The Organization for Economic Cooperation and Development provided the following

Question

The Organization for Economic Cooperation and Development provided the following mean weekly instruction times (hours) for elementary and high school student in various countries: 22.2 (United Sum); 24.8 (France); 24.2 (Mexico); 26.9 (China): 23.8 (Japan). Use the five given limes for the following. a. Find the mean. b. Find the median c. Find the range. d. Find the standard deviation. e. Find the variance. f. Use the range rule of thumb to identity the range of usual values. g. Baked on the result from pan (f), are any of the times unusual? Why or why not? h. What is the level of measurement of the data: nominal, ordinal, interval, or ratio? i. Are the data discrete or continuous? j. There is something fundamentally wrong with using the given times to find statistics such as the mean. What is wrong?

Explanation / Answer

The five observations are:

1. 22.2 (United States)

2. 24.8 (France)

3. 24.2 (Mexico)

4. 26.9(China)

5. 23.8(Japan)

So, the number of given times : 5.

a) Mean is calculated as : (Sum of the given observations) / (Total number of observations)

So, here mean= ( 22.2 + 24.8 + 24.2 + 26.9 + 23.8 ) / 5 = 121.9/5 = 24.38 (approximately)

b) To find the median, there are several steps that we follow.

First we arrange the observations in ascending order.

So , our arranged observations are: 22.2 , 23.8 , 24.2 , 24.8 , 26.9

Now, if 'n' be the number of observations and n is odd ,then median = (n+1)/2 th observation,among the arranged ones.

Here, n= 5 (odd)

So, median = (5+1)/2th ie the 3rd observation..

So, median = 24.2.

c) range = maximum value - minimum value

So,range= 26.9 - 22.2 = 4.7

d) and e) are basically connected. If we can find the variance,then positive square root of the variance will be the standard deviation,so let us solve e) first.

If n is the total number of observations, then variance = 1/n * Summation ( Valuei - mean)2

Valuei denotes the ith value among the 5 values, so, i = 1,2,3,4,5

So, mean = 24.38 ( from a)

So, variance = 1/5* [(22.2-24.38)2 + (24.8 - 24.38)2 + (24.2 - 24.38)2 + (26.9 - 24.38)2 + (23.8- 24.38)2]

=2.3296

So, standard deviation = squareroot( 2.3296) = 1.5263 (approx)

So, d and e are solved.

f) Rule of thumb: To identify the range of usual values.

According to the rule of thumb , a usual range of values would be between ( Mean - 2*standard deviation) and (Mean + 2*standard deviation)

Mean= 24.38

Standard deviation= 1.5263

So, the usual range is : (24.38-2*1.53) to (24.38+2*1.53) i.e from 21.33 to 27.43.

g. All the values lie in the usual range,so none are unusual.

h. This is nominal data.

i. Time is a continuous variable in statistics, so the data is continuous.

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