1. Question 1
   Maximum safe temperature (so ≤1 in 10,000 products soften):

✔️ 1539.0497
❌ 39.0497 F
❌ 1460.9503 F
❌ 1496.281

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2. Question 2
   Distribution of sample means (n = 180, uniform error between -11 and +29; population mean 109):

✔️ A Gaussian Distribution that, in Phi notation, is written ϕ(109, .7407).
❌ A Gaussian distribution that, in Phi notation, is written, ϕ(109, 133.33).
❌ A Uniform Distribution with mean = 109 microns and standard deviation = 11.54 microns.
❌ A Uniform Distribution with mean = 109 microns and standard deviation = .8607 microns.

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3. Question 3
   Probability that observed sample mean 112 (n=45) from pop mean 120 (σ=15) occurred by chance:

✔️ .0173%
❌ 99.9827%
❌ 29.690%
❌ 1.73%

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4. Question 4
   Covariance of Oil and IT (R = -0.22):

✔️ -0.00715
❌ -0.0286
❌ -0.00219
❌ -0.00573

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5. Question 5
   If standardized (mean 0, sd 1), covariance between Oil and IT:

✔️ -0.22
❌ -0.00573
❌ 0
❌ -1

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6. Question 6
   Std dev of portfolio 70% Oil, 30% IT:

✔️ 10.44%
❌ 17.93%
❌ 12.68%
❌ 11.79%

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7. Question 7
   Minimum volatility (Markowitz optimizer) for two-stock portfolio:

✔️ 10.36%
❌ 10.43%
❌ 11.58%
❌ 9.5%

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8. Question 8
   90% CI around point forecast with model error σ = $130,000 (range above & below):

✔️ The 90% confidence interval is from $213,831 below to $213,831 above the point forecast.
❌ $316,831 below/above
❌ $83,831 below/above
❌ $164,831 below/above

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9. Question 9
   Given R = .33, point forecast $5.1M, σ_error = $130k, probability order > $5.3M:

✔️ 6.2%
❌ 93.8%
❌ 12.4%
❌ 4.3%

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10. Question 10
    Portion of uncertainty eliminated by model (R = .33):

✔️ 5.2%
❌ 4.2%
❌ 3.5%
❌ 4.5%

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11. Question 11
    Restaurant receipts: mean $75.25 (σ = 13.50), sample n=85, sample mean $77.20 → probability receipts are better by chance:

✔️ 9.05%
❌ 90.85%
❌ 9.15%
❌ 8.30%

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12. Question 12
    P.I.G. when original σ = 9s and model error σ = 4s:

✔️ 48.2%
❌ 61.5%
❌ 57.2%
❌ 53.3%

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🧾 Summary Table

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