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An ME student claims that BMEs make on average less than S66,000 a year. We collect a sample of 100 BME students and find that their average salary is S64,000 a yea...

Question

An ME student claims that BMEs make on average less than S66,000 a year. We collect a sample of 100 BME students and find that their average salary is S64,000 a year. The population standard deviation for their salary is S10,000 year:With a significance level of a-0.05 what can we say about the ME student's claim?What is the effect size?

An ME student claims that BMEs make on average less than S66,000 a year. We collect a sample of 100 BME students and find that their average salary is S64,000 a year. The population standard deviation for their salary is S10,000 year: With a significance level of a-0.05 what can we say about the ME student's claim? What is the effect size?



Answers

Study more! A student group claims that first-year students at a university study 2.5 hours per night
during the school week. A skeptic suspects that they study less than that on average. He takes a random
sample of 30 first-year students and finds that $\overline{x}=137$ minutes and $s_{x}=45$ minutes. A graph of the data shows no outliers but some skewness. Carry out an appropriate significance test at the 5$\%$ significance level. What conclusion do you draw?

In this question. We want to compare the average earnings of college graduates whether they're men or women, and is there a difference? And if there is a difference in income, what is that different in income between men and women? When he was hypothesis testing specifically the Z test presented at page 4 91 of the book because started Step one gather over data so we know that it started with the simple sites for one. So we know we simple the 100 men. So 100 men and 100 women and those simple we The average earning for men was 90 1002 $100. For women, it was lower. It was 57,000 $800. The standard deviation for all mental not just for a sample for the population of men is $15,000. For women, it's 12,000 $800. We're gonna right now, our hypotheses. So the no hypothesis is based on the question should be that the difference in price the difference in the present earnings or income is exactly $30,000. What is claimed in the question is that this difference is not $30,000 So we will have a two tailed tests as soon as I finish writing this. So the difference in income or earnings and not exactly sure the difference. I'm not economy major. So, uh, the difference is not $30,000 So we have a two tailed tests. Therefore, rejection area is this thing this area in red. So to tell our normal distribution now that we have our hypotheses, you have Oliver data our data that's going to step two. So step to want to find our critical value. We know that Alfa is 0.1 Therefore, Alfa over to is 0.0 Ah, another 05 So are critical values are plus minus yes. Close or minus two 0.575 minus 2.575 on the left of our graph and 2.575 on the right hand side of our graph ever critical value. Now we need a zeke to compare those critical values to. So Step three, figure out Z Z is the difference in earnings for our samples. Ah, I wrote it s because I was thinking simple, but it's w four women. Mine is the difference in earning for all men and all women divided by the variants in earnings for men divided by the number of men. Simple. Plus that the variants in earnings for women divided by the number of women. Simple. All of that inside a square root. Remember, H zero tells us that this quantity is difference for you men and women Home. Oh, minute on women is $30,000. So what we have for Z world be 90,000 200 minus 57,000 800 minus 30 1000 over 15,000 squared, divided by 100 plus 12,800 squared, Also divided by 100 Everything inside a square root. I will let you plug all those values in your calculator or your computer, and you will get a value of 1.21. Let's go compare debt Z value to the critical value found that step to so Z is 1.21 1 point. Anyone that say it's right here in any case is smaller than 2.575 So we're not in the rejection area, so we do not reject age zero. We accept with those samples we accept h zero. So since one since 1.21 is smaller, then 2.575 and we do not reject 80 which means the difference in mean income. Four man and women is 30 $1000 sell quick recap. We gathered all of her data for problem. We rolled down her hypotheses and figured out critical values to define a rejection area. Computed rz two see if it was in their rejection area and we decided, since it wasn't in the rejection area, we did not reject age zero and we concluded that the difference in mean and come for men and women is $30,000.

The following is a solution for number 17. And we're looking at the difference in average starting salaries for grads with master's degree degrees in engineering or business, so we're seeing if there's a difference. Now notice it doesn't say whether one is bigger or smaller, just that there's a difference. So that's why if you kind of skipped down, the alternative hypothesis would be not equal to, because it doesn't specify whether one is greater or less than now, it looks like from this sample, the engineering is greater, but Um but again, it doesn't specify, so we're giving this data here, there are 15 engineers started out making $68,535. That's not a bad starting salary. If you're thinking about going in engineering engineering, that's pretty good. It's more what I make at least. And in the standard deviation of 16, and then business, the there were 11 business majors or master's degree in business, they're starting salary was still not too shabby 63,230 with a standard deviation of 2033. And we're asked to find a 90% confidence interval for the difference of those two means, so I'm gonna use the T. I. T. Four here. And if we go to stat and then tests because those sample sizes are too small, I want you to go all the way down to zero where it says two, sample T. Interval. Uh The magic number is 30. If they were both sample sizes of at least 30 we could use the z. Interval, but we can't, right, So it's the sample sizes are too small. So here you see the data, the X. One bar the s. One and the N. One and then the X. To bar the S. Two and the end to and then the sea levels 20.9 that stands for 90%. And we're still pulled. So click on yes on pooled because those standard deviations are soon to be equal. So whenever we calculate you can see the difference here between engineering and businesses About 4,078 bucks and $6,532. So it looks like a significant difference there. So the confidence in the 90% confidence intervals between 40 77 I'll go and go to the sense 770.60 and then 65 32 uh huh. And 40 cents. So that's the 90% confidence interval for the difference between engineering and business. Okay so now we look at the hypothesis test in part B and it says that the null hypothesis is that there is no difference. And then the alternative is saying that there is a difference and we're testing at the 10% level of significance. So I need a T. Value and I need a p value. And what we're gonna do with that p values, we're going to compare it to the alpha value. So fortunately for us we already have this data ready to go if you go back to stat. And then tests, we can go down to two sample t test stats is highlighted and everything should be pre populated in there because we haven't changed it. The only thing that we need to change is that alternative hypothesis? It should be new one, not equal to mewtwo pulled is still yes. So we can go and calculate from here and we get a T value of very very large 7.39 and we have a p value of essentially zero. So it's not a p value of 1.23. It's the p value of 1.23 times 10 to the negative seventh. So that's point, you know, six zeros and then a one. So it's essentially zero, that's statistically zero. Right? So t equals 7.395. And then the p value is essentially 0.0 I guess, which is less than 0.10 obviously. So we're going to reject the null hypothesis anytime the p values that small, especially we reject pretty thoroughly the null hypothesis that there is no difference. And we can say pretty safely that there is a difference between start starting salaries for graduates with master's degree in engineering compared to business

Were asked to find the probability that a randomly selected teacher turns between $80,000 so they can express. That is the probability that teacher and 35 spend 25,000 dollars a year. And we're given that the average annual salary. Uh, you, too. 47 dozen. So 50 others. That is just delicious trends that we can use from season two. Thanks. This you signal were also given signal or the standard deviation is equal to it doesn't 686 today we're gonna attempt this problem is to find the Z value spertus for these separate conditions. 5000. Perfect. So let's set this one C one this on C two Dan, we're gonna get is Z one is equal to 35 1000. Nice. You? Which is? I have 7 7050 over signal. That doesn't 600 a. I mean, this value is equal to negative 2.24 Now again to the same thing for easy to see to Jersey. Going to 45 dozen minus said 50 to first 1006 100 18. Go Mean that that is in a cool negative zero four eight. So now that we have this, we can rewrite this original expression as from Billy the negative, too. 24 Just Z value for that doesn't is less than Jeanne just less than negatives. You for eight. Now, what we're gonna do is check out the Z table, find the corresponding probabilities of these values for negative 2 to 4. Get your print you to five for natives, use for bait. Digging is your point three, but six. So now we condemn view it. This as follows as the probability that Z is less than negatives your point for and minus the probability that C is less than neck It is, too to four. We know that this is the same thing. Is this so you can just put that and share substitute values in 05 six minus You're sure 12 Which gives us ju 2.3. You're to me. What? That's fine. Apart. Be slightly different. It's asking us probability that teacher earns more than $40,000 a year so you can write. That is the probability that eggs is greater done. 40 1000 dollars. So to solve this, we're just going to use the same from love. Just Z is equal to extra for signal. We know that mule and sing Listen Same useful 50 signal respect in 680. So if you just go ahead substitute 40,000 50 uh, 6 5080 80. They're gonna get value off negative one point 36 However, I usually use this formula distance to be less than inside. And since it's not have toe do one minus, we're gonna have to do probability of said this crater 1.36 is equal to one, minus the probability probability. I see it's less than negative one from 36 That is the equivalent statement. And it's it's useful for us to write it like this because, okay, we can find this value right here using this formula and just one minus to get greater than questions. So now if you find the corresponding value for negative 36 from our see table, we're going to get zero a six. But so done. Iffy. Substitute back into here. Get one minus is your hey six, which is the corresponding probability to this. See value on that gives us a final answer. Serial 0.9 one one for Part C. The question is asking us that if you were offered there you $1000 a year for this position, how would you feel basic information? And so there's no there's no real calculation. You need to do this for this part, but you just need to look at your parts A and B on used logical thinking to terminate. Answer. So here we can tell that the probability that a teacher earns more than $40,000 a year is open 9131 which is the same thing as 91 0.3 1%. No 1.1% of teachers more than $40,000 a year. Therefore, if you were offered doing so 31 dozen dollars, you wouldn't be too happy because most teachers would be making a lot more than you

Hello. So these are the solutions. The question has been next, and, uh, I've also under the reason why I'm giving these answers. Okay, So I hope you find this useful. Thank you very much.


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