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Consider the ollawing set ungrouped sample dala. Answer pants Inrouon(A) Find Ihe mean and standard duviationungrauped Lampiu data(Type Inlegetdecimal )(Type intege...

Question

Consider the ollawing set ungrouped sample dala. Answer pants Inrouon(A) Find Ihe mean and standard duviationungrauped Lampiu data(Type Inlegetdecimal )(Type integer decimal rounded Ingun dc6mlclacer ~needed )(B) Wha proportionthe mua surements Iles nanslaroaro dvialionthe mean? Witrin standard deviations? Wvithin = Rhannnno devialions?Iho data valueswilhin - Wandard deviatlontne rrigan0l Ihu data valuesMkinatunduro cavialions of the MeAn[ne dala vallucstalll withinFtandardivlatons oliha meanQus

Consider the ollawing set ungrouped sample dala. Answer pants Inrouon (A) Find Ihe mean and standard duviation ungrauped Lampiu data (Type Inleget decimal ) (Type integer decimal rounded Ingun dc6mlclacer ~needed ) (B) Wha proportion the mua surements Iles nan slaroaro dvialion the mean? Witrin standard deviations? Wvithin = Rhannnno devialions? Iho data values wilhin - Wandard deviatlon tne rrigan 0l Ihu data values Mkin atunduro cavialions of the MeAn [ne dala vallucstalll within Ftandardi vlatons oliha mean Qusod on YoUr ansi0t} par (0), Nculd You coniciurd Innt tnc narortam uonroximaltbell #pod? Explain, tecaue mcal thu data valuos Iall nuar Iho Moan data valuos farthe tram tho (oan bacaube aboul thu Ntnrlmnurg Uat valuna fall noar (he Monn Aat huttAho Below Ihe meruh bacaui moit Uihe dal Vallusalall unIhar Ihan atnndn d doxldllons Irnm nean cucausB Most dula valuus bulueun dandArd devnllon: Irom the mean Gunumn Your conmcturn conanc hubtoornm wath tha claua WKorvnl Miln Thuctno Which hislogram bolow enown Ihu d4tu?



Answers

Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the $10 \%$ trimmed mean for a data set, first arrange the data in order, then delete the bottom $10 \%$ of the values and delete the top $10 \%$ of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 30 "Aluminum Cans" in Appendix B) for cans that are 0.0111 in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, 10\% trimmed mean, and 20\% trimmed mean. $\begin{array}{rrrrrrrrrrr}247 & 260 & 268 & 273 & 276 & 279 & 281 & 283 & 284 & 285 & 286 & 288\end{array}$20$\begin{array}{rrrrrrrr}289 & 291 & 293 & 295 & 296 & 299 & 310 & 504\end{array}$

When tackling this problem, hopefully have some software at hand whether it's ah, graphing calculator or software online. But really, all you need to do is input these data values into your software. It will calculate the main and standard deviation for you. So in this case, I found a standard deviation calculator online, and I'm going to go ahead and input all of these pieces of data in 89 may push calculating and will give me all the information I need. So the standard deviation between just called SD or you can call it, um, the Scripps Simple Omega. It's 5.6 and then you look around, you see this symbol here, This is mu or in math, that means Mean mu is kind of a fun simple to try to rate of 7.7 and you are all done

Some of those problem once again, you know, probably plug in the values that you have from any tab. Maybe you can use a mini tab. Um If you're doing it online otherwise if you're doing on paper you'll have to copy this into a calculator like a T. I. 83. Um Or you could probably just use Excel, make sure that they get all lined up and calculate the median and average values ourselves. Been very helpful with that. It's what I've used personally for statistics but just as a refresher, the average is going to be the sum of every value that you have in your dataset divided by the number of observations which is just your sample size. Um And the median. It's going to be the arrangement of the largest, the greatest values in ascending order on this way. And it's going to be the middle value of you're going to have an equal number of numbers on this side as you will between that this implies that the sample size size needs to be odd. However, if the sample sizes even all you have to do is take the value of this, the points or the data sets the data between the median. So you just have to take the average between the two and that will be your medium. That's for the observations that you can make. Uh you can just say, oh look at this, the average and A. And set A. Is greater than the median. And then if you want to compare between sets, you could look at the averages for both sets of A. And B and B. Like all this one's bigger. Huh? I wonder why that is. And uh that's pretty much it.

The following is a nova test based on the mean salaries for different metropolitan areas. So the alternative or the null hypothesis is that all the means are the same. So there are six metropolitan areas, I think it goes Chicago, Dallas Miami, Denver san Diego and Seattle. Uh So the null hypothesis is that all the means are the same. And then the alternative is that at least one of them is different. The second step is to find the critical value and you can do that using either software or a table, But they're essentially three things you need. The first thing is your alpha value, your significance level and that's usually given to you the problem and that's .05. Then you need the degrees of freedom for the numerator and the degrees of freedom for the denominator. And the way you find that Is the degrees of freedom for the numerator is the number of categories -1. So there were six cities that we looked at our metropolitan areas, so 6 -1° of freedom would be five for the numerator. And then for the denominators, the total number of data values minus the number of categories. So there were 36 data values minus the six metropolitan areas. So 30 is your degrees of freedom for the denominator. So that should be enough to use a table. But I use a calculator and I wrote a program in here called inverse. F. I'm not going to show you how to how to write the program. You can youtube it if you wish. Um But this is what I do. So um I put in my area which is my alpha value, my degrees of freedom is five and then my degrees of freedom for the denominator is 30 and That gives me my critical value. About 2.534 2534 is my critical value. I call f. star. So 2.534. Okay so anything greater than 2.534. We reject the annual hypothesis that all the means are the same And anything less than 2.534. We failed to reject meaning the h not is true. Okay so the second step is to find the F statistic and there's a formula but it's a bit of a mess. I always use software you know technology is a great thing. So if you go to stat and you can type in your data values. So these are the mean salaries um So again L1 I think was Chicago and then this is the mean salary for Dallas Miami Denver San Diego and Seattle. So there are six categories. And if you go to stat tests and then we're gonna go to the Unova test and then you just type in your columns separated by commas remember there were six columns, six data columns that we used and we need to make sure that all of them are in there and last one and then also you know make sure you separate those by commons, otherwise it's going to read it wrong. So then um that gives us everything we need. So the F. Is the F statistic, that's the third step. So we're looking at this it's about 2.281 as our F. Value. So two point 281 is our f statistic Which is actually barely in the non rejection region 2.281. So that means we fail to reject. Okay and also we can verify that with this p value here. So the p values 0.7 which is a pretty small p value, but it's still in this case greater than the alpha value. So the alpha value remembers point oh five, so it's barely greater than the alpha value. And whenever it's greater than the alpha value, uh we failed to reject, I should probably put H not there, so we failed to reject H not whenever the P values greater than the alpha. Okay. So then the last step is to summarize everything with actual words. So what does this all mean? It just means that there is not sufficient evidence, there is not sufficient. I guess you could say statistical evidence to suggest that the mean salaries from the different metropolitan areas are different. Okay. And that's the five step process for an Innova one way and over test

As we're looking at the mean price of agricultural books at the 10% significance level. The mean is supposed to be 8.45 So that's gonna be our hypothesis. $8.45 to be more specific. But that's it. And then we want to find out if there is a difference so that is a does not equal to tail test your hypothesis. And our green is gonna give us our calculated chi square statistics. So it's gonna be 28 minus one. So 27 divided by the deviation squared. So 8.45 mhm squared times 9.29 squared. And that is going to get us a value of put into my calculator because divided by 8.45 squared. And we get 32.63 Yeah. Yeah. And that's our chi square graph right? There does not equal means we chop it off here and here for a good dose. If we land in the shade of reach right here, we will rejecting all hypothesis at the 10% significance level. That means we're looking at half 5% on both ends. So we're gonna be looking at 5% for the right end and 95% for the left end at a degree of freedom of 27. So we're looking at 27 here and 5% is going to get us 40.113 Mhm. And for the left side, gonna get us at the 95% level. Yeah, and that's gonna be 29 degrees of freedom. So 17.7 all eight 32 is smack dab in the middle there. So we're good. We fail to reject the hypothesis and that is all that was required of us. Um mm The standard deviation is indeed 8045 cents.


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