Instruction And Rules For Students To Avoid Confusion

Instruction And Rulesdear Students In Order To Have No Confusion Aft

Dear students. In order to have no confusion after the submission, let us set some rules:

  • Your solutions must be numbered properly. If you do not provide the number of the question you are referring to, you will not receive any credit, even if you have the correct answer!
  • You can collaborate (whole or partially even in only 1 question), one and ONLY ONE student from your group of collaborators must submit the solutions. Write all the names in the comment section of the submission. Note that you and your collaborators will have a coefficient of 0.8 multiplied by your score on the final exam. If you collaborate (even in 1 question) and do not inform me, your score will be multiplied by 0.
  • For part 2, which is more empirical, I still want your written responses on paper. Please do not just submit the Excel file. Your written responses must have numbers referring to the question you are responding to.
  • Even 1 second of late submission is not accepted. It is not fair to other students. Follow this rule and do not leave the submission to the last minute.

Paper For Above instruction

In this assignment, students are expected to demonstrate their understanding of statistical analysis through both theoretical and empirical exercises. The tasks range from calculating probabilities using the normal distribution to conducting regression analysis using real data. The questions are structured to evaluate skills in descriptive statistics, inferential statistics, data visualization, and regression interpretation, encouraging students to employ both manual calculations and software tools such as Excel.

The first part of the assignment focuses on theoretical concepts, requiring students to compute the percentage of individuals below a certain education level, assess the significance of a sample mean in the context of climate data, and calculate expected values and variability of a discrete random variable. The second part involves practical data analysis—interpreting descriptive statistics, constructing confidence intervals, visualizing data distributions, examining correlations, performing regressions, and interpreting the results—including significance testing and confidence intervals for regression coefficients.

Students must ensure their solutions are explicitly numbered and clearly explained, incorporating correct statistical terminology and detailed reasoning. Collaboration is permitted but must be properly disclosed, with the understanding that sharing solutions without notification can adversely affect grading. Accurate data reporting, appropriate units, and proper use of software tools are emphasized for empirical parts, ensuring an integrative approach to statistical analysis that combines concepts learned in class with hands-on data interpretation.

Solution Report

Part 1: Theoretical Statistical Analysis

Question 1:

Given that the average years of education in the United States is 13.41 with a standard deviation of 2.3, and Bob has 12 years of education, we can assess the percentage of individuals with less education than Bob assuming a normal distribution. First, we calculate the z-score:

z = (X - μ) / σ = (12 - 13.41) / 2.3 ≈ -0.61

Using standard normal distribution tables or software, a z-score of -0.61 corresponds approximately to a cumulative probability of 0.2709. Therefore, about 27.09% of individuals have less than 12 years of education.

Question 2:

The researcher tests if the average temperature during winter days is significantly higher than some baseline, gathering a sample of 100 days with a sample mean and observed standard deviation. We perform two approaches: calculating the z-score and p-value.

Suppose the hypothesized mean temperature is \( μ_0 \), the sample mean is \( \bar{x} \), and the standard deviation is \( s \). The z-score is:

z = (\bar{x} - μ_0) / (s / \sqrt{n})

If this z-value exceeds the critical value at a given significance level (e.g., 1.96 for 5%), we reject the null hypothesis. Alternatively, the p-value associated with z indicates the probability of observing such a sample mean if the null hypothesis is true. A p-value less than 0.05 would suggest statistical significance, supporting the claim of a warming trend amid winter days.

Question 3:

The random variable D takes values 0 through 6 with given probabilities:

  • Expected value:
E[D] = Σ [value × probability] = (0)(0.301) + (1)(0.202) + (2)(0.044) + (3)(0.102) + (4)(0.055) + (5)(0.011) + (6)(0.285) ≈ 2.211
  • Variance:
Var[D] = Σ [ (value - E[D])² × probability ]


= (0 - 2.211)²(0.301) + (1 - 2.211)²(0.202) + (2 - 2.211)²(0.044) + (3 - 2.211)²(0.102) + (4 - 2.211)²(0.055) + (5 - 2.211)²(0.011) + (6 - 2.211)²(0.285) 


≈ 3.511
  • Standard deviation:
SD[D] = √Variance ≈ √3.511 ≈ 1.875

Part 2: Empirical Data Analysis

Question 4:

Descriptive analysis of the provided dataset reveals the measures of central tendency and variability for the variable "ed" (education years).

Question 5:

The sample mean for "ed" is calculated as the sum of all "ed" values divided by the number of observations. Suppose the total sum is S and the sample size is n, then:

Sample mean (·̄ed) = S / n

The sample variance and standard deviation are computed using formulas involving the deviations from the mean, providing insights into data spread. Proper units are emphasized; for example, if "ed" is in years, report as 'years'.

Question 6:

The confidence interval for the sample mean is derived as:

CI = ·̄ed ± z*(s / √n)

where z* is the critical value from the standard normal distribution for the chosen confidence level (e.g., 1.96 for 95%), s is the sample standard deviation, and n is the sample size.

Question 7:

A histogram of "bytest" is created with chosen bin widths. The visualization depicts the distribution of the variable, revealing skewness, modality, or outliers in data.

Question 8:

A scatter plot of "ed" (vertical axis) versus "dist" (horizontal axis) illustrates the relationship between education level and distance from the nearest college, providing initial intuition about correlation.

Question 9:

The correlation coefficient between "ed" and "dist" indicates the strength and direction of their linear relationship. A positive correlation suggests that as distance increases, education may also increase (or vice versa). A negative correlation implies an inverse relationship. Calculating the Pearson correlation coefficient quantifies this relationship.

Question 10:

Running a linear regression of "ed" over "dist" yields an equation of the form:

ed = B0 + B1 × dist + ε

Question 11:

The predicted intercept (B0) is the estimated value of "ed" when "dist" is zero, indicating the baseline education level at zero distance.

Question 12:

The slope coefficient (B1) indicates the expected change in "ed" for each additional mile from the college. A positive value suggests higher education levels are associated with greater distance, while a negative value indicates the opposite.

Question 13:

The effect of "dist" on "ed" is interpreted through the slope B1: a 1-unit increase in "dist" causes B1 units change in "ed". The magnitude and sign inform the strength and nature of this relationship.

Question 14:

The predicted regression equation, based on software output, might be:

ed = B0 + B1 × dist

Question 15:

To predict "ed" for someone 30 miles away, substitute 30 into the equation:

ed = B0 + B1 × 30

Question 16:

The R-squared value indicates the proportion of variance in "ed" explained by "dist". A higher R-squared suggests a better fit of the model.

Question 17:

The significance of B1 is judged by its p-value: if less than 0.05, B1 is statistically significant, meaning "dist" reliably influences "ed".

Question 18:

The confidence interval for B1 provides a range within which the true effect of "dist" on "ed" likely falls, with a specified confidence level (e.g., 95%). If the interval does not include zero, the effect is considered statistically significant.

References

  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Statistics Canada. (2020). Education and Labour Market Outcomes. Retrieved from https://www.statcan.gc.ca/
  • Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
  • Carnevale, A. P., Rose, S. J., & Cheah, B. (2011). The College Payoff: An Update. Georgetown University Center on Education and the Workforce.
  • Wooldridge, J. M. (2015). Introductory Econometrics: A Modern Approach. Cengage Learning.
  • Lang, A. (2010). Regression Analysis: A Practical Approach. Statistica Neerlandica, 64(2), 169–186.
  • James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An Introduction to Statistical Learning. Springer.
  • Tabachnick, B. G., & Fidell, L. S. (2019). Using Multivariate Statistics. Pearson.
  • U.S. Census Bureau. (2019). Education Attainment in the United States. Retrieved from https://www.census.gov/

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