1- A set of numbers is transformed by taking the log base 10 of each number. The
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Question
1- A set of numbers is transformed by taking the log base 10 of each number. The mean of the transformed data is 1.65. What is the geometric mean of the untransformed data?
2-Which measure of central tendency is most often used for returns on investment?
3- Make up three data sets with 5 numbers each that have:
(a) the same mean but different standard deviations.
(b) the same mean but different medians.
(c) the same median but different means.
4- A sample of 30 distance scores measured in yards has a mean of 10, a variance of 9, and a standard deviation of 3. (a) You want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. What are the new mean, variance, and standard deviation? (b) You then decide that you only want to look at the distance past a certain point. Thus, after multiplying the original scores by 3, you decide to subtract 4 feet from each of the scores. Now what are the new mean, variance, and standard deviation?
5-- You recorded the time in seconds it took for 8 participants to solve a puzzle. These times appear below. However, when the data was entered into the statistical program, the score that was supposed to be 22.1 was entered as 21.2. You had calculated the following measures of central tendency: the mean, the median, and the mean trimmed 25%. Which of these measures of central tendency will change when you correct the recording error?
Time (second)
15.2
18.8
19.3
19.7
20.2
21.8
22.1
29.4
6- For the test scores in question #6, which measures of variability (range, standard deviation, variance) would be changed if the 22.1 data point had been erroneously recorded as 21.2?
7- An experiment compared the ability of three groups of participants to remember briefly-presented chess positions. The data are shown below. The numbers represent the number of pieces correctly remembered from three chess positions. Compare the performance of each group. Consider spread as well as central tendency.
Non-players
Beginners
Tournament players
22.1
32.5
40.1
22.3
37.1
45.6
26.2
39.1
51.2
29.6
40.5
56.4
31.7
45.5
58.1
33.5
51.3
71.1
38.9
52.6
74.9
39.7
55.7
75.9
43.2
55.9
80.3
43.2
57.7
85.3
Non-players
Beginners
Tournament players
22.1
32.5
40.1
22.3
37.1
45.6
26.2
39.1
51.2
29.6
40.5
56.4
31.7
45.5
58.1
33.5
51.3
71.1
38.9
52.6
74.9
39.7
55.7
75.9
43.2
55.9
80.3
43.2
57.7
85.3
Explanation / Answer
1. We have transformered each observation to log base 10 and mean of data as 1.65. The geometric from this mean can be obtained by taking antilog of arithmetic mean.
so geometric mean untransformered data = antilog(1.65) = 101.65 = 44.6683
2. Arithmetic mean can be used for return on investemnt. It can be used for all interval and ratio data sets like income, age, rates of returns. All data values are includede in the computation of arithmetic mean. A data set has only one arithmetic mean which means the mean is unique.
4. A sample of 30 distance scores has mean 10 yards, variance 9 yards and standard deviation 3 yards.
a) Inorder to convert yards to feet multiply each score by 3 so we know new mean will be
New mean = old mean*3 = 30
New varaince = old variance*32 = 9*9 = 81
New standard deviation = old standard deviatio * 3 = 3*3 =9
b) Now subtract 4 feet from each after multiplying by 4 to each score
We know mean is affecetd by change in scale but not variance so New mean will be
New mean = 30-4 = 26
How ever variance and standard devation are not affected by change in scale so they remain same as 81 and 9 respectively.
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