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Exam2. A learning experience…. Scores. Raw Scores went from 68 to 147 As percentage of total….40% to 86% Scaled scores went from 60.5 to 100 Some still left to be graded…. Question by Question. Data for Q1 to Q3. Numerical. Categorical. Numerical. Categorical. n=60. Q1.

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Exam2

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Exam2

A learning experience….

Scores
• Raw Scores went from 68 to 147
• As percentage of total….40% to 86%
• Scaled scores went from 60.5 to 100
• Some still left to be graded…
Data for Q1 to Q3

Numerical

Categorical

Numerical

Categorical

n=60

Q1
• expect that the size of the car engine (measured by displacement) would change based on car class (compact, midsize, large)
• H0: MU(compact)=MU(mid)=MU(large)
• Ha: not all equal
• ANOVA single factor (3 samples)
• Unstack the data, excel data analysis
Q2
• expect to see a relationship between car class and recommended fuel type
• Relationship between two categorical variables (car class and fuel type)
• Chi-sq independence test
• 3x2 contingency table of counts…summing to 60
Q3. Fuel type and mpg
• expect that because premium gasoline is higher quality, cars for which it is recommended will get higher gas mileage (on average) than cars for which regular fuel is recommended
• Ho: MU(prem) = MU(reg)
• Ha: MU(prem) > MU(reg)
• Unstack, T-test two sample
• NOTE: We guessed the wrong tail.
• Do not reject HO in favor of THIS Ha.

R got higher sample mean

The wrong p value

The correct p value

Q4a Aspirin and Heart Attack
• Relationship between two 0/1 variable.
• 2x2 contingency table from the facts in the question (like lights and myopia).
• Chi-sq independence test for 2T alternative.
• Half the pvalue if you want a 1T alternative (Paspirin < Pplacebo)
Q4b. How many heart attacks using new design (given Ps)
• It is easy to calculate the mean (most likely) of 250.5.
• Tell me that the actual number is a random variable
• Provide a probability distribution for that random variable

Normal approx to binomial

Q4c. Will new design affect p-value?
• Yes. We will be more certain about Aspirin’s effect and LESS certain about Placebo’s effect.
• The test is focused on the difference.
• The gain in accuracy for aspirin is not as great as the loss in accuracy for placebo (diminishing returns)
• Our test will be less powerful.
• P-value will go up.
• 50/50 v 75/25 v 100/0

Best design

Worst design

Q5. Is Di significantly better than El?
• Not about whether P=0.5
• About whether P(di)=P(el)
• 2x2 chi-squared independence test

2 tailed p-value

1 tailed p-value

Q6. Rportfolio

R1, R2, R3

Will not be

Independent.

• Rportfolio = (R1+R2+R3)/3

Sum of variances (independent)

.414/3

Q7. Total (Avg) weight of n=20
• Mean = 20*μ
• Variance = 20*σ2
• Normal (sum of normals)

Family hotel means…..

Weights in elevator not independent.

More likely to be under 3500.

Pr(total<3500) =

NORMDIST(3500,3000,178.9,true)

= 0.9974

Q8. Al and Bo
• Neither knows σ
• Both get the same
• Al uses t.dist, Bo uses normdist
• The t correctly reflects extra uncertainty…giving Al a higher p-value
• Bo’s cheating is rewarded with a lower p-value.
Q9
• If students don’t cheat, then their IQs are independent identically distributed N(100,15)
• The null hypothesis (mean men = mean women) IS TRUE!!!
• When H0 is true, and we do any test correctly, we reject with probability 0.05.
• We will reject H0 with probability 0.05 and fail to reject with probability 0.95
• What will happen under H0 is “easy”
• What will happen under Ha is very difficult…