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A sample of size 40 yields the following sorted data. Note that I have x-ed out x(39) (the second largest number). This fact will NOT prevent you from answering the questions below.

14.1 46.0 49.3 53.0 54.2 54.7 54.7 54.7 54.8 55.4 57.6 58.2 58.3 58.7 58.9 60.8 60.9 61.0 61.1 63.0 64.3 65.6 66.3 66.6 67.0 67.9 70.1 70.3 72.1 72.4 72.9 73.5 74.2 75.3 75.4 75.9 76.5 77.0 x 88.9

(a) Calculate range, IQR, and median of these data.

(b) Given that the mean of these data is 63.50 (exactly) and the standard deviation is 12.33, what proportion of the data lie within one standard deviation of the mean?

(c) How does your answer to (b) compare to the empirical rule approximation?

(d) Ralph decides to delete the smallest observation, 14.1, from these data. Thus, Ralph has a data set with n = 39. Calculate the range, IQR, and median of Ralph’s new data set.

(e) Refer to (d). Calculate the mean of Ralph’s new data set.
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a. Range= Largest value - Smallest value

=88.9 - 14.1

= 74.8

Q1= (55.4+57.6)/2

=56.5

Q3= (72.4+72.9)/2

=72.65

IQR= Q3 - Q1

=72.65 - 56.5

= 16.15

Median= (63.0+64.3)/2

=63.65

b. Mean= 63.50 and Standard deviation= 12.33

Mean + or - Standard deviation= 68%

63.50+ 12.33= 75.83 and 63.50 - 12.33= 51.17

Which meas that the values that lie between 51.17 and 75.83 in our dataset are 32( from 53,0 to 75,4)

Therefore, (32/40) x 100= 80%

c. The empirical rule of approximation states that Mean + or - Standard deviation= 68% and in our case, it is 80%

d.

Range= Largest value - Smallest value

=88.9 - 46.0

= 42.9

Q1= 57.6

Q3=72.9

IQR= Q3 - Q1

=72.9 - 57.6

= 15.3

Median=64.3

e. Mean= Sum of observation/ Total number of obsevations

according to (b) Mean=63.50

Therefore; Sum of observation= Mean x Total number of observations

=63.50 x 40

=2540

If we remove 14.1, we will be left with 39 observations

Therefore; 2540-14.1= 2525.9

The new mean = Sum of observation/ Total number of obsevations

=2525.9/39

= 64.77
by Wooden (182 points)

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