Question
In this exercise we look at single predictor linear regression models for a continuous quantitative response variable (dependent variable) of interest as a function of a single quantitative predictor variable (independent, explanatory variable). Visualization of the potential linear relationship between the two variables using a scatterplot with the response variable on the vertical axis and the predictor variable on the horizontal axis. Also need to determine the strength and the direction of that linear relationship by computing a correlation between the two variables. We will partition the variation in the response variable into a regression model (explained) variation, and an error or unexplained variation. The associated ANOVA table will give us the explanatory power of the model as the coefficient of determination, or R-squared as before. We will compute the statistical significance of the model parameters (intercept and slope) and compare those p-values to the desired level of significance.

For Problem statement 1 and 2 Use R software perform an analysis using a sample as specified in the problem statement and an analysis using the entire dataset as well as an SRS of size 30.
Include a scatterplot, correlation between the two variables, regression analysis including ANOVA table. Discuss the predictive power (coefficient of determination) and statistical significance of the model parameters (intercept and slope) assuming a level of alpha of 0.05. Comment on results based on the entire data set as well as those from the SRS.

Problem statement 1 (Attached CEREBRAL data in an excel spreadsheet)
Refer to the data for 1050 subjects with cerebral edema (CEREBRAL). Cerebral edema with consequent increased intracranial pressure frequently accompanies lesions resulting from head injury and other conditions that adversely affect the integrity of the brain. Available treatments for cerebral edema vary in effectiveness and undesirable side effects. One such treatment is glycerol, administered either orally or intravenously. Of interest to clinicians is the relationship between intracranial pressure and glycerol plasma concentration.
Suppose you are a statistical consultant with a research team investigating the relationship between these two variables. Select a simple random sample from the population and perform the analysis that you think would be useful to the researchers. Present your findings and conclusions in narrative form and illustrate with graphs using R where appropriate

Problem Statement 2(Attached NCBIRTH800 data in an excel spreadsheet)

Using the data from the Unit 1 assignment (NCBIRTH800), perform the following analysis including your analysis, again for the entire data set as well as an SRS of size 30, comparing results
a. Fit a single predictor linear regression model using variable tgrams as the response variable and variable weeks as the predictor variables.
b. Include a scatterplot, correlation between the two variables, regression analysis including ANOVA table. Discuss the predictive power (coefficient of correlation) and statistical significance of the model parameters (intercept and slope) assuming a level of alpha of 0.05.
c. Comment on the average change in tgrams per week.
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## Problem 1

Exploring relationship between Intracranial Pressure and Glycerol
Plasma Concentration of the overall data.

```{r}
dat <- read_excel("CEREBRAL.xlsx", 1)
names(dat) <- c("Subject",
                "Intracranial Pressure",
                "Glycerol Plasma Concentration")

plot(dat$`Intracranial Pressure`,
    dat$`Glycerol Plasma Concentration`,
    xlab = "Intracranial Pressure",
    ylab = "Glycerol Plasma Concentration",
    main = "Relationship between Intracranial Pressure and Glycerol Plasma Concentration")
```

From the plot it is clear that there is a negative correlation between
Intracranial Pressure and Glycerol Plasma Concentration.

### Exploring relationship on sample dataset

```{r}
dat.sampled <- dat[sample(nrow(dat), 50, replace = FALSE),]
plot(dat.sampled$`Intracranial Pressure`,
    dat.sampled$`Glycerol Plasma Concentration`,
    xlab = "Intracranial Pressure",
    ylab = "Glycerol Plasma Concentration",
    main = "Relationship between Intracranial Pressure and Glycerol Plasma Concentration")
```
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        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        1.86mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        2.5mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        1.29mstwenty4_siteHomeworkLibrary.php#201
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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        34.53mstwenty4_siteHomeworkLibraryRepository.php#30
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        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (299)
        5.17mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 50761 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.12mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
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        1.1mstwenty4_siteHomeworkLibrary.php#260
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
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