1. Data Analysis and Statistical Software I (323-21-403) Quarter: Autumn 02/03
Daniela Stan, PhD
Course homepage:
Office hours: (No appointment needed)
M, 3:00pm - 3:45pm at LOOP, CST 471
W, 3:00pm - 3:45pm at LOOP, CST 471
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Daniela Stan - CSC323

2. Outline Chapter 4: Probability – The Study of Randomness
Means and Variances of Random Variables
Chapter 5: Sampling
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3. Outline Chapter 4: Probability – The Study of Randomness
The Standard Error
Chapter 5: Sampling Distributions
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Daniela Stan - CSC323

4. A measure of the chance error: the standard error
Ex: Assume that a coin is tossed for a repeated number of times. The actual number of heads will be off the expected value for some amount. How big is that amount on average?
The number of heads in 4 tosses of a coin will be:
number of heads X= expected value +chance error
So if we observe 3 heads, the chance error is +1; if we observe 1 head, the chance error is –1.
1/15/2019
Daniela Stan - CSC323

5. A measure of the chance error: the standard error
The chance error is measured by the standard error.
The standard error is calculated by the following 3 steps:
Calculate the deviations of each value of X from the expected value X
x1 – X, x2 – X ,…, xn – X.
Square the deviations and multiply each square deviation by its probability.
Add all the products.
Take the square root.
1/15/2019
Daniela Stan - CSC323

6. The expected value of the number of heads in 4 tosses
of a coin will be:
The expected value is X =2 X 1 2 3 4
Probability
0.0625
0.25
0.375
Deviations
X– X
0–2= –2
1–2 = –1
2–2=0
3–2=1
4–2=2
Square Deviations
(-2)2=4
(-1)2=1
(0)2=0
(1)2=1
(2)2=4
Products=
Dev2* Probability
4*0.0625= 0.25
1*0.25
=0.25
0*0.375=0
1*0.25=0.25
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7. Step 3: sum the products: (0.25+0.25+0+0.25+0.25)=1
Step 4: Take the square root
The standard error of X is =1 = 1.
The observed number of heads in 4 tosses of a coin is likely to be around 2, give or take 1.
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Daniela Stan - CSC323

8. Standard Error & Probability Histograms
The standard error measures the spread of the probability histogram.
Chance 40
(%) 30 20 10
X–1s.e.=2 – 1=1
X+1s.e.=2+1=3
1 s.e.
1 s.e.
X=2
Remark: Observed values are rarely more than 2 or 3 standard errors away!!
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9. Mathematical Expressions
Given a random variable X with probability table X x1 x2 x3 x4 … xk
Probability p1 p2 p3 p4 pk
The expected value is
The standard error is
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Daniela Stan - CSC323

10. Remarks on random processes
An observed value should be somewhere around the expected value; the difference is chance error.
The likely size of the chance error is the standard error.
Observed values are rarely two or three standard errors away from the expected value.
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11. Remarks on random processes (cont.)
The standard deviation is defined for a list of numbers.
The standard error is defined for random processes and measures the chance error. (Subtle difference)
The standard error “makes more sense” if the probability histogram of the random variable is bell-shaped, (similar to the normal distribution).
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Daniela Stan - CSC323

12. Recommended problems Problems 4.60, 4.61/page 334
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13. Chapter 5: Sampling Distributions
The probability distribution of a statistic from a random sample or randomized experiment is its sampling distribution.
Ex: A sample survey asks 2500 adults whether they agree that “I like buying clothes, but shopping is often frustrating and time consuming.”
The number of those who say “Yes” is a random variable X. The random variable X is a count of the occurrences of some outcome in a fixed number of observations.
If the number of distributions is n, than the sample proportion is P=X/n. For example, if 1650 of the 2500 shoppers say “yes”, the sample proportion is 1650/2500=.66
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Daniela Stan - CSC323

14. The binomial distributions for sample counts
The binomial setting:
There are a fixed number n of observations.
The n observations are all independent.
Each observation falls into one of just two categories: “success” or “failure”
The probability of success, call it p, is the same for each observation.
Example:
Toss a coin for n times
Each toss gives either head or tail.
The outcomes of successive tosses are independent
The probability p of getting heads is .5
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Daniela Stan - CSC323

15. The binomial distributions for sample counts
The distribution of the count X of successes in the binomial setting is called the binomial distribution with parameters n (number of observations) and p (probability of success on any one observation). The possible values of X are from 0 to n.
X ~ B (n, p)
When the population is much larger than the sample, the count X of successes in an SRS of size n has approximately the B (n ,p) distribution if the population proportion of successes is p.
How large is large?
- we will use the binomial sampling distribution for counts when the population is at least 10 times as large as the sample.
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16. Finding Binomial Probabilities: Tables
Table C: the entries in the table are the probabilities P(X=k) of individual outcomes for a binomial random variable X.
Example: A quality engineer selects an SRS of 10 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. What is the probability that no more than 1 of the 10 switches in the sample fails inspection?
X = count of bad switches ~ B(10,.1)
P(X<=1)=P(X=1)+P(X=0)= =.7361
Therefore, about 74% of all samples will contain no more than 1 bad switch.
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Daniela Stan - CSC323

17. Finding Binomial Probabilities
Recommended problems: 5.5, 5.6/page 385
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Daniela Stan - CSC323