Statistics in Science ### Statistical Analysis of Two Random VariablesSuppose 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} $.### SamplingYou sample both and observe:\[ (x_1, y_1), \ldots, (x_n, y_n) \]Assume $ n $ is very large.### Constructing a Test StatisticConstruct a test statistic, $ Z $, for the difference in means $ \mu_X - \mu_Y $, by:\[ Z = \frac{(\bar{x} - \bar{y}) - \mathbb{E}(\mu_X - \mu_Y)}{\sqrt{\text{Var}(\bar{X} - \bar{Y})}} \]### AssignmentComplete this equation in terms of the parameters of the original random variables and the sample size $ n $.### Example CalculationSuppose 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 TestingTest a two-sided hypothesis that there is no difference in the means of $ X $ and $ Y $.

The provided information is: n=81x¯=50y¯=55σ2x=25σ2y=144σxy=34.50

Answered: Suppose you have two random variables…

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

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Question

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 $.

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