Economics 391 Spring 2013 Professor Lamarche University Of K

Economics 391 Spring 2013 Professor Lamarche University Of Kentucky

Economics 391 (Spring 2013) Professor Lamarche, University of Kentucky Problem Set 2 Due day: February . The temperature in January is estimated to have a mean of 34 and a standard deviation of 6, in degrees Fahrenheit. Sally estimates that her heating bill can be predicted using the following formula: X = 300 − 5 × temp, where temp is temperature. Find how much she is expected to pay in January. Moreover, obtain the variance and standard deviation of the heating bill.

2. An instructor graded a large number of midterm exams and she considers that the test scores are normally distributed with a mean of 70 and a standard deviation of 10. (a) What is the portion of students obtaining scores between 85 and 95? (b) What is the score needed to be at the top 10% of the class?

3. Response time at an online site can be modeled with an exponential distribution with a mean service of 5 minutes. John knows this and it is not sure to wait for an answer. What is the probability that for the reply to John’s request (a) Will take longer than 10 minutes? (b) Shorter than 10 minutes? (c) If the mean service is now 2 minutes, what is the probability that it takes longer than 10 minutes?

4. James W. has a portfolio that includes 20 shares of Disney and 30 shares of Amazon. The price of Disney stock is normally distributed with a mean of 25 and a variance of 80. The price of Amazon is also normally distributed with a mean of 40 and a variance of 119. James finds out that these stocks are negatively correlated, with ρ = −0.4. (a) Find the mean and standard deviation of James W.’s portfolio. (b) Would you advise James to sell Amazon and buy Disney? (c) What is the probability that the value of the portfolio is greater than $2,000? (d) What is the mean and standard deviation of James’ portfolio if stocks are not correlated?

5. Use the computer to calculate the following probabilities: (a) P(t64 > 2.12) (b) P(t27 2.5) (e) P(F34.62 > 1.8) (f) P(χ² 3 > 1.) Suppose you have an investment A whose return is normally distributed with mean 8% and standard deviation of 5%. An alternative investment B gives an the same return with a standard deviation of 8%. (a) What is the probability of losing money in investments A and B? (b) Would you rather construct a portfolio that uses A and B (e.g., C = 10A + 10B)? Why?

Paper For Above instruction

The following paper provides comprehensive solutions and analyses to the given economic and statistical problems, integrating relevant theories, formulas, and data interpretations to elucidate the key concepts and practical applications involved.

Expected Heating Bill Based on Temperature Distribution

The problem revolves around estimating Sally's expected heating bill based on the temperature in January, which follows a normal distribution with a mean (μ) of 34°F and a standard deviation (σ) of 6°F. The formula for Sally’s heating bill is X = 300 − 5 × temp, where temp is the temperature. To find the expected bill, we need the expected value of X, which directly depends on the expected temperature.

The expected value of X, E[X], can be derived using the linearity of expectation: E[X] = 300 − 5 × E[temp]. Given that E[temp] = 34°F, we have:

  • E[X] = 300 − 5 × 34 = 300 − 170 = 130 dollars.

This indicates that Sally is expected to pay $130 in January, based on the average temperature estimate.

To compute the variance and standard deviation of the heating bill, we recognize that X is a linear function of the temperature variable:

Var[X] = (−5)^2 × Var[temp] = 25 × σ² = 25 × 36 = 900.

So, the standard deviation of X is:

SD[X] = √900 = 30 dollars.

In summary, Sally's expected heating bill is $130, with a variance of 900 and a standard deviation of $30, reflecting the variability tied to temperature fluctuations.

Normal Distribution of Midterm Scores

The instructor observes that midterm scores are normally distributed with a mean (μ) of 70 and a standard deviation (σ) of 10. We analyze two questions: the portion of students scoring between 85 and 95, and the score needed for the top 10% of the class.

Part (a): To find the proportion of students scoring between 85 and 95, we convert these scores into z-scores:

  • z for 85: (85 − 70)/10 = 1.5
  • z for 95: (95 − 70)/10 = 2.5

Using standard normal distribution tables or software, the probabilities are:

  • P(Z
  • P(Z

Therefore, the proportion of students between 85 and 95 is:

0.9938 − 0.9332 = 0.0606, or approximately 6.06%.

Part (b): To find the score needed to be in the top 10%, we determine the 90th percentile (z-score ≈ 1.28). Solving for score:

score = μ + z × σ = 70 + 1.28 × 10 = 70 + 12.8 = 82.8.

Hence, a score of approximately 82.8 or higher qualifies a student for the top 10%.

Response Time Analysis Using Exponential Distribution

The response times follow an exponential distribution with a mean (μ) of 5 minutes, implying the rate parameter λ is:

λ = 1/μ = 1/5 = 0.2.

Part (a): Probability response takes longer than 10 minutes:

P(T > 10) = e^(−λ × 10) = e^(−0.2 × 10) = e^(−2) ≈ 0.1353.

Part (b): Probability response takes shorter than 10 minutes:

P(T 10) ≈ 1 − 0.1353 = 0.8647.

Part (c): If mean service time is now 2 minutes, λ becomes 1/2 = 0.5. Then, P(T > 10):

P(T > 10) = e^(−0.5 × 10) = e^(−5) ≈ 0.0067.

This demonstrates a significantly reduced likelihood of waiting longer than 10 minutes with faster service.

Portfolio Analysis Including Normal Distributions and Correlation

James W.'s portfolio comprises 20 Disney shares and 30 Amazon shares. The stock prices are normally distributed:

  • Disney: mean = 25, variance = 80 (so SD ≈ √80 ≈ 8.94).
  • Amazon: mean = 40, variance = 119 (SD ≈ √119 ≈ 10.91).

Correlation coefficient: ρ = −0.4.

(a) To find the mean and standard deviation of the portfolio:

The total value of Disney stocks:

20 × mean = 20 × 25 = $500.

The total value of Amazon stocks:

30 × mean = 30 × 40 = $1,200.

Total mean of the portfolio:

μ_portfolio = $500 + $1,200 = $1,700.

Variance of the portfolio:

  • Var_total = (20² × 80) + (30² × 119) + 2 × 20 × 30 × Covariance.

The covariance is:

Cov = ρ × SD_D × SD_A = (−0.4) × 8.94 × 10.91 ≈ −0.4 × 97.54 ≈ −39.02.

Variance calculation:

  • Var_portfolio = 400 × 80 + 900 × 119 + 2 × 20 × 30 × (−39.02) = 32,000 + 107,100 − 46,824.8 = 92,275.2.

Standard deviation:

SD_portfolio ≈ √92,275.2 ≈ 303.99.

(b) Advice to sell Amazon and buy Disney is subjective but regulatoric analysis suggests diversifying or minimizing risk; due to negative correlation, combining stocks may reduce portfolio risk, favoring diversification.

(c) Probability that the value exceeds $2,000:

Z-score = (2000 − 1700)/303.99 ≈ 0.987. Using standard normal tables:

P(Z > 0.987) ≈ 0.162.

(d) If stocks are uncorrelated, covariance term drops to zero, and the variance simplifies to:

Var = 400 × 80 + 900 × 119 = 32,000 + 107,100 = 139,100, leading to SD ≈ √139,100 ≈ 373.0.

Statistical Probabilities Using t, F, and Chi-Square Distributions

Using statistical software or tables, the probabilities are as follows:

  • (a) P(t₆₄ > 2.12): Approximately 0.018 (p-value for t around 2.12 with df = 64).
  • (b) P(t₂₇
  • (c) P(t₁₂₁
  • (d) P(F₇.₂₀ > 2.5): Approximately 0.009 (using F-distribution tables or software).
  • (e) P(F₄₃.₆₂ > 1.8): Approximately 0.075.
  • (f) P(χ²₃ > 1): For chi-square with df=3, P(χ² > 1) ≈ 0.80.

Investments with normally distributed returns:

  • Investment A: mean = 8%, SD = 5%. The probability of losing money (return

Z = (0 − 8)/5 = −1.6. Using standard normal tables, P(Z

  • Investment B: mean = 8%, SD = 8%. Z = (0 − 8)/8 = −1.0. P(Z

Portfolio constructed as C = 10A + 10B:

Expected return:

μ_C = 10 × 8% + 10 × 8% = 160%.

Variance of C assuming correlations:

  • Var_C = 10² × 25 + 10² × 64 + 2 × 10 × 10 × Covariance.

The combined variance depends on the covariance, which involves both variances and correlations, leading to a risk-profile analysis. Diversification considerations suggest that combining negatively correlated assets can reduce overall portfolio risk, aligning with investment theory promoting diversification for risk mitigation.

Conclusion

This comprehensive analysis employs probabilistic models, statistical distributions, and financial principles to address the varied questions posed. Understanding the interplay of mean, variance, correlation, and probability distributions is essential for making informed decisions in economics and finance. These strategies demonstrate how fundamental statistical tools are invaluable for economic forecasting, risk assessment, and portfolio management.

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