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Quantitative Analysis of Data - Bio-statistics

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2.A group of 25 subjects have their diastolic blood pressures measured. The results, in SPSS are:

|-----------|-------|--------|
|N |Valid |25 |
|-------|--------|
| |Missing|0 |
|-----------|-------|--------|
|Median |85.00 |
|-------------------|--------|
|Mode |82.00 |
|-------------------|--------|
|Minimum |55.00 |
|-------------------|--------|
|Maximum |110.00 |
|-----------|-------|--------|
|Percentiles|25 |71.00 |
| |-------|--------|
| |50 |85.00 |
| |-------|--------|
| |75 |98.00 |
|--------|-------|--------|

Don't worry about values exactly at the endpoints of these intervals. Do the calculations roughly.
(1 point each)

a. What percentage of subjects were from 55 to 85?
b. What percentage of subjects were < 85?
c. What percentage of subjects were from 71 to 85?
d. What percentage of subjects were > 71?
e. What percentage of subjects were > 98?
f. Is there one value more common than the rest, and if so, what is it?

3. Assume you have already been give a Z value. This saves you a step. Consider and determine the following probabilities (1 point each).

A. Pr (-1 < Z < 1)
B. Pr (0 < Z < 1)
C. Pr (Z > 1)
D. Pr (-1 < Z < 0)
E. Pr (Z < -1)
F. Pr (Z > -2)
G. Pr (-1 < Z < 2)

4. Suppose the mean systolic blood pressure in a group of individuals is 150 mmHg, with a standard deviation of 15. Assuming SBP follows a normal distribution in this population, compute (1 point each):

A. Pr (135 < value < 165)
B. Pr (value > 165)
C. Pr (value < 135)
D. Pr (138.75 < value < 161.25)

5 Compute the 5th, 50th, and 95th percentiles of SBP from the previous question. (3 points: 1 each).

In questions 6 - 8, use the 1 and 2 SD rules, without the table.

6.In general, what percentage of a Gaussian data set is within 1 SD of the mean? What percentage is within 2 SD's of the mean? (2 points: 1 each)

7.If the mean grade on an exam was 80, SD = 6, where did about 68% of the grades fall? How about 95%? Assume the grades are Gaussian. (2 points).

8.Consider the following data: 1, 1, 2, 2, 4, 5, 6, 9, 40, 200

Use the 68% and 95% rules to test the normality of these data. (2 points).

9.A researcher studying a subtype of lymphocytes obtains a sample mean of 100 per mL, and a standard deviation of 20, with 25 subjects. Within what interval can you be roughly 68% sure the population mean number of these cells per mL lies? How about 95% sure? (2 points)

10.A researcher has a sample of 500 subjects. The mean is 40, median is 20, range 10-100. (2 points each)

a.Could this researcher calculate a useful interval with 95% probability of containing the population mean (using the mean and SEM)? Explain

b.Could the researcher use the mean and SD to usefully estimate where 95% of the individual subject values were? Explain

c)If there were 10 subjects, would your answers to a and b change?

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Solution Preview

a. What percentage of subjects were from 55 to 85?
55 is minimum and 85 is the median. The median divides the data into two parts with each part containing 50% of the data points. Therefore we have 50% of subjects from 55 to 85.
b. What percentage of subjects were < 85?
The median divides the data into two parts with each part containing 50% of the data points. Therefore we have 50% of subjects < 85.

c. What percentage of subjects were from 71 to 85?
71 is 25 percentile and 85 is 50 percentile, so we have 50-25=25% data points between 71 to 85.
d. What percentage of subjects were > 71?
71 is 25 percentile, so 25% data lies below 71 and 75% above 71. Hence there are 75% subjects with >71.

e. What percentage of subjects were > 98?
98 is 75 percentile, so 75% data lies below 98 and 25% above 98. Hence there are 25% subjects with >98.

f. Is there one value more common than the rest, and if so, what is it?
Yes. This value is mode for the data. This value is 82.

3. Assume you have already been give a Z value. This saves you a step. Consider and determine the following probabilities (1 point each).

A. Pr (-1 < Z < 1)
Look at the z table look for the value z=1. As the distribution is symmetric, P(-1<z<+1) is twice the table value. Else you can use Excel to find the value.
Without table also we can work the approximate probabilities. We know that 68.3% data points lies with +/- 1 standard deviation, 95.4% within +/-2 standard deviation and 99.7% within +/-3 standard deviation.

B. Pr (0 < Z < 1)
Answer=0.683/2 =0.3415 (only one side we are looking at)

C. Pr (Z > 1)