Question
## Instructions

Please work these problems on your own. You may use web searches, but not interactive methods such as asking others online or in person.

Please complete the questions on this template and upload your solutions in a single knitted Word or pdf document. Please also upload your completed template.

In light of the exam context, the data sets for the questions have been generated clearly to satisfy or obviously to violate the requirements of the statistical procedures. If reasonable exploratory analysis is done, there should be little ambiguity as to whether the given data satisfy the requirements. This is unrealistic, but less stressful for students and graders alike.

## Questions

1. Student's t, one sample

load("dat_one_sample.RData")

1.a. Please make a histogram of dat_one_sample with informative bin widths.
1.b. Please generate a Normal qq plot for dat_one_sample.
1.c. Please perform a Student's t-test of the null hypothesis that dat_one_sample is drawn from a Normal population with mean equal to 1. Report the 90% (not 95%) confidence interval for the mean. Please do this whether or not your work in 1.a and 1.b indicates that the hypotheses making the one sample Student's test a test of location of the mean are satisfied.
1.d. Considering your work in 1.a and 1.b, how do you interpret the results (p-value and confidence interval) in 1.c?

2. Wilcoxon signed rank
2.a. Please perform a Wilcoxon signed rank test of the null hypothesis that dat_one_sample is drawn from a population symmetric around its mean with mean equal to 1.
2.b. Considering your work in 1.a and 1.b, how do you interpret the results in 2.a?

3. The data set dat_pre_post simulates pre-intervention measurements for 20 individuals together with their post-intervention measurements. Carry out the most powerful test we have learned of the null hypothesis that the intervention is not associated with any systematic increase or decrease in the measurement. Please justify your choice of test. (Note that to have evidence that any change was caused by the intervention, a controlled experiment would be required.) To help you decide which test to use, please generate a scatter plot of the pre values against the post values.

4. The data dat_two_sample simulate independent, identically distributed samples from a population with the samples from $X$ in the "val" column, labeled with "gp"="x" and independent, identically distributed samples from a population with the distribution $Y$ in the "val" column, labeled with "gp"="y"
4.a.Please visually assess the Normality of the x's and the y's.
4.b. Please display histograms or density plots of the x's and the y's.
4.c. Please carry out an F test of the equality of the variance of the x's and y's.
4.d. Please carry out Welch's test of the null hypothesis that the means of x and y are equal. Please interpret the result using the work in 4.a-4.c.

5. Please carry the Mann Whitney U test on x and y. Please interpret the result using the work in 4.a-4.c.

6. Please carry out a $\chi^2$ test of independence on the contingency table in "mat". Please interpret the results.

load("mat.Rdata")

7. Please carry out Fisher's exact test on "mat" and interpret the results.

8.a. Please fit a linear regression model giving post as a linear function of pre from the dat_pre_post data set. Display the coefficients with their p-values.

8.b. Please use your work in question 3 and any additional plots you find informative to address the validity of the model, the coefficients, and the p-values. Please address the linearity of the relation between pre and post, the Normality of the residuals, and the issue of influential observations.
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## Questions

1. Student's t, one sample

```{r}
load("dat_one_sample.RData")
```

1.a. Please make a histogram of dat_one_sample with informative bin widths. (2 points)
```{r}
hist(dat_one_sample)
```
   
1.b. Please generate a Normal qq plot for dat_one_sample.(2 points)

```{r}
qqnorm(dat_one_sample)
qqline(dat_one_sample)
```

1.c. Please perform a Student's t-test of the null hypothesis that dat_one_sample is drawn from a Normal population with mean equal to 1. Report the 90% (not 95%) confidence interval for the mean. Please do this whether or not your work in 1.a and 1.b indicates that the hypotheses making the one sample Student's test a test of location of the mean are satisfied. (3 points)

```{r}
x_bar <- mean(dat_one_sample)
se <- sd(dat_one_sample)/sqrt(length(dat_one_sample))
t_stats <- (x_bar - 1)/se
alpha <- .10
z.alpha <- qnorm(1-alpha)
conf_int <- x_bar + c(-1, 1)*z.alpha*se
p_values <- 2*pt(-abs(t_stats),df=length(dat_one_sample)-1)

cat("Confidence Interval: ", conf_int)
cat("\np-value: ", p_values)
```

1.d. Considering your work in 1.a and 1.b, how do you interpret the results (p-value and confidence interval) in 1.c? (3 points)

Since mean value 1 is within the range of confidence interval -0.4687665 1.316097, it shows null hypothesis that dat_one_sample is drawn from a Normal population with mean equal to 1 holds.

p-value also support the null hypothesis, as it is 0.4146397 which is above 0.10 (90% confidence level). It is failed to reject the null hypothesis.
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      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
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      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (367μs)
      • Illuminate\Cache\Events\CacheHit (284μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (338μs)
      • Illuminate\Cache\Events\CacheHit (235μs)
      • Illuminate\Cache\Events\CacheHit (355μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (267μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (488μs)
      • Illuminate\Cache\Events\CacheHit (265μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (499μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (260μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (577μs)
      • Illuminate\Cache\Events\CacheHit (326μs)
      • Illuminate\Cache\Events\CacheHit (268μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (294μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (290μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (227μs)
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      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (226μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (289μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (305μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (263μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (157μs)
      • Illuminate\Cache\Events\CacheHit (381μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (287μs)
      • Illuminate\Cache\Events\CacheHit (225μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheMissed (336μs)
      • Illuminate\Database\Events\StatementPrepared (774μs)
      • Illuminate\Database\Events\QueryExecuted (939μs)
      • eloquent.retrieved: App\Models\SubjectCat (59μs)
      • Illuminate\Cache\Events\KeyWritten (299μs)
      • Illuminate\Cache\Events\CacheHit (1.01ms)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (226μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (337μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (367μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (295μs)
      • Illuminate\Cache\Events\CacheHit (247μs)
      • Illuminate\Cache\Events\CacheHit (263μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • creating: site.layouts.app (527μs)
      • composing: site.layouts.app (26μs)
      • creating: components.canonical (537μs)
      • composing: components.canonical (103μs)
      • creating: components.open-graph (211μs)
      • composing: components.open-graph (73μs)
      • creating: site.headers.header (374μs)
      • composing: site.headers.header (104μs)
      • Illuminate\Cache\Events\CacheHit (1.84ms)
      • creating: components.footer (81μs)
      • composing: components.footer (92μs)
      • Illuminate\Cache\Events\CacheHit (929μs)
      • Illuminate\Cache\Events\CacheMissed (365μs)
      • Illuminate\Database\Events\StatementPrepared (843μs)
      • Illuminate\Database\Events\QueryExecuted (1.35ms)
      • eloquent.retrieved: App\Models\SubjectCat (57μs)
      • Illuminate\Cache\Events\KeyWritten (354μs)
      • Illuminate\Cache\Events\CacheHit (241μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (304μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (299μs)
      • Illuminate\Cache\Events\CacheHit (293μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • creating: components.forms.contact-us (281μs)
      • composing: components.forms.contact-us (166μs)
      • creating: components.forms.get-started (196μs)
      • composing: components.forms.get-started (83μs)
      • creating: components.forms.free-tool-download (191μs)
      • composing: components.forms.free-tool-download (82μs)
      • creating: components.forms.claim-free-worksheet (147μs)
      • composing: components.forms.claim-free-worksheet (76μs)
      • creating: components.forms.tutor-subscription-waitlist (140μs)
      • composing: components.forms.tutor-subscription-waitlist (75μs)
      • creating: components.forms.tutor-subscription-join (139μs)
      • composing: components.forms.tutor-subscription-join (74μs)
      • creating: components.forms.tutor-support (140μs)
      • composing: components.forms.tutor-support (74μs)
      • 311 x Illuminate\Cache\Events\CacheHit (13.19%)
        91.68ms
        20 x Illuminate\Database\Events\QueryExecuted (7.31%)
        50.83ms
        20 x Illuminate\Database\Events\StatementPrepared (5.72%)
        39.78ms
        11 x Illuminate\Cache\Events\CacheMissed (1.5%)
        10.42ms
        11 x Illuminate\Cache\Events\KeyWritten (1.34%)
        9.32ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.84%)
        5.83ms
        1 x Illuminate\Routing\Events\Routing (0.21%)
        1.49ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.2%)
        1.40ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.15%)
        1.03ms
        1 x eloquent.booting: App\Models\SubjectCat (0.09%)
        627μs
        1 x Illuminate\Routing\Events\RouteMatched (0.09%)
        613μs
        1 x creating: components.canonical (0.08%)
        537μs
        1 x creating: site.layouts.app (0.08%)
        527μs
        1 x creating: homework.show (0.07%)
        462μs
        1 x creating: components.breadcrumbs (0.06%)
        420μs
        16 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        381μs
        1 x creating: site.headers.header (0.05%)
        374μs
        1 x eloquent.booting: App\Models\Subject (0.05%)
        317μs
        1 x creating: components.forms.contact-us (0.04%)
        281μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        265μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        233μs
        1 x creating: components.open-graph (0.03%)
        211μs
        2 x eloquent.retrieved: App\Models\Subject (0.03%)
        205μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        199μs
        1 x creating: components.forms.get-started (0.03%)
        196μs
        1 x creating: components.forms.free-tool-download (0.03%)
        191μs
        1 x composing: homework.show (0.03%)
        179μs
        1 x composing: components.breadcrumbs (0.02%)
        168μs
        1 x composing: components.forms.contact-us (0.02%)
        166μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        148μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        147μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        140μs
        1 x creating: components.forms.tutor-support (0.02%)
        140μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        139μs
        1 x composing: site.headers.header (0.01%)
        104μs
        1 x composing: components.canonical (0.01%)
        103μs
        1 x composing: components.footer (0.01%)
        92μs
        1 x composing: components.forms.get-started (0.01%)
        83μs
        1 x composing: components.forms.free-tool-download (0.01%)
        82μs
        1 x creating: components.footer (0.01%)
        81μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        76μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        75μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        74μs
        1 x composing: components.forms.tutor-support (0.01%)
        74μs
        1 x composing: components.open-graph (0.01%)
        73μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        63μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        58μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        41μs
        1 x composing: site.layouts.app (0%)
        26μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated65.86ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '50887' limit 1
        8.93mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 50887
        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
      • select * from `subjects` where `subjects`.`id` in (56)
        7.17mstwenty4_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
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (50887)
        3.83mstwenty4_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
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.07mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        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
      • select * from `solutionslibrary` where `id` <> 50887 and `subject` = 56 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        20.64mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 50887
        • 1: 56
        • 2: published
        • 3: 0
        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 (56)
        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` = 50887 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        930μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 50887
        • 1: question
        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
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 50887 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.1mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 50887
        • 1: teaser
        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
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 50887 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        3.19mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 50887
        • 1: solution
        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
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.03mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.14mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.03mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.94mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.71mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
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      App\Models\HomeworkLibrary\HomeworkLibrary
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