2 of 12
1.1 Cautions
Question 1. The paper only gives the above data and states that the new chemotherapy agent is more effective in decreasing tumor mass in comparison to the standard agent. You are reviewing the paper for the journal. What do you think
2 of 13
1.1 Introducing R
Figure 1: The Univac II, circa 1958.
Figure 2: The program Z(1) = Y + W (1) on a punch card.
Statistics and Computers The U.S. Bureau of the Census used the UNIVAC computer to process the 1950 Census data. In 1951, a report state
2 of 10 DEFINITIONS
1.1 Fundamentals
Rare event rule. Definition 1.1 If we observe an event that has an small probability of being observed, we conclude that our assumptions about its probability are wrong. It wasn't random chance. (Very Important!) Examp
6 of 10
1.2 Probabilities of a single trial
Total Area = 1
Total Area = 1
P (A)
P (B)
P (A)
P (B)
P (A or B)
P (A or B) = P (A) + P (B) - P (A and B)
When events are not disjoint, we must subtract P (A and B) to only count the overlapped area once. Determ
Introduction to Hypothesis Testing
9 of 14
If 1. 2. 3.
Tells us the likelihood of supporting a true alternative hypothesis (making the correct decision). Good tests have powers of at least 0.8-0.9. Generally not simple to calculate, depends on (1) level,
2 of 14 Question 1. How could you support your claim?
1.1 Introduction
Question 2. You conduct a study of our class and find the proportion of students who wear corrective lenses is 55.6%. Does this support our hypothesis that the proportion of people in
8 of 13
1.3 Testing a claim when is unknown
One Sample t-Test, alpha=0.05, Two Tailed
1.0 0.9 0.8 0.7 0.6 Power 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
n = 1000
n = 250
n = 100
n = 50
n = 30
n = 20
n = 10
n= 5
h (NOTE: CLT must
2 of 13
1.2 Testing a claims when is known REVIEW
What is a hypothesis test? A hypothesis test calculates the probability of observing our sample data assuming the null hypotheses H0 is true. If the probability is low enough (p-value ), we reject H0 and h
SOLUTIONS MAT 167: Statistics Final Exam Instructor: Anthony Tanbakuchi Spring 2009
Name: Computer / Seat Number:
No books, notes, or friends. Show your work. You may use the attached equation sheet, R, and a calculator. No other materials. If you choose
SOLUTIONS MAT 167: Statistics Final Exam Instructor: Anthony Tanbakuchi Spring 2008
Name: Computer / Seat Number:
No books, notes, or friends. Show your work. You may use the attached equation sheet, R, and a calculator. No other materials. If you choose
SOLUTIONS MAT 167: Statistics Final Exam Instructor: Anthony Tanbakuchi Fall 2008
Name: Computer / Seat Number:
No books, notes, or friends. Show your work. You may use the attached equation sheet, R, and a calculator. No other materials. If you choose to
2 of 11 Galaxy SMC LMC NGC 6822 NGC 598 NGC 221 NGC 224 NGC 5357 NGC 4736 NGC 5194 NGC 4449 NGC 4214 NGC 3031 NGC 3627 NGC 4826 NGC 5236 NGC 1068 NGC 1055 NGC 7331 NGC 4258 NGC 4151 NGC 4382 NGC 4472 NGC 4486 NGC 4649 Vel.m.year 5.36E+12 9.15E+12 -4.10E+1
Relative Standing
7 of 12
Student heights (with Mini-Me)
Box plots and distributions
Modified box plot. Definition 1.8 Boxplot where whiskers have a maximum length of 1.5 IQR. Useful for identifying outliers. Box plot: boxplot(x) Where x is a vector of da
2 of 12
1.1 Relative Standing
Histogram of height
15 Frequency 65 70 height 75 0 5 10 5
Histogram of height.skewed
Frequency
0
1
2
3
4
30
40
50
60
70
80
height.skewed
Now we will look at methods for measuring the relationship of individual data points to
2 of 7
1.2 Probability distributions
Types of questions we want to answer "Find the probability that the mean is . . . " english statement mathematical notation "equal to fifty" P ( = 50) "not equal to fifty" P ( = 50) "not fifty" P ( = 50) "greater than
2 of 15
R: R: R: R: R: par ( mfrow = c ( 2 , 2 ) ) c u r v e ( x ^ 2 , -10 , 1 0 ) c u r v e ( x ^ 3 , -10 , 1 0 ) c u r v e ( s i n ( x ) , -2 pi , 2 p i ) c u r v e ( s i n ( p i x ) / ( p i x ) , -5, 5 )
1.1 Introduction
20 40 60 80
x^2
x^3 -10 -5 0 x
2 of 6
1.2 Permutations and Combinations n items can be arranged in n! ways.
R Command
Factorial: factorial(x) Finds x! (There is a limitation on how large x can be.a )
a The factorial function cannot compute values beyond x 170 due to how it's implemente
6 of 8
1.3 Summary
Question 16. At a large university, there are 32 males and 660 females in a statistics class, find the probability of selecting 8 males without replacement.
PROBABILITY OF "AT LEAST ONE" Definition 1.6 Probability of "At least one". To
2 of 8
1.2 Probabilities of multiple trials
1.2
Probabilities of multiple trials
Single trial We previously looked at single trials and learned how to use the addition rule (OR) to calculate probabilities. Multiple trials We will now learn how to calculat
2 of 8
1.2 Measures of variation
Figure 1: Image and its histogram.
Please welcome our special visitor to the class today: Add Mini-Me (Verne Troyer) 2' 8" to the class:
R: R: R: R: R: R: l o a d ( "C l a s s D a t a . RData " ) h e i g h t = c l a s s .
2 of 11
1.1 Introduction
1.1
Introduction
Measures of center Robert Pershing Wadlow (February 22, 1918 - July 15, 1940) is the tallest person in medical history for whom there is irrefutable evidence. He is often known as the "Alton Giant" because of his
Linear regression
[ 1 ] 1 . 1 7 6 7 e+13 R: y . bar = mean ( y ) R: y . bar [ 1 ] 2 . 8 1 6 e+22
7 of 11
(b) Find the slope using equation 7:
R: b1 = sum ( ( x - x . bar ) ( y - y . bar ) ) /sum ( ( x - x . bar ) ^ 2 ) R: b1 [ 1 ] 1339313079
(c) Find the
SOLUTIONS MAT 167: Statistics Final Exam Instructor: Anthony Tanbakuchi Fall 2007
Name:
Computer / Seat Number:
Multiple choice part: ll in answer on the scan form. Do not attach work for this section (no partial credit is awarded). Written part: Write al
SOLUTIONS MAT 167: Statistics Test II Instructor: Anthony Tanbakuchi Spring 2009
Name: Computer / Seat Number:
No books, notes, or friends. Show your work. You may use the attached equation sheet, R, and a calculator. No other materials. If you choose to
8 of 14
1.3 One-Way Analysis of Variance
Test statistic To test the hypothesis H0 : 1 = 2 = = k , the test statistic is: F = M S(treatment) M S(error) (9)
having an F distribution with two degrees of freedom: (numerator) (denominator)
2
df1 = k - 1 df2 =