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(1) Use Chebyshev’s theorem to find what percent of the values will fall between 206 and
296 for a data set with a mean of 251 and standard deviation of 15
Work:
Chebyshev´s theorem:
Percent of values between mean-k(sd) and mean + k(sd) is at least (1 -1 /k^2)100%
Here mean = 251
mean – 3sd = 251-3(15) = 206 and mean + 3sd = 251+3(15) = 296
So percent of values between between 206 and 296 is at least (1-1/(3^2))100% = 88.89%
Answer: 88.89% (rounded to 2 dp)
(2) Use the Empirical Rule to find what two values 68% of the data will fall between for a
data set with a mean of 293 and standard deviation of 19
Work:
Using the empirical rule 68% of the data will fail between mean-sd and mean+sd
Mean = 293, sd=19
Then 68% of the data values fall between 293-19 = 274 and 293+19 =312
Answer: Values are 274 and 312
(3) Nine college students had eaten the following number of times at a fast food restaurant
for dinner in the last ten days:
4, 10, 1, 7, 1, 8, 1, 5, 2
Find the mean, median, mode, range, and midrange for these data.
Work:
Sample size = n = 9
Mean = sum(x)/n = (4+10+1+7+1+8+1+5+2)/9 = 39/9
Ordered data is: 1,1,1,2,4,5,7,8,10
Median = middle value (located at the 5th place) = 4
Mode = most frequent value = 1
Range = Max – Min = 10-1 = 9
Midrange = (Max+Min)/2 = (10+1)/2 = 5.5
Answer:
Mean = 39/9
Median = 4
Mode = 1
Range = 9
Midrange = 5.5
(4) A Math test has a mean of 51 and standard deviation of 5.0. Find the corresponding z
scores for:
I. a test score of 64
II. a test score of 47
Work:
I) z-score = (64-mean)/sd = (64-51)/5 = 13/5 = 2.6
II) z-score = (47-mean)/sd = (47-51)/5 = -4/5 = -0.8
Answers:
I) 2.6
II) -0.8
(5) Rank the following data in increasing order and find the position and value of the 58th
percentile. Please show all of your work.
551387770098
Work:
Ordered data: 0,0,1,3,5,5,7,7,7,8,8,9
n = 12 p = 58
(p/100)*(n+1) = (58/100)*(13) = 7.54 , 7th value and 8th value are the same (7)
Then 58th percentile is 7
Answer: 58th percentile is 7
(6) The following data lists the average monthly snowfall for January in 15 cities around
the US:
13 12 24 44 32 28 12 17
39 47 11 33 45 17 22
Find the mean, variance, and standard deviation.
Work:
Sample size = n = 15
Mean = sum(x)/n = (13+12+….+17+22)/15 = 396/15 = 26.4
Variance = sum[(x-mean)2]/(n-1) = [(13-26.4)2+...+(22-26.4)2]/14 =167.83
(rounded to 2 dp)
Standard deviation = V(x) = 167.83 = 12.95
Answers:
Mean = 26.4
Variance = 167.83
Standard deviation = 12.95
(7) Starting with the data values 70 and 100, add three data values to the sample so that the
mean is 79, the median is 80, and the mode is 80. Please show all of your work.
Work:
Let the remainng values x ,y and z
If the mode is 80 then 2 values (at least) must be 80 then we select: x = y = 80
Mean = (70+100+80+80+z)/5 = 79
330+z = 5*79 = 395
Then : z = 395-330 = 65
Ordered data is 65,70,80,80,100 (mean is 80)
Answer: We must add the values: 80, 80 and 65
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