can write this 0.53 These many. This is the proportion off the people who have lost 25% off more of their portfolio value. Now what we have to do is we have to develop in 95% confidential double, which means our Alfa is equal to 0.5 Okay, what is the formula for finding the confidence interval? The formula for finding a confidence interval is fever plus minus rude over fever into one minus fever. This is a sample proportions upon and where n happens to be our sample size. But this thing is multiplied. This thing is multiplied by a critical value. Z star Now what is the star? Zee Star is nothing but see Alfa by doing this case. Alfa by two z Alfa by what is our Alfa Alfa 0.5? So what will be Alfa by two Alfa by two This is going to be 0.25 for E. So how do we get the value of the Alfa? I do. You can either use a calculator or any other statistical tool, and you will find that the value for Z Alfa by two happens to be 1.96 All right, so let us substitute the values. This is going to be 0.53 plus minus rude over 0.53 multiplied by one minus 0.53 which is nothing but 0.47 upon. And and over here I have 1.96 So let me just use a calculator in order to find this. Okay, so this is route over 0.53 multiplied by 0.47 divided by 1500 because N is 1500 and I multiply this thing by 1.96 So this happens to be 0.25 to so this value over here is 0.25 to So this is plus minus 0.53 Right now, this is going to give me a lower value and and upper value. So 0.53 minus 0.252 which is 0.50 for it. Zero point 5048 comma. This is just a woman. 0.53 plus zero point 0 to 5 to what happens to be 0.5552 0.555 to. So this is our confidence interval 95% confidence interval for a part eight. Okay, this is our first answer. Moving on toe part B. What's part B c In part B. It is said that 31% of the respondents feel that they have to save more for retirement to make up for what they lost. And again, we want to develop a 95% confidence interval for the population proportion since we have 95% are Alfa is the same. Which means our Alfa by two is against same that 0.25 and rz Alfa by two is 1.96 We already know the formula that we wrote above, so we're just going to substitute the values here. This is going to be 0.31 plus minus rude over 0.31 multiplied by 0.69 upon 1500 and I multiply this thing by 1.96 Right now I'm going to use the calculator again. So this is route over 0.31 multiplied by 0.69 divided by 1500 multiple 11.96 So this time I have 0.234 So 0.234 This over here is plus minus 0.31 So what I do is 0.31 minus 0.234 with a 0.28660 point 2866 comma. This is 0.3 even, plus 0.2340 point +3334 0.3334 So this is where 95% confidence interval in this case s O. This is the 195% confidence in trouble for the population proportion. Okay. They're not even percent of the respondents feel that they have to save more for the retirement to make up for what they have lost. All right, so this is our answer to part. Be moving on to part, see what is but see. In this case, 5% of the respondents have given $25,000 or more to charity. So again, we want to develop a 95% confidence and double. Okay, so this time R P bar 0.50 point 05 plus minus root over. We have 0.5 multiplied by 0.95 upon +1500 upon 1500 And this is again were deployed by 1.96 right, 1.96 Whatever we have here in plus minus, this is known as the margin of error. So this is Rudo was 0.5 multiplied by 0.95 divided by +500 and I multiply this by 1.96 which happens to be 0.11 So this is 0.11 and I have over here plus minus 0.5 Okay, a soul 0.5 minus 0.11010390 point 039 Coma. What do we have? 0.5 plus 0.110 point 061 Like this is my population proportion. Confidence in trouble. Okay, so this is the 95% confidence in trouble for, you know, the proportion off the respondents who gave $25,000 or more to charity over the previous year. Now, there is also a part D and in part D. They're asking us, how does the margin of error for the different interval estimates is related to Bieber? Okay, So what is the margin of error? The thing that is over here in plus minus. So for the first case, it is 0.252 for part A. It was 0.252 then I think it was 0.234 0.234 And, uh, in the last part, it was 0.10 point 011 If I look carefully in the first part, it was 53% right. RP borrows 53%. Then it came down to 31%. That is point even then 25% 0.5 So I can see that my e My margin of error is decreasing. Right Is this is decreasing with this is decreasing with p bar, okay. And also, another very important thing over here is what they're asking is Why do we choose? Generally, people are as, uh, 0.5, right. This is the question. Why do we choose P bar as 0.5? Well, because it leaves the maximum room for error. That is why I p p. Bar is chosen as zero point five. Okay, so the margin off error is highest. For example, proportions when peak up or pee bar is actually close to 0.5. And I think this completes a problem. Yes. So these would be our ancestors. So I just write This error is maximum error is maximum or I could see that error maximizes error maximizes when r P bar happens to be close to when P bar is close to is close to 0.5 and this is our answer.


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