1. Random Sampling and Data Description
Chapter 6
Random Sampling and
Data Description

2. Learning Objectives
Compute and interpret the sample mean, sample variance, sample standard deviation, sample median, and sample range
Explain the concepts of sample mean, sample variance, population mean, and population variance
Construct and interpret visual data displays
Explain the concept of random sampling
Construct and interpret normal probability plots

3. Data Summary and Display
Essential to good statistical thinking
Focus on important features of the data
Provide insight about the type of model that should be used
Computer has become an important tool in the presentation and analysis of data
User enters the data and then selects the types of analysis
Packages are available for both mainframe computers as well as personal computers

4. Sample Mean Useful to describe data features numerically
Can characterize the location or central tendency
Refer to this arithmetic mean as the sample mean
Where the n observations in the sample are denoted by x1, x2,…, xn
Sample mean as a reasonable estimate of the population mean, 

5. Sample Standard Deviation
Sample mean does not provide all of the information
Variability in the data may be described by the sample variance or the sample standard deviation
Sample standard deviation, s, is the positive square root of the sample variance

6. Sample Range
Difference between the largest and smallest observations, or the sample range, is a useful measure of variability
Sample range
r=max(xi)-min (xi)
As the variability in sample data increases, the sample range increases

7. Sample Median and Sample Mode
Two more measure of central tendency
Median divides the data into two equal parts, half below the median and half above
If the number of data points is even, the median is halfway between the two central values
If the number data points is odd, the median is the central value
Mode is the most frequently occurring data point (s)

8. Example
The data below are the joint temperatures of the O-rings (degrees F) for each test firing or actual lunch of the space shuttle rocket motor (from Presidential Commission on the Space Shuttle Challenger Accident):
Compute the sample mean, sample median, sample range, and sample standard deviation

9. Solution Sample mean is 65.85,or Sample median is 67
……[67.5]……
Sample range is 53, or
r = 84-31
Sample standard deviation is 12.16

10. Random Sampling
Interested to work with a sample of observations selected from a population
Relationship between the population and the sample
Impossible or impractical to observe the entire population
Use a probability distribution as a model for a population
Sample from the population to make decisions about the population

11. Understand Random Sampling
Wish to reach a conclusion about the proportion of people who earn at least $35,000 in a specific year
Let p represent the unknown value of this proportion
Impractical to question every individual
Make inference regarding the true proportion p
Select a random sample
Use the observed proportion of people
is computed by dividing the number of individuals in the sample by the total n
Many random samples are possible
Value of will vary. That is, is a random variable

12. Statistic Random sample is called a statistic
Statistic is any function of the observations in a random sample
Sample mean , the sample variance S2, and the sample standard deviation s are statistics

13. Data Display Graphical displays of sample data are very powerful
Many techniques

14. Stem-and-Leaf Diagrams
Stem-and-leaf diagram is a good way to represent the data
Steps
Divide each data point into two parts: a stem and a leaf
List the stem values in a vertical column
Record the leaf for each observation beside its stem
Write the units for stems and leaves on the display

15. Example
Consider 21, 24, 24, 26, 27, 27, 30, 32, 38, 41
Select as stem values the numbers 2,3, and 4
Record the leaf for each observation beside its stem
Last column in the diagram is a frequency count of the no. of leaves associated with each stem
Frequency 6 3 1 24

16. Frequency Distributions
More compact summary of data than a stem-and-leaf
Must divide the range of the data into intervals
Called class intervals, cells, or bins
Number of bins depends on the number of observations
Equal to the square root of the number of observations

17. Histograms Visual display of the frequency distribution
Gives insight about possible choices of probability distribution
Stages for constructing
1) Label the bin boundaries on a horizontal scale
2) Mark and label the vertical scale with the frequencies
3) Draw a rectangle where height is equal to the frequency corresponding to that class

18. Cumulative Frequency Plot
Variation of the histogram
Useful in data interpretation
Height of each bar is the total number of observations that are less or equal to the upper limit of the class
Illustrated in the right graph

19. Example
Consider the following data on the motor fuel octane ratings of several blends of gasoline.
Construct a frequency distribution and histogram
Use 8 classes

20. Solution
Illustrated in the right

21. Probability Plots
Graphical method for determining whether sample data points conform to a hypothesized distribution
Very simple and can be constructed quickly
Uses special graph paper, known as probability paper
Focus primarily on normal probability plot

22. Constructing a Probability Plot
Sample data points are first ranked from smallest to largest
x1, x2,..., xn is arranged
x(1),x(2),…, x(n)
Plotted against their observed cumulative frequency [(j -0.5)/n] on the probability paper
Plotted points fall approximately along a line
Constructed on ordinary graph paper by plotting the standardized normal scores zj against x(j)
Standardized normal scores satisfy
[(j-0.5)/n]= P(Zzj)=(zj)

23. Example
A soft-drink bottler is studying the internal pressure of 1-liter glass bottles. A random sample of 16 bottles is tested, and the pressure strength (psi) are obtained. The data are shown below.
Does it seem reasonable to conclude that pressure strength is normally distributed?

24. Solution Use the steps to construct a probability plot
Assumption of normality appears reasonable
Data falls along a straight line

25. Next Agenda Discusses point estimation of parameters
Introduces some of the important properties of estimators, the method of maximum likelihood, sampling distributions, and the central limit theorem