Suppose you have two random variables X,Y ux,Hy,o o3,0xY where Cov(X, Y) = 0x.Y You sample both (1poume n is very very very large.) 5
Suppose you have two random variables X,Y ux,Hy,o o3,0xY where Cov(X, Y) = 0x.Y You sample both (1poume n is very very very large.) 5
MATLAB: An Introduction with Applications
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Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Statistics in Science
![### Statistical Analysis of Two Random Variables
Suppose you have two random variables \( X, Y \) with the following properties:
\[ X, Y \sim \mu_X, \mu_Y; \sigma_X^2, \sigma_Y^2, \sigma_{XY} \]
where \( \text{Cov}(X, Y) = \sigma_{XY} \).
### Sampling
You sample both and observe:
\[ (x_1, y_1), \ldots, (x_n, y_n) \]
Assume \( n \) is very large.
### Constructing a Test Statistic
Construct a test statistic, \( Z \), for the difference in means \( \mu_X - \mu_Y \), by:
\[ Z = \frac{(\bar{x} - \bar{y}) - \mathbb{E}(\mu_X - \mu_Y)}{\sqrt{\text{Var}(\bar{X} - \bar{Y})}} \]
### Assignment
Complete this equation in terms of the parameters of the original random variables and the sample size \( n \).
### Example Calculation
Suppose you observe sample data:
- \( n = 81, \bar{x} = 50, \bar{y} = 55 \)
- Known values: \( \sigma_X^2 = 25, \sigma_Y^2 = 144, \sigma_{XY} = 34.5 \)
### Hypothesis Testing
Test a two-sided hypothesis that there is no difference in the means of \( X \) and \( Y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7ef3e23-5577-4236-ac2c-d553e333dcb9%2F6ffcf35a-7461-4e6a-adc5-1bb53a4ac7b1%2Fg5hjima_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Statistical Analysis of Two Random Variables
Suppose you have two random variables \( X, Y \) with the following properties:
\[ X, Y \sim \mu_X, \mu_Y; \sigma_X^2, \sigma_Y^2, \sigma_{XY} \]
where \( \text{Cov}(X, Y) = \sigma_{XY} \).
### Sampling
You sample both and observe:
\[ (x_1, y_1), \ldots, (x_n, y_n) \]
Assume \( n \) is very large.
### Constructing a Test Statistic
Construct a test statistic, \( Z \), for the difference in means \( \mu_X - \mu_Y \), by:
\[ Z = \frac{(\bar{x} - \bar{y}) - \mathbb{E}(\mu_X - \mu_Y)}{\sqrt{\text{Var}(\bar{X} - \bar{Y})}} \]
### Assignment
Complete this equation in terms of the parameters of the original random variables and the sample size \( n \).
### Example Calculation
Suppose you observe sample data:
- \( n = 81, \bar{x} = 50, \bar{y} = 55 \)
- Known values: \( \sigma_X^2 = 25, \sigma_Y^2 = 144, \sigma_{XY} = 34.5 \)
### Hypothesis Testing
Test a two-sided hypothesis that there is no difference in the means of \( X \) and \( Y \).
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