course Mth 152 ÎvÙl‰À²Â²„û”ûr»¤¼šà¿´ØÝâÞÏassignment #016
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20:32:43 query probl 13.3.6 range, std dev of {67, 83, 55, 68, 77, 63, 84, 72, 65}
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RESPONSE --> 55+63+65+67+68+72+77+83+84 = 634 634 / 9= 70.4 mean deviation from the mean squared deviation 70.4 - 55 = 15.4 15.4 x 15.4 = 237 .16 70.4 - 63 = 7.4 7.4 x 7.4 = 54.76 70.4 - 65 = 5.4 5.4 x 5.4 = 29.16 70.4 - 67 = 3.4 3.4 x 3.4 = 11.56 70.4 - 68 = 2.4 2.4 x 2.4 = 5.76 70.4 - 72 = 1.6 1.6 x 1.6 = 2.56 70.4 - 77 = 6.6 6.6 x 6.6 = 43.56 70.4 - 83 = 12.6 12.6 x 12.6 = 158.76 70.4 - 84 = 13.6 13.6 x 13.6 = 184.96 237.16 + 54.76 + 29.16 + 11.56 + 5.76 + 2.56 + 43.56 + 158.76 + 184.96 = 728.24 728.24 / 9 = 80.91 square root of 80.91 = 8.99 the range would be 29
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20:32:55 ** x dev. from mean squared dev. 55 15.4 237.16 63 7.4 54.76 65 5.4 29.16 67 3.4 11.56 68 2.4 5.76 72 1.6 2.56 77 6.6 43.56 83 12.6 158.76 84 13.6 184.96 634 728.08 mean = 634 / 9 = 70.4 std. dev. = `sqrt (728.08 / 8) = 9.54 range = 84 - 55 = 29 **
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RESPONSE --> o.k.
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21:13:13 **** query probl 13.3.12 freq dist 14,8; 16,12; 18,15; 20,14; 22,10; 24,6; 26,3
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RESPONSE --> 14 x 8 = 112 16 x 12 = 192 18 x 15 = 270 20 x 14 = 280 22 x 10 = 220 24 x 6 = 144 26 x 3 = 78 mean = 1296 / 68 = 19.05 19.05-14 = 5.05 5.05x 5.05 =25.502 x 8 =204 19.05 -16 = 3.05 3.05 x 3.05 =9.302 x 12 = 112 19.05- 18 = 1.05 1.05 x 1.05 =1.102 x 15 = 16 19.05 - 20 = .95 .95 x .95 =.902 x 14 =13 19.05 - 22 =2.95 2.95 x 2.95 =8.70 x 10 =87 19.05 - 24 = 4.95 4.95x4.95=24.50 x 6 =147 19.05 - 26 = 6.95 6.95 x 6.95 =48.30 x 3 =144 204+112+16+13+87+147+144 = 723 723/68=10.63 square root of 10.63 =3.26
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21:14:04 ** Value freq Value * Freq Dev^2 * F 14 8 112 204.80 16 12 192 112.32 18 15 270 16.80 20 14 280 12.32 22 10 220 86.40 24 6 144 146.40 26 3 78 144.48 Total 68 1296 723.52 Total squared dev is 723.5 so ave squared dev is 723.5 / 68 = 10.6, approx. Std dev is sqrt(ave squared dev) = sqrt(10.6) = 3.3 approx. **
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RESPONSE --> o.k. some of my numbers did not match the numbers in the answer but the end result was approximately the same.
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21:20:04 **** query probl 13.3.18 chebyshev for z=5 What is the least possible number of elements of a sample which lie within 5 standard deviations of the mean?
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RESPONSE --> 1 - 1/5^2 = 1 - 1/25= 24/25 = .96 used the formula in example 6 on page 813
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21:20:10 ** The formula 1 - 1/k^2 gave you .96. That's the proportion which must under any circumstances lie between mean and 5 std dev from the mean. So the number is .96 n, where n is the number of elements in the sample. **
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RESPONSE --> o.k.
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21:23:28 query probl 13.3.48 mean length of stay 2.7 days, std dev 7.1 days.
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RESPONSE --> mean length = 2.7 standard deviation = 7.1 2.7 + 7.1 = 9.8
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21:23:36 ** A sketch of a normal distribution will be a normal, or ‘bell-shaped’ curve with its peak at the mean, dropping to about 60% of peak value at 1 std dev from the mean and to about 14% of peak value at 2 std dev from the mean. For a normal distribution with mean 2.7 and std dev. 7.1 one std dev from the mean occurs at 2.7 + 7.1 = 9.8 and at 2.7 – 7.1 = -4.4; two std dev from the mean occurs at 16.9 and -11.5. The corresponding normal curve cannot represent length of stay, since length of stay must not be less than zero. As a result we obtain a curve which ‘tails off’ for large values of x, but whose area is concentrated mostly between 0 and 2.7. This curve is not symmetric like the bell curve but is very skewed, ‘bunched up’ on one side of the mean 2.7 and more spread out for larger values. **
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RESPONSE --> o.k.
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21:31:20 **** Describe your sketch of the distribution of lengths of stay.
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RESPONSE --> I'm really not sure what is wanted on this one. but on the previous problem I knew that if the mean stay is 2.7 and the standard deviation is 7.1 then I had to add 2.7 and 7.1 to get the distribution of 9.3 2.7 would be the middle stay
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21:31:25 2.7 is in the center with each number within 7.1 of the right or left of 2.7 and each additonal number on the left or right within 7.1 of each other. I see the curve as not being skewed.
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RESPONSE --> o.k.
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21:31:35 **** Is your distribution skewed? If so why, and if not why do you think it shouldn't be?
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RESPONSE --> ???????????????????
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21:31:48 STUDENT RESPONSE AND INSTRUCTOR COMMENT: I said no, but I'm not really sure how to determine this if I draw it myself, becasue I place all the numbers at an equal distance from each other. It just hit me while typing this that it must be skewed to the right because I can't really have negative days stay in the hospital. I think I have confused myself ** You didn't confuse yourself. That's exactly the point. You can't stay fewer than 0 days; since even 1 standard deviation is way below 0 the deviations must be primarily to the right of the mean. So the distribution must be skewed significantly to the right. GENERAL SUGGESTION: In general to understand the graphs of various distributions, try to understand in terms first of the bell-shaped curve with max height at the mean, dropping to about 60% height at a distance of 1 std dev from the mean and to about 14% at 2 std dev from the mean. Then understand that this distribution can be distorted, or skewed, as in this problem. This occurs when most of the distribution lies close to the mean on one side, with a smaller part of the distribution spread out further from the mean on the other. **
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RESPONSE --> o.k.
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