Chapter 7: Inference for Distributions
A visual comparison of normal and paranormal distribution Lower caption says
'Paranormal Distribution' - no idea why the graphical artifact is occurring. http://stats.stackexchange.com/questions/423/what-is-your-favorite-data-analysis-cartoon 2

7.1: Inference for the Mean of a Population - Goals
• Be able to distinguish the standard deviation from the standard error of the sample mean.
• Be able to construct a level C confidence interval
(without knowing ) and interpret the results.
• Perform a one-sample t significance and summarize the results. • Be able to determine when the t procedure is valid.

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Conditions for Inference (Chapter 6)
1. The variable we measure has a Normal distribution with mean  and standard deviation σ.
2. We don’t know , but we do know σ.
3. We have an SRS from the population of interest. 4

Shape of t-distribution

http://upload.wikimedia.org/wikipedia/commons/thumb/4/41/Student_t_pdf.svg/1000
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px-Student_t_pdf.svg.png

t-Table
(Table D)

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Table A vs. Table D
Table A
Standard normal (z)
P(Z ≤ z) df not required

Table D t-distribution P(T > t) df required

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Example: t critical values
What is the t critical value for the following:
a) Central area = 0.95, df = 10
b) Central area = 0.95, df = 60
c) Central area = 0.95, df = 100
d) Central area = 0.95, z curve
e) Upper area = 0.99, df = 10
f) lower area = 0.99, df = 10
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Summary: CI
Confidence Interval x ± t*(df)

s n Upper Confidence Bound  < x + t*(df)
Lower Confidence Bound  > x - t*(df)
Sample Size

 t ' s n 
 m

2

s n s n 9

Example: Sample size
You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes. The average weight from your four boxes is 222 g with a sample standard deviation of 5 g.
a) What sample size is required to obtain a margin of error of 2 g at a 95% confidence level? 10

Single mean test: Summary
Null hypothesis: H0: μ = μ0 x  0
Test statistic: t  s/ n

One-sided: upper-tailed
One-sided: lower-tailed two-sided Alternative
Hypothesis
Ha: μ > μ0
Ha: μ < μ0
Ha: μ ≠ μ0

P-Value
P(T ≥ t)
P(T ≤ t)
2P(T ≥ |t|)
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Robustness of the t-procedure
• A statistical value or procedure is robust if the calculations required are insensitive to violations of the condition.
• The t-procedure is robust against normality.
– n < 15 : population distribution should be close to normal.
– 15 < n < 40: mild skewedness is acceptable
– n > 40: procedure is usually valid.
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Inferences for Non-Normal Distributions
• If you know what the distribution is, use the appropriate model.
• If the data is skewed, you can transform the variable. • Use a nonparametric procedure.

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7.2: Comparing two Means - Goals
• Be able to construct a level C confidence interval for the difference between two means and interpret the results. • Perform a two-sample t significance and summarize the results. • Be able to construct a level C confidence interval for a matched pair and interpret the results.
• Perform a matched pair t significance and summarize the results.
• Be able to determine when the t procedure is valid.
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Conditions for Inference: 2 - sample
1. Each group is considered to be a sample from a distinct population.
• We have an SRS from the population of interest for each variable.
2. The responses in each group are independent of those in the other group.
3. The variable(s) we measure has a Normal distribution with mean  and standard deviation σ.
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Df for 2-sample t test
2

s s 
  
n1 n 2  df 
2
2
2
2
1 s1 
1 s2 
  
  n1  1 n1  n 2  1 n 2 
2
1

2
2


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Two-sample Test (independent): Summary
Null hypothesis: H0: μ1 – μ2 = Δ
Test statistic: t  x1  x2  
2
2 s1 s 2

n1 n2

Upper-tailed
Lower-tailed
two-sided

Alternative
Hypothesis
Ha: μ1 – μ2 > Δ
Ha: μ1 – μ2 < Δ
Ha: μ1 – μ2 ≠ Δ

P-Value
P(T ≥ t)
P(T ≤ t)
2P(T ≥ |t|)

Note: If we are determining if the two populations
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Example: