Utah Election: 66% Of Voters Supported Candidate

In The 2004 Election In Utah 66 Of All Utah Voters Voted For Amendme

In the 2004 election in Utah, 66% of all Utah voters voted for Amendment 3, the marriage amendment. Suppose a statistics major decided to predict the sampling distribution of p̂ for samples of size 100 for this situation. What is the mean of the sampling distribution of p̂ ? In this assignment, you are to estimate a security market line. You have current information about beta and expected annual return on 5 assets, as given in the following table: Beta Expected return 0..........149 (Note that expected return is a decimal, so the first asset has an expected return of 4.6%.) Please do the following and turn in a 2-4 page, double-space typed report to answer the following: 1. Run a regression line to determine the slope and y-intercept of the security market line. If you need help figuring out the regression line to run, you can refer to Chapter 11 and find the graph that shows this. Include a printout of the regression work done as an appendix (which will not count against the 4-page limit). You can use any computer program you want to run your regression. 2. Explain how to interpret your regression, including what the slope and y-intercept mean in terms we have used in this class. Calculate 95% confidence intervals for the slope and y-intercept. (Note: Even if a computer printout shows a 95% confidence interval, you must still show how it was calculated.) 3. Look up the current return on one-year Treasury bills/notes as your risk-free rate. Two good sources are and the date used. In your report, please state your source and the date. 4. Given the current risk-free rate, does the regression estimate of your risk-free rate match the actual current risk-free rate? Use confidence intervals to help answer this question. 5. What is the current expected market rate of return (based on your regression)? Please show all of your work and/or explain how you get your answers. If you use a statistical package to calculate any of the above answers, you still need to show how these numbers are calculated in your report. Note on academic misconduct: It is academic misconduct on this homework assignment to use anybody else’s computer output to answer these questions. You must complete the regressions for your report by yourself (or within your group), but you can get help on how to do regressions in general from any person or source you desire. Each report must be done independently of other reports.

Paper For Above instruction

This assignment encompasses two distinct topics: the estimation of a sampling distribution for a proportion based on Utah's 2004 voting data, and the analysis of a security market line (SML) through regression. Both segments involve statistical reasoning and interpretation, requiring a comprehensive understanding of fundamental concepts.

Part 1: Sampling Distribution of p̂

The first task involves analyzing the proportion of Utah voters who supported Amendment 3. Given that 66% of Utah voters voted for the amendment and a sample size of 100 voters, we are asked to determine the mean of the sampling distribution of the sample proportion p̂.

In statistical terms, the sampling distribution of p̂ (the sample proportion) follows a binomial distribution that can be approximated by a normal distribution when the sample size is sufficiently large, according to the Central Limit Theorem. The mean of this sampling distribution is equal to the true population proportion, p, which in this case is 0.66 (or 66%).

Mathematically, the mean of the sampling distribution of p̂ is:

μ_p̂ = p = 0.66

This means that, on average, the sample proportion p̂ will tend to be around 66% for repeated samples of size 100 from this population.

Part 2: Estimating the Security Market Line

The second component of the assignment involves estimating the security market line (SML) using regression analysis based on expected returns and beta coefficients for five assets. The SML depicts the relationship between systemic risk (beta) and expected return, a cornerstone in asset pricing theory.

Regression Analysis and Interpretation

The first step is to regress the expected return (dependent variable) on beta (independent variable). The regression model generally takes the form:

Expected Return = α + β * (Market Risk Premium) + ε

where α (alpha) estimates the y-intercept, and the slope estimates the market risk premium. This regression provides an empirical estimate of the SML, with the y-intercept indicating the risk-free rate when beta is zero, and the slope representing the additional return required per unit of beta (systematic risk).

Suppose that the regression output yields a slope of 0.149 and an intercept of approximately 0.046 (or 4.6%), reflecting the data provided, where the first asset’s expected return is 4.6%. The regression results inform us about how well the assets’ returns relate to their risk and help us interpret the market expectations.

Calculating Confidence Intervals

The 95% confidence intervals for both the slope and the intercept are computed using the standard error estimates obtained from the regression output and the critical value associated with a 95% confidence level, typically 1.96 for large samples.

The formula for a confidence interval:

Estimate ± (Critical value) * (Standard error)

where the critical value (approximately 1.96) is determined from the standard normal distribution.

For the slope:

CI_slope = slope ± 1.96 * SE(slope)

Likewise, the same applies for the intercept:

CI_intercept = intercept ± 1.96 * SE(intercept)

Current Risk-Free Rate and Comparison

Next, the task involves retrieving the current one-year Treasury bill rate from reputable sources such as the Federal Reserve or financial data providers, including the source and date of retrieval. This rate approximates the risk-free rate used in CAPM and asset pricing models.

Comparing the regression estimate of the risk-free rate (the intercept) to the actual current rate involves examining the confidence interval. If the actual rate falls within this interval, it suggests the regression provides a consistent estimate; if not, discrepancies may exist due to market conditions or model limitations.

Expected Market Return

The regression’s slope and intercept also facilitate the calculation of the current expected market return, often interpreted as the annual return of the market portfolio. This can be approximated by the regression's estimate of the market risk premium plus the risk-free rate.

Mathematically:

Expected Market Rate = Intercept + Slope * 1

or, more simply, from the regression — the expected return for a portfolio with beta = 1.

Conclusion

This analysis combines foundational concepts in statistics and finance to interpret data, estimate probabilities, and derive insights into market behavior. Understanding the sampling distribution enhances knowledge about population proportions, while regression-based analysis of the SML elucidates the interplay between risk and return. Such insights are essential in financial decision-making and market analysis, emphasizing the importance of accurate statistical estimation and interpretation.

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