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The mean salary of employees at a company is $32,500,with a standard deviation of $2,500.Calculate the z-score for an employee who makes $38,750.b. What does this Z...

Question

The mean salary of employees at a company is $32,500,with a standard deviation of $2,500.Calculate the z-score for an employee who makes $38,750.b. What does this Z-score tell you?According to Chebyshev's Theorem, what percent of employees have salaries between $26,250 and $38,750?d. According to the empirical rule; what percent of employees have salaries that are within $5,000 of the mean?Suppose that the distribution of annual salaries is bell-shaped: Would a salary of $42,000 be consider

The mean salary of employees at a company is $32,500,with a standard deviation of $2,500. Calculate the z-score for an employee who makes $38,750. b. What does this Z-score tell you? According to Chebyshev's Theorem, what percent of employees have salaries between $26,250 and $38,750? d. According to the empirical rule; what percent of employees have salaries that are within $5,000 of the mean? Suppose that the distribution of annual salaries is bell-shaped: Would a salary of $42,000 be considered an outlier? Why or why not?



Answers

Sample annual salaries (in thousands of dollars) for employees at a company are listed. $\begin{array}{lllllll}40 & 35 & 49 & 53 & 38 & 39 & 40 \\ 37 & 49 & 34 & 38 & 43 & 47\end{array}$ (a) Find the sample mean and the sample standard deviation. (b) Each employee in the sample receives a $\$ 1000$ raise. Find the sample mean and the sample standard deviation for the revised data set. (c) Each employee in the sample takes a pay cut of $\$ 2000$ from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)?

So in this question, we are given sample salaries in thousands of dollars for employees, and in part, they were asked to find the sample mean and the sample standard deviation. So for the sample mean we basically take the some effects divided by the number of elements. Sample standard deviation. We take the summer off squares, we divide by and minus one and take the square root. So if we take all of the's values here, our sample mean yes, 41.5385 So Mm hmm. 0.5 green 85 the unit ISS thousands off dollars and our sample standard deviation. So we basically take each value. We subtract the mean from it, we square it and then we summit and then we divide by the number of elements. So the sample standard deviation is 5.3169 same unit off thousands off dollars. So this is our answer to part. So now, in part B were asked to, we're told that each employee receives a 5% race and were asked to find the sample mean and standard deviation. So with the 5% race for us to find the sample mean and standard TV should. So for 5% rates were basically going to take each of these values, and we're going to multiply that by 1.5 and then we're going to use our sample mean and standard deviation formulas and calculate the values. So when we multiply everything by 1.5, for example, 42 come 44.1 and just another example 45 will become 47.25 So we do that for all the values and here, our sample mean is going to be 43.615 thousands off dollars and our sample standard deviation using this formula is going to be 5.583 thousands of dollars. So that's our answer to part. Think now in Part C were asked to take the salaries and divided by 12 to get the monthly salary, whereas to find the mean and standard deviation for the salaries. So, for example, if we take 42 and we divide by 12, we get 3.5. So we're going to do the same thing for all our values here, and after doing that, we will calculate again the same using same formula. The sample mean and simple center, activations or sample. Mean here is 3.46 15 So 3.4615 thousands off dollars and our sample standard deviation is point 0.44 mhm in the unit ears again, thousands off dollars. So we have our answer for part C Now in party, what we're asked is what we can conclude from the results off A, B and C. So basically what we can conclude from our results is when there was a 5% raise in salaries, there was also actually a 5% raise in the mean and standard deviation. So 5% brace in the data for salaries also caused it 5% increase in the sample mean and the sample standard deviation and mean the same thing when the salaries were divided by 12. Then the sample standard deviation and the mean we're also divided by 12, which we can actually see here. So and we actually take our original samples mean and center division, and we divide by 12. We actually get thes values, and so that is our conclusion for parties. Whatever change in the day. If all of the data changes consistently in a particular manner, so does the sample mean and standard division. They basically changed by that same consistent amount.

Problems? Well the annual employer customer participant is normally distributed with the standard deviation equals $625 for birthday. If we know that 73.89 of the values is greater than $1,700. We won't find them in to get them in first. Let's draw the normal distribution which is centered at the mean And we have 73.9 is greater than 1700 and 700. And she means We have here one 1700 to the left of the mean and we have 74 is greater than which means the area and the the normal distribution to the right of x equals 1700 is 4.7389. Let's see how to find them in. We will convert this into a school Which means that centered at zero and we have a corresponding value to this value that has the same area to the right of it. How to get this value? This value equals x minus. You divided by six Which means it equals 1700 minus mu divided by Sigma which is given by 625. This means to get new from this equation we know we need to get the value of the we can get the value of using the standard normal distribution tables which gives us the area to the left of the city schools. This means we need to get the area to the left of the score Which equals 1 -0.7389 which equals 4.261 on 1-. Open 7389. Of course we're going to 611 by entering the table to search for this area. We will get the corresponding Z value. Let's enter the table. They search for 4.26 11 .26 here. Which means 4.611 is here. It's Ed That equals minus 0.601234 adds. It Equals -0.64. This means we have here. Is that is known Is it Equals -0.2. Sorry, -0.64 which equals this exploration. And from this equation, the only unknown is mu we can get new as -4.64. To blow it by 625. It's a plus Plus 100 1700. Then it equals 2100 dollars. Of course for about three we suppose that the mean equals 2000 And $258. And the center of the division is the same. We want to find the cost. The cost At which 31.56%. 31.56 person is greater than this would. Let's imagine the distribution again which is centered at new. We need to find the value at which 31.56 are greater than this very or greater. Which means this person is here And this area is all .3156. To get X. We should get that first by converting this into that school. We have year zero and we have fears that. And to get dessert school, we were searching the tables by the area to the left of the score, which means we want to get this area Enea Equals 1 -0.3156. Because the total area under the normal distribution equals what It's all boy in six it for full. Let's enter the tables to get That corresponding to this area. 4.6844. We can find that here At 1.6 eight is 69 6844 is here At that equals 0.4. It a point for it. And we know that Z equals X minus mu divided by sigma. It equals X minus mu, divided by sigma, wow. This means we can get X as All .48, multiplied by 625 plus 2258 equals 2000 558 dollars.

Look expert. A salary expenses, Judy five day would be Uh huh, the flyby. The salary experience. Let's bring this duty weekends. Given the mean and standard deviation values for X and Y and the standard deviation value says 1 29 and 57 will find a radiance ing Mosque away. That this 6 16,041 and 3000 2 49 No, we have to find a meeting Standard deviation office Total weekly salaries that will be for X and Y X place by so we'll find you've expressed by there is equal to your face lazy over I though there's 1000 to 50. Let's 4 50 That was 1007 100. No, we fight the Stein in division for experts. Wife Sigma X Plus y converted nuts squared off millions Morph Expressway. Since X plus X and Y are independent way have bearings off explains. Why is equal toe really interface less variant off? Why that is giving There is 6 15,041 flesh. 3000 2 49 It's 8 19,090 that is your little 1 41 point $03

Okay, let's start. So here we have. Ah, the question is given and we have 12 people in total. So let's start with the me. So the mean basically is trust all the values added up the white by the number in this case is tough. So this is found to be 5 25 and media. In this case, we have I even number. So we gotta take the average right here. Take the Irish off the six place and the seventh place. So 400 plus 500. Do I buy too? This 4 50 Okay, so how many? Ah, workers earn more than the mean right army. It's 5 25 How many do you have? Right, Right. You're only two. So only two That gets more than the mean So which one mean where media that her referred. Ah, the better described a typical wage at this company, right? It has to be the media in this case because most workers earns around here. Most workers, right. And lastly, we've spread with best described the payroll the range like er or the standard deviation. Okay, so first, let's look at I kjr so I key are once again looks for the placements, right? So this placements we'll be about Well, no will, probably not change even if Ah, the all lighters. Or you should say where the top qualities change. So the I k r is rather stable. Ah, standard deviation in this case, um, is kind of inflated because you have such a high number here, right? The spread, the spread, it's high. So this is not good. What about rage? Once again, this number here, the top never here. It's really high. So the max minus min it's gonna be were high, right? But this also does not reflect the true or I should say the most people has earned right here. So the range is not good. I Keilar is good because eight reflects what is happening here. And it's rather stable. And I'm changed. Whether the top gets a bit more with less. That's it


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