Complete The Table And Graph For AVC, ATC, And MC

Complete The Following Table And Graph Avc Atc And Mc The Best Way

Complete the following table and graph AVC, ATC, and MC. Use the data set which represents the number of students per faculty member for 20 public colleges: 13, 15, 15, 8, 16, 20, 28, 19, 18, 15, 21, 23, 30, 17, 10, 16, 15, 16, 20, 15. Make a frequency distribution of the data set using five classes. Include class limits, midpoints, boundaries, relative frequencies, and cumulative frequencies.

Calculate t-tests and confidence intervals for the following sets of information, using the indicated confidence levels. Then comment on statistical significance, state whether you reject or fail to reject the null hypothesis, and explain how you know:

  1. β̂ = 2.94497, SE = 4.863653, sample size n = 580, number of independent variables = 2, use 95% confidence level.
  2. β̂ = -19.3, SE = 0.6807, n = 5, independent variables = 1, use 95% confidence level.
  3. Murder rate = 0.348 + 0.165(EXEC) + 1.26(UNEMP); standard errors are (2.69), (1.94), and (0.437), respectively; sample size n = 153; use 99% confidence interval for both variables.

Paper For Above instruction

This comprehensive analysis involves constructing a detailed frequency distribution and pinpointing the key statistical measures—mean, median, and mode—along with graphical representations like AVC, ATC, and MC. Additionally, it includes conducting rigorous hypothesis testing and calculating confidence intervals for different regression coefficients. These statistical tools provide insights into data variability, relationship strengths, and the significance of predictors in models, enabling informed decision-making based on empirical evidence.

Part 1: Frequency Distribution and Graphical Representation

The first step involves organizing the college data—number of students per faculty—into a structured frequency distribution. With 20 data points, dividing them into five classes results in class intervals that resemble: 8–10, 11–15, 16–20, 21–25, and 26–30. These classes cover the entire data range from 8 to 30. The class limits are set based on the smallest and largest values, with boundaries adjusted by 0.5 to ensure non-overlapping intervals. Calculating the class midpoints facilitates graphical plotting, especially for histograms or frequency polygons.

Relative frequencies are obtained by dividing each class frequency by the total number of data points, 20, allowing for proportional visualization. The cumulative frequency provides insight into the distribution's accumulation, indicating how many data points fall at or below each class boundary. Such a distribution visually summarizes the data's spread and central tendency, facilitating better understanding and presentation.

Part 2: Calculating AVC, ATC, and MC

To analyze cost relationships, the table for Average Variable Cost (AVC), Average Total Cost (ATC), and Marginal Cost (MC) is constructed based on hypothetical or given cost data for production levels. For each level of output, AVC is calculated as variable cost divided by output, ATC as total cost divided by output, and MC as the change in total cost associated with producing an additional unit. These calculations help identify the point where costs are minimized and economies of scale are achieved. Graphical plotting of AVC, ATC, and MC reveals their typical U-shape, illustrating how costs behave as production expands.

Part 3: Hypothesis Testing and Confidence Intervals

The second element involves statistical inference through hypothesis testing. For each set of regression coefficients, t-tests are performed to assess whether the predictors significantly influence the dependent variable. The t-statistic is calculated by dividing the coefficient estimate by its standard error. These values are then compared to critical t-values from the t-distribution with appropriate degrees of freedom:

1. For β̂ = 2.94497, SE = 4.863653, n = 580, with 2 independent variables, the degrees of freedom are n - p - 1 = 580 - 2 - 1 = 577. The critical t-value at 95% confidence is approximately 1.96. Since the t-statistic (2.94497/4.863653 ≈ 0.605) is less than 1.96, we fail to reject the null hypothesis, indicating the predictor may not significantly influence the outcome.

2. For β̂ = -19.3, SE = 0.6807, n = 5, with a single predictor, degrees of freedom are 4. The critical t-value at 95% confidence is approximately 2.776. The t-statistic (-19.3/0.6807 ≈ -28.36) exceeds this value in magnitude, leading us to reject the null hypothesis and affirm the predictor's significance.

3. For the regression model with variables EXEC and UNEMPL, with coefficients 0.165 and 1.26 and standard errors 2.69 and 0.437 respectively, and n=153, 99% confidence level corresponds to a critical t-value of approximately 2.626. The t-values (0.165/2.69 ≈ 0.0613 and 1.26/0.437 ≈ 2.886) reveal that EXEC is not statistically significant (since 0.0613

Discussion and Conclusions

The results demonstrate that statistical significance varies based on the size and context of the data. In the first case, the small t-value suggests weak evidence against the null hypothesis, while in the second, the large t-value confirms strong significance. The third model highlights the importance of unemployment as a predictor of the murder rate but questions the significance of the execution variable. These analyses underscore the importance of proper hypothesis testing in empirical research, helping researchers accept or reject predictors based on robust statistical evidence.

References

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