Homework Assignments Stat 200 7361 Introduction To Statistic

11202017 Homework Assignments Stat 200 7361 Introduction To Statis

Identify the core assignment questions and instructions from the provided mixed content. The main tasks involve calculating predicted scores using regression equations, analyzing X, Y data for correlation, performing a chi-square goodness-of-fit test, testing for association between categorical variables, analyzing descriptive statistics in a chi-square distribution, conducting a hypothesis test on flight delay data, evaluating the possibility of negative coefficients of determination, and analyzing scatter plots and regression lines for a relationship between variables. The assignment encompasses statistical analysis methods including regression, correlation, chi-square testing, hypothesis testing, and data interpretation.

Paper For Above instruction

Introduction

Statistics provide vital tools for analyzing data and making informed decisions. These tools include regression analysis, correlation, hypothesis testing, chi-square tests, and graphical representations. This paper explores various statistical methodologies, applying them to diverse real-world problems and datasets, illustrating their significance in understanding relationships and distributions within data.

Regression Analysis and Prediction

Regression analysis is crucial for understanding the relationship between an independent variable (X) and a dependent variable (Y). The regression equation provided is Y’ = 2X + 9. To predict the score for a person scoring 6 on X, we substitute X=6 into the equation, yielding Y’ = 2(6) + 9 = 12 + 9 = 21. Conversely, if the predicted score (Y’) is 14, solving for X gives 14 = 2X + 9, which simplifies to 2X = 5, resulting in X = 2.5. These calculations exemplify how regression equations enable prediction and reverse-engineering of data points, providing insights into variable relationships.

Correlation Calculation and Significance

Given a dataset of X and Y values, calculating the correlation coefficient (r) evaluates the strength and direction of the linear relationship between variables. Determining if r is significantly different from zero involves hypothesis testing—specifically, testing the null hypothesis that there is no correlation (r=0). A significant r indicates a meaningful linear association, which can be assessed through p-values derived from t-tests with degrees of freedom n-2, where n is the number of data points.

Chi-Square Goodness-of-Fit Test

In evaluating whether the prize winners' distribution across school classes aligns with the expected distribution based on their representation in the school, expected frequencies are calculated. For example, with 36 winners, the expected counts are: freshmen 30% of 36 = 10.8, sophomores 25% of 36 = 9, juniors 25% = 9, and seniors 20% = 7.2.

The chi-square statistic is computed by summing (observed - expected)^2 / expected for each category. Conducting this test assesses whether the actual distribution deviates significantly from the expected frequencies. A high chi-square value with a low p-value indicates significant deviation, suggesting the prize distribution may not be purely random and might be influenced by biases or other factors.

Association Between Texture and Color in Limestone

To assess whether there is an association between color and texture of limestone, the chi-square test of independence is appropriate. A contingency table with categories like light, medium, dark and textures such as fine, medium, coarse is analyzed. The expected counts for each cell are calculated based on marginal totals, and the chi-square statistic measures the deviation of observed from expected counts. A significant result indicates an association between texture and color, implying that certain textures are more common with specific colors.

True/False Statement on Chi-Square Distribution

The statement that "The standard deviation of the chi-square distribution is twice the mean" is false. The chi-square distribution's mean is equal to its degrees of freedom, and its standard deviation is √(2*degrees of freedom).

Hypothesis Testing for Flight Delays

Using a sample of 25 flights with an average delay of 22 minutes and a standard deviation of 15 minutes, we test the airline’s claim that the average delay is at most 15 minutes with variance no more than 150. The hypotheses are:

• Null hypothesis (H0): μ ≤ 15 and σ^2 ≤ 150
• Alternative hypothesis (H1): μ > 15 or σ^2 > 150

The test involves calculating the t-statistic for the mean delay:

t = (sample mean - hypothesized mean) / (standard deviation / √n) = (22 - 15) / (15 / √25) = 7 / (15/5) = 7 / 3 = 2.33.

Comparing this to the critical t-value at α=0.05 and df=24 provides the decision. If the calculated t exceeds the critical value, we reject H0, indicating that delays are significantly longer than claimed.

Coefficient of Determination

The coefficient of determination (r^2) quantifies the proportion of variance in the dependent variable explained by the independent variable. It cannot be negative because it is a squared value of the correlation coefficient, which ranges from -1 to 1. Therefore, r^2 ranges from 0 to 1, representing the percentage of variation accounted for, with higher values indicating a better fit of the regression model.

Scatter Plot and Regression Analysis

Analyzing the relationship between size and cost involves plotting a scatter graph to visually assess their association. If the plot shows a clear pattern, such as an upward trend, it indicates a positive correlation. To quantify this, the least-squares regression line is computed, typically in the form y = a + bx, where 'a' is the intercept and 'b' is the slope. The correlation coefficient (r) is also calculated to measure the strength of the relationship. A significant correlation suggests that size is a good predictor of cost, which can be validated with p-values.

Conclusion

Statistical tools such as regression, correlation, chi-square tests, hypothesis testing, and graphical analysis are instrumental in extracting meaningful insights from data. These methods facilitate understanding relationships, testing assumptions, and evaluating distributions, which are essential in research, business, and scientific investigations. Proper application of these techniques enhances decision-making processes by providing evidence-based conclusions.

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