#$&* course MTH 152 1/25 11 Question: `q001. Note that there are 10 questions in this assignment.
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Given Solution: To get the mean value of the numbers, we first note that there are eight numbers. Then we had the numbers and divide by eight. We obtain 5 + 7 + 9 + 9 + 10 + 12 + 13 + 15 = 80. Dividing by 8 we obtain mean = 80 / 8 = 10. The difference between 5 and the mean 10 is 5; the difference between 7 and the mean 10 is 3; the difference between 9 and 10 is 1; the differences between 12, 13 and 15 and the mean 10 are 2, 3 and 5. So we have differences 5, 3, 1, 1, 0, 2, 3 and 5 between the mean and the numbers in the list. The average difference between the mean and the numbers in the list is therefore ave difference = ( 5 + 3 + 1 + 1 + 0 + 2 + 3 + 5 ) / 8 = 20 / 8 = 2.5. Self-critique: OK ------------------------------------------------ Self-critique rating: ********************************************* Question: `q002 What is the middle number among the numbers 13, 12, 5, 7, 9, 15, 9, 10, 8? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the middle number of (13, 12, 5, 7, 9, 15, 9, 10, 8) all we need to do is find out how many numbers are listed here which we can tell that there are 9 numbers. Now all we need to do is subtract 1 from 9 which leaves us with 8 numbers the reason we do the subtraction first is so that we can remember to leave out a number which is the middle number. Now we need to divide 8 by 2 which leave us with 4 numbers. Now all we need to do is take the first 4 and the last 4 numbers off which leaves us with the answer of 9. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: It is easier to answer this question if we place the numbers in ascending order. Listed in ascending order the numbers are 5, 7, 8, 9, 9, 10, 12, 13, and 15. We see that there are 9 numbers in the list. If we remove the first 4 and the last 4 we are left with the middle number. So we remove the numbers 5, 7, 8, 9 and the numbers 10, 12, 13, and 15, which leaves the second '9' as the middle number. Self-critique: OK ------------------------------------------------ Self-critique rating: ********************************************* Question: `q003. On a list of 9 numbers, which number will be the one in the middle? Note that the middle number is called the 'median'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since there are not any numbers listed we know that if we have 9 numbers we need to subtract 1 then take the first and last 4 numbers off which will leave us with the middle number. Thus out of a group of 9 numbers our answer will be the 5th number. For exsample: (10, 12, 500, 89, 6541, 1, 2, 80, 2137, 7, 5) no matter what the numbers are or how many there are it is easier than it looks. Step 1 count how many numbers there are in this case there are 11 numbers so we subtract 1 from 11 which leaves us with 10 Step 2 divide the numbers in half 10/2=5 Step 3 take off the first 5 numbers leaving 1, 2, 80, 2137, 7, 5 Step 4 Take off the last 5 number which will leave us with 1 Thus the answer to this problem is 1 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: If the 9 numbers are put in order, then we can find the middle number by throwing out the first four and the last four numbers on the list. We are left with the fifth number on the list. In general if we have an odd number n of number in an ordered list, we throw out the first (n-1) / 2 and the last (n-1) / 2 numbers, leaving us with the middle number, which is number (n-1)/2 + 1 on the list. So for example if we had 179 numbers on the list, we would throw out the first (179 - 1) / 2 = 178/2 = 89 numbers on the list and the last 89 numbers on the list, leaving us with the 90th number on the list. Note that 90 = (179 - 1) / 2 + 1, illustrating y the middle number in number (n-1)/2 + 1 on the list. Self-critique: I should have listed my answer in terms of n instead of going into an example. ------------------------------------------------ Self-critique rating: ********************************************* Question: `q004. What is the median (the middle number) among the numbers 5, 7, 9, 9, 10, 12, 13, and 15? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First we count the numbers which tells us there are 8 numbers. Which in this case it works a little differently. What we would need to do is take the first number and last number off leaving us with 7, 9,9,10,12,13. Which leaves us with 6 number knowing this we will continue taking the first and last number off till we get to the middle 2 numbers which will be 9, and 10. Knowing the two middle numbers we take an average of those two which 9+10=19 then we divide 19/2=9.5 Which gives us the middle number of this set which is 9.5 Confidence Rating: 3
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Given Solution: There are 8 numbers on this list. If we remove the smallest then the largest our list becomes 7, 9, 9, 10, 12, 13. If we remove the smallest and the largest from this list we obtain 9, 9, 10, 12. Removing the smallest and the largest from this list we are left with 9 and 10. We are left with two numbers in the middle; we don't have a single 'middle number'. So we do the next-most-sensible thing and average the two numbers to get 9.5. We say that 9.5 is the middle, or median, number. Self-critique: OK ------------------------------------------------ Self-critique rating: ********************************************* Question: `q005. We saw that for the numbers 5, 7, 9, 9, 10, 12, 13, and 15, on the average each number is 2.5 units from the average. Are the numbers in the list 48, 48, 49, 50, 51, 53, 54, 55 closer or further that this, on the average, from their mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 48+48+49+50+51+53+54+55= 408/8 = 51 51-48=3, 51-48=3, 51-49=2, 51-50=1, 51-51=0, 53-51=2, 54-51=3, 55-51=4 3, 3, 2, 1, 0, 2, 3, 4 3+3+2+1+0+2+3+4=18/8= 2.25 In answer to the question the number in the list 48, 48, 49, 50, 51, 53, 54, 55 are closer on average from their mean. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: The mean of the numbers 48, 48, 49, 50, 51, 53, 54, and 55 is (48 + 48 + 49 + 50 + 51 + 53 + 54 + 55) / 8 = 408 / 8 = 51. 48 is 3 units away from the mean 51, 49 is 2 units away from the mean 51, 50 is 1 unit away from the mean 51, and the remaining numbers are 2, 3 and 4 units away from the mean of 51. So on the average the distance of the numbers from the mean is (3 + 3 + 2 + 1 + 0 + 2 + 3 + 4) / 8 = 18 / 8 = 2.25. This list of numbers is a bit closer, on the average, then the first list. Self-critique: I should have went into more detail about the average of the second set of numbers. ------------------------------------------------ Self-critique rating: ********************************************* Question: `q006. On a 1-10 rating of a movie, one group gave the ratings 1, 8, 8, 9, 9, 10 while another gave the ratings 7, 7, 8, 8, 9, 10. Find the mean (average) and the median (middle value) of each group's ratings. Which group would you say liked the movie better? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The 1st group median: 8.5 The 1st group mean: 7.5 The 2nd group median: 8 The 2nd group mean: 8.1 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: The mean of the first list is (1 + 8 + 8 + 9 + 9 + 10) / 6 = 45 / 6 = 7.5. The median is obtained a throwing out the first 2 numbers on the list and the last 2 numbers. This leaves the middle two, which are 8 and 9. The median is therefore 8.5. The mean of the numbers on the second list is (7 + 7 + 8 + 8 + 9 + 10) / 6 = 49 / 6 = 8 .16. The median of this list is found by removing the first 210 the last 2 numbers on the list, leaving the middle two numbers 8 and 8. The median is therefore 8. The first group had the higher median and the lower mean, while the second group had the lower median but the higher mean. Since everyone except one person in the first group scored the movie as 8 or higher, and since everyone in both groups except this one individual scored the movie 7 or higher, it might be reasonable to think that the one anomalous score of 1 is likely the result of something besides the quality of the movie. We might also note that this score is much further from the mean that any of the other scores, giving it significantly more effect on the mean than any other score. We might therefore choose to use the median, which limits the otherwise excessive weight given to this unusually low score when we calculate the mean. In this case we would say that the first group liked the movie better. Self-critique: I should have explained in detail how the answers were obtained. I should have also given explanations of the scores. ------------------------------------------------ Self-critique rating: ********************************************* Question: `q007. Suppose that in a certain office that ten employees make $700 per pay period, while five make $800 per pay period and the other two make $1000 per pay period. What is the mean pay per period in the office? What is the median? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First we need to list the numbers of emplyees and their pay to find the correct answer. According to the statement we have 10+5+2= 17 employees 10 * $700 + 5 * $800 + 2 * $1000 = $13,000 13,000/17 = $764.70 Thus the mean pay is $765 To find the meadian we need to list the pay. 700, 700, 700, 700, 700, 700, 700, 700, 700, 700, 800, 800, 800, 800, 800, 1000, 1000 Now we know that there are 17 numbers total here so to find the median we need to subtract one from 17 which leaves 16 now we need to divide by 2 so 16/2=8 now we need to take off the first and last 8 numbers which leaves us with. 700 Thus the median pay for this office is $700.00 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: There are a total of 10 + 5 + 2 = 17 employees in the office. The total pay per pay period is 10 * $700 + 5 * $800 + 2 * $1000 = $13,000. The mean pay per period is therefore $13,000 / 17 = $823 approx.. The median pay is obtained by 'throwing out' the lowest 8 and the highest 8 in an ordered list, leaving the ninth salary. Since 10 people make $700 per period, this leaves $700 as the median. STUDENT QUESTION: Is it typical to use the median value if there are ‘oddball’ scores in a group? INSTRUCTOR RESPONSE A few 'oddball' scores have little effect on the median, but can have a great effect on the mean. Other factors can also be important depending on the situation, but if a lot of 'oddball' scores, or 'outliers', are expected the median is often the better indication of average behavior than the mean. Self-critique: I have checked my work about 8 different times and I cannot get to answer of 823. I have no clue what I have done wrong. ------------------------------------------------ Self-critique rating:
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Given Solution: The mean was found in the preceding problem to be $765. The deviation of $700 from the mean is therefore $65, the deviation of $800 from the mean is $35 and the deviation of $1000 from the mean is $135. Since $700 is paid to 10 employees, $800 to five and $1000 to two, the total deviation is 10 *$65 + 5 * $35 + 2 * $235 = $1295. The mean deviation is therefore $1295 / 17 = $76.18 , approx.. ********************************************* Question: `q009. What is the mean of the numbers 1.05, 1.03, 1.06, 1.08, 1.06? On the average by how much do these numbers deviate from the mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1.05 + 1.03 + 1.06 + 1.08 + 1.06 = 5.28 5.28 - 1.05 = 4.23, 5.28 - 1.06 = 4.22, 5.28 - 1.08 = 4.20, 5.28 - 1.06 = 4.22 4.23 + 4.22 + 4.20 + 4.22 On average each number deviates from the mean by 4.22 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3 ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q010. What is the mean of a set of numbers in which 1.05 occurs 4 times, 1.03 occurs 3 times, 1.06 occurs 10 times and 1.08 occurs 3 times? On the average by how much do the numbers in this set deviate from their mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1.05 * 4 + 1.03 * 3 + 1.06 * 10 + 1.08 * 3= 21.13 / 20 = 1.06 The mean is 1.06 1.06 - 1.05 = 0.01, 1.06 - 1.03 = 0.03, 1.06 - 1.06 = 0.00, 1.08 - 1.06 = 0.02 0.01 + 0.03 + 0.00 + 0.02 = 0.06 / 4 = 0.02 On average the numbers deviate from their mean by 0.02 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3 ------------------------------------------------ Self-critique Rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!