Investigate Distributions with Numbers

Part 1 (10 points)

1. Describe three different ways to measure the center of a data set. Give an example where one measure of the center is preferred over another.

The three different ways to measure the center of a data set include the mean, mode, and median. First, the mean is equivalent to the summation of all the values in the data set divided by the number of values in the data set. Secondly, the median happens to be the mid-range data for a set of data that has been arranged in ascending order. Lastly, mode is the data that occurs most frequently in the data set.

Median is preferred over the others because it is less impacted by skewed data and outliers.

2. Explain the quartiles of a distribution in terms of percentiles

The quartiles of a distribution include the first percentile, second quartile which is the median and the third percentile. The first quartile is equivalent to the 25th percentile, second quartile equivalent to the 50th percentile and third quartile equivalent to the 75th percentile.

3. Describe the different components of a box plot. Use the items included in the five-number summary.

The components of a box plot include the following:

1. Minimum This is the smallest number in the data set

2. First Quartile When the data set is arranged in ascending order from the least to the highest, and the data is split into four groups, the first quartile is the data at the lower fourth mark of the data

3. Median When the data set is arranged in ascending order from lowest to highest, the median happens to be the data in the middle of the data set

4. Third Quartile - When the data set is arranged in ascending order from least to the highest, and the data is split into four groups, the third quartile is the data at the upper fourth mark of the data

5. Maximum This is largest number in the data set

4. Describe the IQR rule for identifying outliers. Then, create a mock data set with at least 12 data points and with at least two outliers. Justify the outliers by applying the IQR rule.

IQR is calculated by subtracting the 1st Quartile from the 3rd Quartile

The rule for identifying outliers is as follows:

Multiply IQR by 1.5 and subtract the 1st Quartile

Multiply IQR by 1.5 and add the 3rd Quartile

Any numbers that lie outside these figures are outliers

Consider the following data set

1, 12, 18, 27, 29, 31, 33, 34, 36, 39, 55, 65

1

Minimum

24.75

1st Quartile

32

Median

36.75

3rd Quartile

65

Maximum

IQR = 36.75 24.75 = 12

12 1.5 24.5 = -6.5

12 1.5 + 36.75 = 54.75

The outliers are 55 and 65

5. Write a short paragraph that defines standard deviation explains its importance. Explain the difference between population standard deviation and sample standard deviation

Standard deviation is a metric that indicates the dispersion of a data set from its mean. This measure is computed as the square root of the variance by ascertaining the variation...…the decision of how much Gatorade to order for each game, the supply manager would like to know the following information.

The probability that the number of gallons consumed will be:

a) Greater than 18 gallons

= 20, ? = 3

For x ? 18 gallons:

z = (18 20) / 3 = -0.6667

The corresponding area = 0.2514

P (x ?- 0.67) = 1 0.2514

= 0.7486

b) Between 22 and 25 gallons

= 20, ? = 3

For 22 ? x ? 25 gallons:

z = (22 20) / 3 = 0.67

The corresponding area = 0.7486

z = (25 20) / 3 = 1.67

The corresponding area = 0.9525

P (0.67 ? x ? 1.67) = 0.9525 0.7486

= 0.2039

c) Less than 16 gallons

= 20, ? = 3

For x ? 16 gallons:

z = (16 20) / 3 = -1.33

P (x ? -1.33) =

= 0.9176

Obtain the probabilities by first finding the appropriate z-score, then use the Standard Normal Cumulative Proportion table in the textbook.

4. Body Mass Index may be determined by taking your weight in kilograms and then dividing by the square of your height in meters. The National Center for Health Statistics found that BMI of American men, ages 20 to 29 follows an approximate normal distribution with mean 26.8 and standard deviation 5.2 (Cheryl D. Fryar et al. , Anthropometric reference data for children and adults: United States, 2007-2010, Vital and Health Statistics, Series 11, Number 252 (October, 2012), at www.cdc.gov/nchs).

a) People with BMI less than 18.5 are often classified as underweight. What…