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Question 42 ptsIf the standard error of the mean is 10 years of experience for a sample of consisting of 40 workers, what is the probability that the sample mean wi...

Question

Question 42 ptsIf the standard error of the mean is 10 years of experience for a sample of consisting of 40 workers, what is the probability that the sample mean will not exceed 22 years for a population with a mean of 17 years experience_ YOUR ANSWER SHOULD BE BETWEEN 0 AND 100 AND ROUNDED TO 4 DECIMAL PLACES. (e.g: 59.5321 would be 59.5321%)

Question 4 2 pts If the standard error of the mean is 10 years of experience for a sample of consisting of 40 workers, what is the probability that the sample mean will not exceed 22 years for a population with a mean of 17 years experience_ YOUR ANSWER SHOULD BE BETWEEN 0 AND 100 AND ROUNDED TO 4 DECIMAL PLACES. (e.g: 59.5321 would be 59.5321%)



Answers

To estimate the mean age for a population of 4000 employees, a simple random sample of 40 employees is selected.
a. Would you use the finite population correction factor in calculating the standard error of the mean? Explain.
b. If the population standard deviation is $\sigma=8.2$ years, compute the standard error both with and without the finite population correction factor. What is the rationale for ignoring the finite population correction factor whenever $n / N \leq .05 ?$
c. What is the probability that the sample mean age of the employees will be within $\quad \pm 2$ years of the population mean age?

The following is a solution to number four. And this says that a random sample of 19 people, I guess uh from a normal distribution. So that's important. The population distribution is normal. That makes that sample size okay. If it weren't normal, then we would need a sample size of at least 30. So the 19 is okay. And the sample mean of that particular uh Uh there is no context of this problem. So the sample mean of this sample is .8 and then the sample standard deviation is .4, and we're supposed to test that alpha equals .01 if the mean is less than one. Um so first off, because we don't know the population standard deviation, we were never given sigma, were only given the ass, we need to use the T test. Alright, We cannot use the Z test because we don't know sigma's we have to use the T test and it's a five step process, like I said. So first off, the uh the first step is to state our Nolan are alternative hypotheses. So the null hypothesis h not always has some sort of equality and we're testing a population means. So you're always gonna use greek symbols here. So mu is 1.0. And then we're testing if it's less than 1.0. So the alternative is that mu is less than 1.0. The 2nd and 3rd step, I'm actually going to write together because we're going to use technology to get it. So the second step is to find the test statistic and you can use a formula for that. But if you have a calculator, if you have, you know, Excel or some sort of software, um it does a lot of the work for you. So that's what I'm going to use just to save some time. So t is the test statistic and then the p value is also going to be given to me. So let's go to the calculator. Now I'm on ATI 84. And if you go to stat on the T- 84 and go down to tests, it's the second option here. The T test. And we need the summary stats here, so make sure summary stats is highlighted And the mute not, that's the hypothesized value. So in this case it's 1.0 or just one. So it's the null hypothesis. The X. bar is your sample mean, and that was given as .8 SX is the sample standard deviation, which was given us four, And then in is the sample size and that was given as 19. And then you get down here, and this is the alternative hypothesis. So it's either not equal to less than or greater than and if you look back that alternative is less than one. So that's all good. And now we can calculate and that gives you some stuff here. But really this is the T. And the P. Is what I need. So the T. Is negative 2.179. So let's go and write that down -2.179. and then the P value Is .02 0.2 And the p value really is probably the most important thing on that what you do. And step forward as you explicitly compare the P value with the alpha value. And remember the alpha value here is 1.2 is greater than one. And whenever the p value is greater than alpha than we fail to reject the null hypothesis. So failed to reject H. Not had that P value been less than alpha than you would reject H not, but since it's greater than alpha than you failed to reject, so any time you failed to reject your conclusion, which is the last step step is going to be something along this line, there is not sufficient evidence to suggest that The population means I'll just use mu is less than 1.0, so there is not sufficient evidence to say that the alternative hypothesis is true, so therefore we're for lack of a better phrase, were accepting that the null hypothesis is true.

The following is a solution to number five and this is a proportion test where were asked the question or 200 people are asked the question. Um these people have a valid driver's license. Do you drive an American made um vehicle? And of the 200 people, 115 of them said Yes. And we're supposed to test at the alpha are at the significance level of .05. If more than half the people drive an american car. So first off this is called a one I'm going to abbreviate here but it's a one prop Z test. So one proportion Z test will be um the name of this and it's a five step process. The first step is to state our hypotheses and our hypotheses, the knoll always assumes some sort of equality. So mean, I'm sorry not the mean it's the population proportion so P equals 50.5 and that's that's where they get the half. So half is 0.5. And then the alternative it's asking for us to test if it's greater than half or more than half. Okay so that's those are first two steps are first step and it's the two hypotheses. The 2nd and 3rd step are kind of the same thing as the calculator is going to give it to us. So we find the test statistic which all this called Z. And then we also find the P. Value. So you can use the formula here to find Z. And then use you know tables and whatnot for the p value. But I think a calculator is a much easier way to go about this. So if you go on a T I. T. For at least so you can use any sort of calculator you wish. But I think the T I A for is the easiest to use. So if you had a stat and then tests And you're going to go down to this fifth option here where it says one prop Z. Test. So go to option five and then the Peanut, that's the hypothesized value. So that's .5 X. was the number of favorable. So that's 115 in this case. And as the sample size and remember there were 200. And we're checking to see if um If it's greater than 50% so greater than half. So that's greater than Peanut. Alright so everything should be good to go and then we can calculate and it gives us you know the P. Hat and it gives us a lot of information but really it's just the Z. And P. That really make a difference. So the Z value is 2.12 and the p value is 0.17 So don't get that mixed up with P happy hat is the sample proportion? The statistic P stands for the p value. Okay so let's write these down and then we'll talk about a little bit. So the The Z. is like I said 2.1-1 and then the p value Is 0.017. So the P value, like I said is probably the most important thing on here. What you do in the fourth step is you explicitly compare the P value with your significant level alpha. And this time we have a P value of point to 1.2 is going around and an alpha value of .5. So in this case the p value is less than alpha anytime. The p values less than alpha than you reject the null hypothesis. Now had this been greater than alpha? The p value you would fail to reject but it's less than the alpha, so you reject h not. So we're rejecting this null hypothesis that it's equal 2.5, and we're for lack of a better phrase, we're accepting that The actual population proportion is greater than 50%. So our conclusion here would be something to this effect. There is convincing statistical evidence to suggest that more than and the scroll down. Bear with me, more than half of individuals with valid driver's license, our licenses dr american cars or vehicles. Okay, so we are uh we do have sufficient statistical evidence to say that more than half the people drive american vehicles.

Um The central limit limit Children states that that if this sample Bye. And is larger that is a creator or it's called a tatty then there's something distribution distribution of sample means Palestinian normal distribution. Um The minnow sam playing distributions of sample means this meal. Well sample I mean yes knew where newest the population. Me and the population means is 100. And the standard division of the sampling distribution of sample meanies. Ah New reforms Rugova. And then they recognized the population mean an energy center. Right. So we have big night Population mean is 16. And the um Sample sizes for that is eight. Now we know that is a call to. Mhm. Is the courage to X minus mu upon stigma. We have X. S. Um 110 -200 upon heat. There's 1.25. And we have again X. Is 150 0 at once. It 3.75. Um Now Observe the values of X. That is 1.25 and 7.35 In the three table to obtain the area of the left side of the respective Cisco. So the area of left side. Yes 0.0764. And area right side is We're A .9236. No, that is so abstracted the areas to evaluate the total area of this region between. So total area is maybe i is point 9999-. So that is um sorry 9236 -0.076. For that is a call to 0.847. Mhm. Well the that is Recalled to 84.72%. Your sister this ahead? No.

All right. So this population we're looking at has a meat of 200 standard deviation of 50 at a sample size of 100 has taken to find a point estimate for the population. Mean now we're supposed to find the probability that the sample mean is within Lester minus five or plus or minus 10 of the population. Mean All right, let's find our stare deviation for our sampling distribution. Ah, because it doesn't give us a population size. We're going to assume it's infinite. So this equals population standard deviation over the square root of, and it's gonna be 50 over the square root of 100. So that's 50 over 10 were just five. All right? What? We could z score and figure out the probabilities and then do some subtraction. We don't need to do that, because keep in mind, this is following a normal distribution by central limit. Yeah, Central limit. Durham. So thus we've established in a previous section that the probability that this is the mean this is one standard deviation. This is two standard deviation. This right here, 68.5% of all data points and I didn't space this well enough. Hold on. We're gonna move us up for the sake of this point. This year is 95% of all date of all the data. So right here, this is one standard deviation. So we could say that part is 68.5% probability in part B. It's two standard deviations, so that's 95% probability, and there you have it.


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