Ma 151 Test 2 Name Directions Show All Work In A Neat And Or

Ma 151 Test 2 Name Directions Show All Work In A Neat And Or

1. I have a data set consisting of the following three points: (1,5), (2,7), (5,15). I propose two different lines as a reasonable fit for this data: and 2. Which line is a better fit, and why? Do not guess or base your answer on visual inspection of a graph; there is a specific procedure you must follow to answer this question.

2. In a model for optimizing revenue, I find that the profit y for a particular item, as a function of price p, is given by . What is the optimal price I should charge for that product, and what is my profit if I charge that price?

3. In an exponential regression model, Excel finds a best-fit curve of 4,845 . Convert this equation into one of the form , and give the value of y when x=3.15.

4. Given the following set of 5 points, use the technique we explored in class to decide whether the data is approximately linear, approximately exponential, or neither. If you find that the data is approximately linear, give the approximate slope; if exponential, give the approximate base. You must show work to get any credit for this question. The data points are: (3,25), (4,33), (5,50), (6,69), (7,96).

5. Based on data we have collected from teens and adults, we determine that a person’s weight w (in pounds) can be estimated from his or her height h (in inches) by the equation 3 40. Use this formula to estimate the weight of someone who is 4’9†tall.

6. According to our formula in question 5, a slightly premature infant whose length is 12†would have a weight of -4 lbs., which of course is nonsense. Explain, based on what we’ve discussed in class, why our formula has apparently failed. Assume the formula is valid as it is described in the earlier question.

7. In a particular linear model which models a set of data points, I calculate the sum of squared error between the estimated points (from the equation of the line) and the actual data, and the sum turns out to be zero. What does that tell me about the points and the line?

Paper For Above instruction

The given assignment encompasses a variety of statistical modeling concepts, such as linear regression, exponential modeling, optimization, and the assessment of data fit. This paper aims to systematically address each question by providing clear and precise explanations rooted in mathematical reasoning and statistical methods.

Question 1: Comparing Two Lines for Fit

The task involves two proposed lines to fit three data points: (1,5), (2,7), and (5,15). To determine which line fits better without relying on visual judgment, the standard approach is to calculate the residuals and the sum of squared errors (SSE) for each line. The residual for each point is the difference between the observed y-value and the y-value predicted by the line. The sum of squared errors, obtained by summing the squares of these residuals, provides a quantitative measure of the fit—the smaller the SSE, the better the line fits the data.

Suppose the two lines are fitted using least squares regression, resulting in equations:

• Line 1: y = a₁x + b₁
• Line 2: y = a₂x + b₂

Calculating each line's parameters involves solving the normal equations based on the data points. Once the equations are derived, residuals at each data point are computed as:

• Residual at (1,5): r₁ = y₁ - (a×1 + b)
• Residual at (2,7): r₂ = y₂ - (a×2 + b)
• Residual at (5,15): r₃ = y₃ - (a×5 + b)

Squaring each residual and summing yields the SSE for each line. Comparing SSE values indicates which line provides a better fit: the line with the smaller SSE. This method aligns with the least squares criterion widely used in regression analysis.

Question 2: Revenue Optimization and Profit Function

The profit function y(p) models profit as a function of price p; typically, y(p) = (p - c)×q(p), where c is the cost per unit, and q(p) is the quantity demanded depending on p. To find the optimal price, we differentiate y(p) with respect to p and set the derivative to zero, solving for p.

Suppose y(p) is given explicitly; then the first derivative y'(p) helps identify maximum profit points. Once the optimal p is found, plug it back into y(p) to compute maximum profit, yielding both the best price to charge and the corresponding profit value.

Question 3: Converting Exponential Model

An exponential regression model often has the form y = a·b^x. If Excel finds a fit of y = 4,845 · some curve, it generally refers to y = a·b^x with given parameters. Converting into exponential form involves identifying the constants a and b.

For instance, if the model is y = 4845·e^{kx}, then taking natural logs results in:

ln y = ln 4845 + kx

This form enables linear regression to find ln y against x, extracting a and b accordingly. To find y when x=3.15, substitute into the equation.

Question 4: Analyzing Data for Linearity or Exponentiation

The data points are: (3,25), (4,33), (5,50), (6,69), (7,96). To determine whether the data are approximately linear or exponential, we analyze the differences and ratios between points.

Calculate the first differences in y for linearity. If the differences are roughly constant, the data is approximately linear, and the slope is the average difference. For exponential behavior, examine the ratios y_{i+1}/y_{i}; if these ratios are approximately constant, the data is exponential, and the base is the ratio value.

Question 5: Estimating Weight from Height

The formula provided is w = 3h + 40, where h is in inches. To estimate weight for someone 4’9”, convert the height from feet and inches to inches:

4 feet 9 inches = (4×12) + 9 = 48 + 9 = 57 inches.

Plug into the formula:

w = 3 × 57 + 40 = 171 + 40 = 211 lbs.

Question 6: Limitations of the Formula

The formula yields a negative weight (-4 lbs.) for a length of 12 inches, which is physically impossible. This outcome highlights that the linear relationship modeled by the formula is only valid within a certain range of heights and weights. When extrapolated beyond this range, especially at very small or large values, the model may predict nonsensical results. The formula's failure in this case underscores the importance of understanding the domain of the regression model and recognizing that biological data often require more complex modeling that accounts for nonlinearities and variability.

Question 7: Zero Sum of Squared Errors

If the sum of squared errors (SSE) is zero, this indicates that the line perfectly fits all the data points—each predicted y-value exactly matches the actual y-value. In other words, the data points lie exactly on the regression line. This typically occurs when the data points are linear or when the model includes parameters that perfectly capture the data variation, resulting in no residuals and an SSE of zero. While such a perfect fit is rare in real-world data, it signifies that the model explains all the variation in the data points without error.

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