PROBLEM 4: Grades received in BUS310-01 course: 66 53 89 92 98 72 71 80 example of how problem should be answered: PROBLEM: #colds / year n=18 subjects 10; 8; 5; 2; 3; 3; 3; 6; 4; 2; 4; 5; 4; 4; 1; 0; 3; 5; For computations done by hand, data must be ordered. 0 1 2 2 3 3 3 3 4 Median 4 4 4 5 5 5 6 8 10 Q1 Q3 Sample mean = =72/18 = 4 colds Sample median = 4 colds (midpoint of data) Sample mode = 3 and 4 (the value that occurs most often; this is a bimodal data set) First Quartile (approximation) = 0.25 * N = 0.25 * 18 = 4.5; If the value is not an integer, we can round it up to the nearest integer 5 à value is 3 Third Quartile (approximation) = 0.75 * N = 0.75 * 18 = 13.5 à 14 à value is 5 Range = max – min =10 – 0 = 10 IQR = Q3 – Q1 = 5 – 3 = 2 Variance = s2 = = 96/17 = 5.64 colds squared Standard deviation = s = = 2.37 colds Coefficient of variation = CV = x 100% =2.37/4 *100% = 59.25% To convert to a Z-score: Zi = (Xi - mean)/s The 0 becomes (0 – 4) / 2.37 = -1.68; the 1 becomes (1 – 4) /2.37 = -1.27; the 2 becomes (2- 4)/2.37 = .84; and the 4 becomes a 0; etc. All values below the mean have negative Z scores and all values above the mean have positive Z scores. This is the output from MS Excel using the descriptive tool. For your homework solution an output as the one below (for the excel solution) will be sufficient. Column1 Mean 4 Standard Error 0.560112 Median 4 Mode 3 Standard Deviation 2.376354 Sample Variance 5.647059 Kurtosis 1.497461 Skewness 0.887652 Range 10 Minimum 0 Maximum 10 Sum 72 Count 18 Quartiles are approximations. A more refined definition is the following: For Lower Quartile (25%): Sort all observations in ascending order Compute the position L1 = 0.25 * N, where N is the total number of observations. If L1 is already a whole number (integer), the lower quartile is midway (average) between the L1-th value and the next one. If L1 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the lower quartile. For Upper Quartile(75%): Sort all observations in ascending order Compute the position L3 = 0.75 * N, where N is the total number of observations. If L3 is a whole number, the upper quartile is midway (average) between the L3-th value and the next one. If L3 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the upper quartile. Please follow the above definitions in calculating the quartiles. There are two distinct situations: set size equal to a power of 4 (L is already a whole number – integer), set size not a power of 4.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
PROBLEM 4: Grades received in BUS310-01 course:
66 53 89 92 98 72 71 80
example of how problem should be answered:
PROBLEM: #colds / year n=18 subjects
10; 8; 5; 2; 3; 3; 3; 6; 4; 2; 4; 5; 4; 4; 1; 0; 3; 5;
For computations done by hand, data must be ordered.
0 1 2 2 3 3 3 3 4
Q1 Q3
Sample
Sample median = 4 colds (midpoint of data)
Sample
First Quartile (approximation) =
0.25 * N = 0.25 * 18 = 4.5; If the value is not an integer, we can round it up to the nearest integer 5 à value is 3
Third Quartile (approximation) = 0.75 * N = 0.75 * 18 = 13.5 à 14 à value is 5
IQR = Q3 – Q1 = 5 – 3 = 2
Variance = s2 = = 96/17 = 5.64 colds squared
Standard deviation = s = = 2.37 colds
Coefficient of variation = CV = x 100% =2.37/4 *100% = 59.25%
To convert to a Z-score: Zi = (Xi - mean)/s
The 0 becomes (0 – 4) / 2.37 = -1.68;
the 1 becomes (1 – 4) /2.37 = -1.27;
the 2 becomes (2- 4)/2.37 = .84;
and the 4 becomes a 0; etc.
All values below the mean have negative Z scores and all values above the mean have positive Z scores.
This is the output from MS Excel using the descriptive tool. For your homework solution an output as the one below (for the excel solution) will be sufficient.
|
Column1 |
|
|
|
|
|
Mean |
4 |
|
Standard Error |
0.560112 |
|
Median |
4 |
|
Mode |
3 |
|
Standard Deviation |
2.376354 |
|
Sample Variance |
5.647059 |
|
Kurtosis |
1.497461 |
|
Skewness |
0.887652 |
|
Range |
10 |
|
Minimum |
0 |
|
Maximum |
10 |
|
Sum |
72 |
|
Count |
18 |
|
For Lower Quartile (25%):
|
For Upper Quartile(75%):
|
Please follow the above definitions in calculating the quartiles. There are two distinct situations: set size equal to a power of 4 (L is already a whole number – integer), set size not a power of 4.
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