Final Review - Department of Statistical Sciencesolgac/stab22_2013/notes/FinalReview_new… ·...
Transcript of Final Review - Department of Statistical Sciencesolgac/stab22_2013/notes/FinalReview_new… ·...
Final Review
What to do:
Read lectures 1-12, chapters 1-9 and 12-25
from the textbook
Go over the midterm review
Do the assigned exercises from the textbook
Go over the quiz questions
Use sample exams to practice
Use extra TAs' office hours
Topics to review:
Random variables:
Discrete
o Probability distribution
o Mean and variance of a discrete random variable
Example: Let and be independent and have the same distribution
given below:
-1 0
2/3 1/3
Let . Find the probability distribution of X, E(X) and Var(X).
o Binomial distribution
X ~ Bin (n, p), X = 0, 1, …, n
√
( ) , where (
)
o Geometric distribution
X ~ Geom (p), X = 1, 2, 3, …
√
, where q = 1 – p
Continuous
o Density curve
o Normal distribution
o Normal approximation (sampling distribution for counts and sample
proportions)
Let X be the count of successes in the sample and be the sample
proportion of successes.
When n is large, the sampling distributions of these statistics are
approximately Normal:
X is approximately √
is approximately √
)
As a rule of thumb, we will use this approximation for values of n and p that
satisfy and .
Sampling distribution of sample mean (CLT):
Draw an SRS of size n from any population with mean and finite standard
deviation . When n is large enough,
√
Example: 20% of customers at a bakery will buy a brownie.
(a) What is the probability that more than 110 customers buy a brownie?
(b) How many brownies does the bakery need to have in stock so that the
probability of selling out in a day is 1%?
500 customers arrive at the bakery in a
day. Assume that individual customers
make their purchases independently.
(a) What is the probability that more than 110 customers buy a brownie?
(b) How many brownies does the bakery need to have in stock so that the
probability of selling out in a day is 1%?
Statistical inference
o Confidence intervals
For population mean :
√ if is known
√ if is unknown
For difference of means : √
Pooled: √
where
For single proportion p: √
For difference of proportions :
√
o Hypothesis testing
Test statistic
To test we find statistic
√ ( is known)
To test we find statistic
√ ( is unknown)
To test we find statistic
√
or
√
if
where
To test we find statistic
√
To test we find statistic
where
√ (
) and
P-value
Significance level
Power
Type I and Type II errors
Matched pairs t-procedures
Non-normal populations: sign test
Example: 50 smokers were questioned about the number of hours they sleep
each day. We want to test the hypothesis that the smokers need less sleep than
the general public which needs an average of 7.7 hours of sleep. The population
standard deviation is 0.5
(a) For what values of we would reject the null hypothesis at significance
level of .05.
(b) If the sample mean is 7.5, what can you conclude?
Example: 1500 randomly selected pine trees were tested for traces of the Bark
Beetle infestation. It was found that 153 of the trees showed such traces. Test
the hypothesis that more than 10% of the trees have been infested. (Use a 5%
level of significance)
Example: Suppose that ten identical twins were reared apart.
The mean difference between the high school GPA of the twin brought up in
wealth and the twin brought up in poverty was 0.07. If the standard deviation of
the differences was 0.5, find a 95% confidence interval for the
difference. Assume the distribution of GPA's is approximately normal.
Example: Do employees perform better at work with music playing. The music
was turned on during the working hours of a business with 45 employees. Their
productivity level averaged 5.2 with a standard deviation of 2.4. On a different
day the music was turned off and there were 40 workers. The workers'
productivity level averaged 4.8 with a standard deviation of 1.2. What can we
conclude at the .05 level?
Example: Is the severity of the drug problem in high school the same for boys
and girls? 85 boys and 70 girls were questioned and 34 of the boys and 14 of
the girls admitted to having tried some sort of drug. What can be concluded at
the .05 level?
Example: Consider the following hypothesis test
vs
Assume , , and . Find the probability of a Type II
error for a particular value .