Download Midterm II Contents Z or Standard Score Finding Probabilities

Midterm II Contents
Thurs, Nov. 13th
My pain and confusion covary
At levels both looming and scary
To pass this exam
I'll be needing some scam Oh statistics! I should have been wary.
Z or Standard Score
The standard- or z-score, represents the number of standard
deviations a random variable x falls from the mean:
Normal Distributions
5.1-5.5
Confidence Intervals
6.1-6.4
Hypothesis Testing
7.1-7.5
Testing with Two
Samples + Variances
8.1-8.4
10.3
- approximately 1 hour long
- format short answer, similar to homework
- bring calculator, 1 sheet allowed
Finding Probabilities
To find the probability that z is less
than a given value, read the
corresponding cumulative area.
-3 -2 -1 0 1 2 3
Test scores for a civil service exam are normally
distributed with a mean of 152 and a standard deviation of
7. Find the standard z-score for a person with a score of:
(a) 161
(b) 148
(c) 152
z
To find the probability is greater
than a given value, subtract the
cumulative area in the table from 1.
-3 -2 -1 0 1 2 3
z
To find the probability z is between
two given values, find the cumulative
areas for each and subtract the
smaller area from the larger.
-3 -2 -1
0 1 2 3
z
1
Application Example
Monthly utility bills in a city are normally distributed with a mean of
$100 and a standard deviation of $12. A utility bill is randomly
selected. Find the probability it is between $80 and $115.
Normal Distribution:  = 100;  = 12
P(80 < x < 115)
P(–1.67 < z < 1.25)
Four Statistics of Interest
Mean large sample
Mean small sample
Proportions
Variance or Standard Deviation
Subtract areas under the curve:
0.8944 – 0.0475 = 0.8469
The probability that a utility bill is
between $80 and $115 is 0.8469.
Maximum Error of Estimate
The maximum or margin of error of estimate E is the greatest possible
distance between the point estimate and the value of the parameter it is
estimating for a given level of confidence, c.
When n  30, the sample standard deviation, s, can be used for .
Example: Find E, the maximum error of estimate based on 35
ticket prices for a one-way plane fare from Atlanta to Chicago at a
95% level of confidence given s = 6.69.
Using zc = 1.96, s = 6.69, and n = 35,
You are 95% confident that the maximum error of the estimate is $2.22.
Sample Size
Given a c-confidence level and a maximum error of estimate,
E, the minimum sample size n, needed to estimate , the
population mean is:
Example: estimate the mean one-way fare from Atlanta to
Chicago. How many fares must be included in your sample if
you want to be 95% confident that the sample mean is within
$2 of the population mean?
You should include at least 43 fares in your sample.
2
Statistical Hypotheses
Confidence Intervals: Normal or t?
A statistical hypothesis is a claim about a population.
Null hypothesis H0
contains a statement of
equality such as , = or .
Alternative hypothesis Ha
contains a statement of
inequality such as < ,  or >
Assume that the equality condition in the null
hypothesis is true, regardless of whether the claim is
represented by the null or alternative hypothesis
Accept or reject null, never prove null is true
Writing Hypotheses
Write the claim about the population. Write its complement.
Either hypothesis, can represent the claim.
Example: A hospital claims its ambulance response
time is less than 10 minutes.
claim
Example: A consumer magazine claims the
proportion of cell phone calls made during evenings
and weekends is at most 60%.
8 Steps in a Hypothesis Test
1.
2.
3.
4.
5.
6.
7.
8.
Write the null and alternative hypothesis
State the level of significance
Identify the sampling distribution
Find the critical value
Find the rejection region
Find the test statistic, P-value
Make your decision
Interpret your decision
claim
3
Hypothesis Testing Metrics
One Sample
Large (≥30)
Small (<30)
Population
Distribution
Metric

Standardized
Normal

t-distribution
Proportion
%
Variance or
Standard
Deviation
2 or 
Standardized
Normal
Chi-squared
Test
Independent vs Dependent Samples
z
Test metrics: mean difference between two samples
̅
̅ ,
t
z
χ2
Hypothesis Testing Metrics
Two Samples Population
Distribution
Metric
Large (≥30)
Standardized Normal
Small (<30)
if dependent,
normal
Proportion
Variance
p1 – p2
F
Two Samples
Two-Sample Hypothesis Testing
for Population Means
Test
z
t-distribution
(check variances)
t-distribution
t
Standardized Normal
if large (np1,np2>5)
F-distribution
z
t
F=s12/s22
4