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I WANN the solution qwith all calculations step by step
 

Let's break down the steps required to solve this problem. I'll address each question in detail and provide all the necessary calculations.

### Step 1: Plot the scatterplot of \( X \) and \( Y \)

#### Data Extraction:
From the contingency table provided:

- \( X \) (Dose) = [1, 2, 3, 4]
- \( Y \) (Time of hospitalization in months) = [1, 4, 7]

And the corresponding counts:

| \( X \) (Dose) | 1 month | 4 months | 7 months | Total |
|:--------------:|:-------:|:--------:|:--------:|:-----:|
| 1 ml           |   4     |    1     |    1     |  6    |
| 2 ml           |   2     |    1     |    0     |  3    |
| 3 ml           |   4     |    2     |    3     |  9    |
| 4 ml           |   8     |    4     |    3     |  15   |
| **Total**      |  **18** |  **8**   |  **7**   | **33**|

For the scatterplot, we plot \( X \) against \( Y \). However, since each combination has a frequency (count), this should be a weighted scatterplot.

### Step 2: Calculate the Pearson Correlation Coefficient \( r \) between \( X \) and \( Y \)

The Pearson correlation coefficient formula is:

\[
r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{\left(n\sum{X^2} - (\sum{X})^2\right) \left(n\sum{Y^2} - (\sum{Y})^2\right)}}
\]

Where:
- \( n \) is the number of observations,
- \( X \) is the dose,
- \( Y \) is the hospitalization time.

I'll calculate this using the values provided.

### Step 3: Plot the regression curve and compute the regression equation of \( Y \) on \( X \). What do you conclude?

The regression equation is typically of the form:

\[
Y = a + bX
\]

Where:
- \( b \) is the slope: \( b = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} \)
- \( a \) is the intercept: \( a = \bar{Y} - b\bar{X} \)

I'll calculate these and then plot the line on the scatterplot.

### Step 4: Calculate the correlation coefficient of \( X \) and \( Y \). Is a linear fit recommended? Construct the regression line for \( Y \) on \( X \).

This is similar to step 3, where we determine if the linear regression line is a good fit based on the correlation coefficient.

### Step 5: Estimate the dose of drug required to keep the hospital stay within 2 months.

Given the regression equation, we can substitute \( Y = 2 \) months and solve for \( X \) to estimate the dose.

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I'll start by calculating the Pearson correlation coefficient and move through each step, but first, I need to extract all the specific values from the contingency table. I'll proceed with these calculations.It appears there was an issue with the dimensions when expanding the \( Y \) values based on the frequency counts. I'll correct this by expanding each combination of \( X \) and \( Y \) individually according to the frequencies in the table, then recalculate the correlation coefficient. Let's proceed with the correct approach.It looks like there is a mismatch in the lengths of the expanded \( X \) and \( Y \) arrays. This happened due to an error in the way the values were expanded. I will carefully reconstruct the arrays based on the correct frequencies and recalculate the Pearson correlation coefficient. Let me correct this now.It seems that there is still a mismatch in the lengths of the arrays. To resolve this, I'll reconstruct the expanded arrays based on the contingency table directly and ensure that the counts are accurate. This will allow us to calculate the Pearson correlation coefficient correctly.

Let's go through this step by step, confirming the count expansions manually.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you still need help with understanding the problem or want to discuss any of the steps conceptually, feel free to ask! I can guide you through the process manually or help with another type of question.

We consider treatment B for patients who had a hospital stay of less than 7 months and vary the dosage, in ml, of the drug given to each patient. The results are given in the following contingency table (data provided). The problem involves plotting scatter plots, calculating Pearson correlation coefficient, regression, and estimation.

Step-by-step solution for analyzing hospital stay data using Pearson's correlation coefficient and linear regression. This problem involves key statistical concepts such as correlation and regression, making it suitable for undergraduate-level students.

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