MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative p…
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### Part 1: Analyzing a Random Sample of Observations The following data represent a random sample of 5 observations. Assume that \( Y \) is a linear function of \( X \) and a normally distributed disturbance term with a mean of zero and a constant variance. In other words: \[ Y = a + bX + u, \quad u \sim N(0, \sigma^2) \] #### Observational Data: | Observation | Y | X | |-------------|---|---| | 1. | 1 | 17| | 2. | 3 | 13| | 3. | 5 | 8 | | 4. | 7 | 10| | 5. | 9 | 2 | ### Tasks 1. **Create a Scatterplot**: - **Objective**: Create a scatterplot of this dataset (either manually, or by using Excel graphing functions). - **Steps**: Plot the values of \( Y \) on the vertical axis and \( X \) on the horizontal axis. 2. **Calculate Statistical Parameters**: - **Objective**: Calculate the sample means, variances, and standard deviations of the variables \( X \) and \( Y \) either “by hand” or using Excel. Write one sentence describing what each of these means in the context of the dataset. - **Definitions**: - **Sample Mean**: The average value of the observations. - **Variance**: A measure of the dispersion of the observations from the mean. - **Standard Deviation**: The square root of the variance, indicating the average distance of the observations from the mean. ### Calculations The calculations below should be done by applying the bivariate regression formulas. In other words, do them “by hand” using a calculator, or in Excel in a manner that clearly reveals the formulas used. Additionally, verify your calculations by entering the data into an Excel sheet and using the regression analysis function under Data Tools. 1. **Regression Coefficients**: - **Show how to calculate \( a \) and \( b \)**. Write one sentence discussing the interpretation of each of them. - \( a \): The intercept of the regression line, representing the expected value of \( Y \) when \( X \) is zero. - \( b \): The
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Transcribed Image Text:### Part 1: Analyzing a Random Sample of Observations The following data represent a random sample of 5 observations. Assume that \( Y \) is a linear function of \( X \) and a normally distributed disturbance term with a mean of zero and a constant variance. In other words: \[ Y = a + bX + u, \quad u \sim N(0, \sigma^2) \] #### Observational Data: | Observation | Y | X | |-------------|---|---| | 1. | 1 | 17| | 2. | 3 | 13| | 3. | 5 | 8 | | 4. | 7 | 10| | 5. | 9 | 2 | ### Tasks 1. **Create a Scatterplot**: - **Objective**: Create a scatterplot of this dataset (either manually, or by using Excel graphing functions). - **Steps**: Plot the values of \( Y \) on the vertical axis and \( X \) on the horizontal axis. 2. **Calculate Statistical Parameters**: - **Objective**: Calculate the sample means, variances, and standard deviations of the variables \( X \) and \( Y \) either “by hand” or using Excel. Write one sentence describing what each of these means in the context of the dataset. - **Definitions**: - **Sample Mean**: The average value of the observations. - **Variance**: A measure of the dispersion of the observations from the mean. - **Standard Deviation**: The square root of the variance, indicating the average distance of the observations from the mean. ### Calculations The calculations below should be done by applying the bivariate regression formulas. In other words, do them “by hand” using a calculator, or in Excel in a manner that clearly reveals the formulas used. Additionally, verify your calculations by entering the data into an Excel sheet and using the regression analysis function under Data Tools. 1. **Regression Coefficients**: - **Show how to calculate \( a \) and \( b \)**. Write one sentence discussing the interpretation of each of them. - \( a \): The intercept of the regression line, representing the expected value of \( Y \) when \( X \) is zero. - \( b \): The
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