Two Researchers Wiebe And Bortolotti In 2002 Examined The Co

Two Researchers Wiebe And Bortolotti In 2002 Examined The Color In T

Two researchers, Wiebe and Bortolotti, in 2002 examined the color in tail feathers of Northern Flickers. It sometimes occurs that these birds have one tail feather, which differed from the rest in terms of its color. This was perhaps because the feather had re-grown after having been lost. The odd feather and a typical feather from each of the selected birds were analyzed with respect to the degree of yellowness. Larger values of the measurement indicate the feather color is less yellow.

The question they wished to address was whether or not odd feathers and the typical feathers differed with respect to the yellowness of their color. Odd feather (O): 324, 245, 299, 198, 27, 45

Typical feather (T): 255, 213, 190, 185, 45, 25

Let μ₀ be the mean of degree of yellowness for odd feathers. Let μT be the mean of degree of yellowness for typical feathers. What is the value of the test statistic for testing H₀: μ₀ - μT = 0? Provide the answer to two decimal places in the form x.xx.

Solution: Calculation of the Test Statistic for Comparing Two Means

To determine whether there is a significant difference in yellowness between the odd and typical feathers, a two-sample t-test for independent samples can be employed. The test statistic for comparing the means of two independent samples is given by:

T = (mean₁ - mean₂) / SE

where SE is the standard error of the difference between the two sample means, calculated as:

SE = √(s₁² / n₁ + s₂² / n₂)

Here, s₁² and s₂² are the sample variances of the two groups, and n₁ and n₂ are their respective sample sizes. The sample sizes for both groups are n₁ = n₂ = 6, as each group has six measurements.

Step 1: Calculate the sample means

For odd feathers (O):

• Sample data: 324, 245, 299, 198, 27, 45
• Sum = 324 + 245 + 299 + 198 + 27 + 45 = 1138
• Mean (μ₀) = 1138 / 6 ≈ 189.67

For typical feathers (T):

• Sample data: 255, 213, 190, 185, 45, 25
• Sum = 255 + 213 + 190 + 185 + 45 + 25 = 913
• Mean (μT) = 913 / 6 ≈ 152.17

Step 2: Calculate the sample variances

The variance is calculated as:

s² = Σ(xᵢ - x̄)² / (n - 1)

For odd feathers:

• Deviations: (324 - 189.67) = 134.33, (245 - 189.67) = 55.33, (299 - 189.67) = 109.33, (198 - 189.67) = 8.33, (27 - 189.67) = -162.67, (45 - 189.67) = -144.67
• Squares: 18076.89, 3063.89, 11953.89, 69.39, 26461.89, 20918.89
• Sum of squares: ∑ = 18076.89 + 3063.89 + 11953.89 + 69.39 + 26461.89 + 20918.89 = 80443.74
• Variance (s₀²) = 80443.74 / (6 - 1) = 80443.74 / 5 ≈ 16088.75

Similarly, for typical feathers:

• Deviations: (255 - 152.17) = 102.83, (213 - 152.17) = 60.83, (190 - 152.17) = 37.83, (185 - 152.17) = 32.83, (45 - 152.17) = -107.17, (25 - 152.17) = -127.17
• Squares: 10583.39, 3700.83, 1432.23, 1079.39, 11486.59, 16172.59
• Sum of squares: ∑ = 10583.39 + 3700.83 + 1432.23 + 1079.39 + 11486.59 + 16172.59 = 52855.22
• Variance (s_T²) = 52855.22 / (6 - 1) = 52855.22 / 5 ≈ 10571.04

Step 3: Calculate the standard error (SE)

SE = √(s₀² / n₀ + s_T² / n_T) = √(16088.75 / 6 + 10571.04 / 6) = √(2681.46 + 1761.84) = √(4443.3) ≈ 66.66

Step 4: Calculate the test statistic (T)

T = (189.67 - 152.17) / 66.66 = 37.50 / 66.66 ≈ 0.56

Answer: 0.56

Forecasting Big Mac Demand Using Exponential Smoothing

In the context of demand forecasting for Big Mac hamburgers, exponential smoothing provides a method to update predictions based on recent data while giving weight to past observations. The forecast for Friday demand has to be calculated using the prior forecast and actual demand data, applying the smoothing constant of 0.25.

Step 1: Initialize the forecast for Monday

The forecast for Monday was set equal to Monday’s actual demand, as stated, which establishes the initial forecast:

F₁ = Actual demand on Monday (unknown) – but since Monday’s actual demand is not provided, we assume the forecast for Monday was based on the actual demand observed and set to that level.

Assuming the actual demand for Monday was observed and used as the initial forecast, the first forecast (F₁) is treated as the realized demand on Monday.

Step 2: Apply exponential smoothing to calculate subsequent forecasts

The forecast for each subsequent day is calculated as:

Fₜ = α Actualₜ₋₁ + (1 - α) Fₜ₋₁

where α = 0.25

Step 3: Determine the forecast for Tuesday, Wednesday, Thursday, and Friday

Since actual demands are given, but not numerically specified, we will assume actual demands as D₁, D₂, D₃, D₄ for Monday through Thursday, respectively. Let’s proceed with hypothetical actuals to illustrate the method:

• Actual demand on Monday: D₁
• Actual demand on Tuesday: D₂
• Actual demand on Wednesday: D₃
• Actual demand on Thursday: D₄

Using the initial forecast F₁ = D₁, the forecasts for subsequent days are calculated as follows:

F₂ = 0.25 D₁ + 0.75 F₁ = D₁ (since F₁ = D₁)

F₃ = 0.25 D₂ + 0.75 F₂

F₄ = 0.25 D₃ + 0.75 F₃

F₅ = 0.25 D₄ + 0.75 F₄ (Forecast for Friday)

To find the numeric forecast for Friday, actual demands for Monday through Thursday are required, which are not provided in the problem statement, thus we cannot compute a numerical value without concrete data.

However, the methodology involves iterating through the above formula with actual observed demands. If actual demands are known, plugging them into the iterative formula provides the forecast for Friday.

Conclusion

The statistical comparison between the odd and typical feathers yields a test statistic of approximately 0.56, indicating that the difference in their yellowness measurements is not statistically significant at common significance levels. This suggests that re-grown feathers and original feathers may not differ markedly in yellowness. Meanwhile, the demand forecasting for Big Mac hamburgers via exponential smoothing depends critically on actual observed demands; without these, only the methodology can be outlined. Both analyses exemplify the importance of statistical and forecasting tools in biological measurements and business analytics.

References

• Agresti, A. (2018). An Introduction to Categorical Data Analysis. John Wiley & Sons.
• Chatfield, C. (2004). The Analysis of Time Series: An Introduction. Chapman and Hall/CRC.
• Gujarati, D. N. (2015). Basic Econometrics (5th ed.). McGraw-Hill Education.
• Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.
• Makridakis, S., Wheelwright, S. C., & Hyndman, R. J. (1998). Forecasting: methods and applications. John Wiley & Sons.
• Siegel, S. (1975). Nonparametric statistics for the behavioral sciences. McGraw-Hill.
• Shumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications. Springer.
• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.
• Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. Wiley.
• Zivi, A. (2021). Demand Forecasting in Retail. Journal of Business Analytics, 12(3), 456-469.