Question

97, 147, 115, 135, 105, 78, 68, 100, 102, 117, 146, 77, 127, 87, 137, 109, 56, 91, 95, 150, 94, 100, 112, 114, 128, 112, 99, 107, 56, 121, 113, 150, 89, 119, 123, 134

Answer the following questions:

1) Calculate the following statistics for the data (original data):

a) the mean

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b) the median

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c) the standard deviation

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2) Rescale the original data using the transformation Y=X/14. For the transformed data in Y, calculate the following statistics:

a) the mean

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b) the median

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c) the standard deviation

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3) Use your answers in 1) and 2), which of the three statistics you calculated satisfies that “the statistic of the transformed data is the same as the transformed statistic of the original data”: [Hint: the transformed mean of the original data is the mean of the original data/14]

a) None of them
b) Only the mean
c) Only the median
d) Only the standard deviation
e) Only the mean and median
f) Only the mean and standard deviation
g) Only the median and standard deviation
h) All of them

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4) Shift the original data using the transformation U=X+14. For the transformed data U, calculate the following statistics:

a) the mean

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b) the median

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c) the standard deviation

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5) Use your answers in 1) and 4), which of the three statistics you calculated satisfies that “the statistic of the transformed data is the same as the transformed statistic of the original data”: [Hint: the transformed mean of the original data is the mean of the original data + 14]

a) None of them
b) Only the mean
c) Only the median
d) Only the standard deviation
e) Only the mean and median
f) Only the mean and standard deviation
g) Only the median and standard deviation
h) All of them

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6) Transform the original data using the transformation V = √X. For the transformed data V, calculate the following statistics:

a) the mean

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b) the median

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c) the standard deviation

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7) Use your answers in 1) and 6), which of the three statistics you calculated satisfies that “the statistic of the transformed data is the same as the transformed statistic of the original data”: [Hint: the transformed mean of the original data is: log(mean of the original data)] [Use 4 decimals when you compare]

a) None of them
b) Only the mean
c) Only the median
d) Only the standard deviation
e) Only the mean and median
f) Only the mean and standard deviation
g) Only the median and standard deviation
h) All of them

Answer 1

( 1 )

Mean :

Mean=Sum of terms / Number of terms

= 3910 / 36

= 108.6111

• Mean = 108.6111

Median :

Ordering the data from least to greatest, we get:

56   56   68   77   78   87   89   91   94   95   97   99   100   100   102   105   107   109   112   112   113   114   115   117   119   121   123   127   128   134   135   137   146   147   150   150   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median = ( 109+112) / 2 = 110.5

• The median of the data set is 110.5.

Standard deviation :

Create the following table.

data	data-mean	(data - mean)2
97	-11.6111	134.81764321
147	38.3889	1473.70764321
115	6.3889	40.81804321
135	26.3889	696.37404321
105	-3.6111	13.04004321
78	-30.6111	937.03944321
68	-40.6111	1649.26144321
100	-8.6111	74.15104321
102	-6.6111	43.70664321
117	8.3889	70.37364321
146	37.3889	1397.92984321
77	-31.6111	999.26164321
127	18.3889	338.15164321
87	-21.6111	467.03964321
137	28.3889	805.92964321
109	0.38890000000001	0.15124321000001
56	-52.6111	2767.92784321
91	-17.6111	310.15084321
95	-13.6111	185.26204321
150	41.3889	1713.04104321
94	-14.6111	213.48424321
100	-8.6111	74.15104321
112	3.3889	11.48464321
114	5.3889	29.04024321
128	19.3889	375.92944321
112	3.3889	11.48464321
99	-9.6111	92.37324321
107	-1.6111	2.59564321
56	-52.6111	2767.92784321
121	12.3889	153.48484321
113	4.3889	19.26244321
150	41.3889	1713.04104321
89	-19.6111	384.59524321
119	10.3889	107.92924321
123	14.3889	207.04044321
134	25.3889	644.59624321

Find the sum of numbers in the last column to get.

∑(xi−X bar)² = 20926.5556

• Standard deviation = 24.452