# Tutor profile: Zachary R.

## Questions

### Subject: Latin

Translate the follow sentence Poetae fabulas de natura narrabant quod naturam amabant.

oetae fabulas de natura narrabant quod naturam amabant. It looks like there are 2 verbs in this sentence, "narrabant" and "amabant." We will likely have a conjunction connecting the two clauses. Because of their -nt endings, we know our verb is plural. Since "fabulas" is accusative and "natura" follows a preposition (making it the object of the preposition, that means "Poetae" is the plural subject. The first clause flows like this: The poets tell stories of nature... Quod can either be a demonstrative pronoun or the conjunction "because." Since we have a second verb, it is likely a conjunction. Since naturam is accusative, there is no other subject in the second clause, so we will use the subject from the first clause. The whole sentance could be translated, The poets tell stories of nature because they love nature.

### Subject: Calculus

Find the derivative of f(x): $$ f(x) = x^{3}cos (x) $$

This is function is a product of two functions, so we will need to use the formula if $$ f(x) = u(x)v(x) $$ $$ f'(x) = u'(x)v(x) + u(x) v'(x) $$ where $$ u(x) = x^{3}$$ and $$v(x) = cos (x)$$ Using power rule and derivatives of trigonometric functions, our derivative will look like $$ f'(x) = 3x^{2}cos (x) - x^3sin(x) $$

### Subject: Statistics

A professor wants to know if taking her tests in the morning or night will result in a better score. She plans on using the test data from her morning and night underwater-basket-weaving classes, which both have 20 students each. On first test, the students in the morning class collectively scored an average of 91 out of 100, with a standard deviation of 3. The students in his night class scored an average of 85 out of 100, with a standard deviation of 4. Can this professor be at least 95% confident that taking tests in the morning will result in a higher test score?

In this problem, the professor is testing the hypothesis that taking tests in the morning will result in a greater score than at night. We will use a hypothesis testing the difference in means to answer her question! The 2 hypotheses are: $$ H_{0} : d = 0 $$ $$ H_{1} : d > 0 $$ where d is $$M_{morning} - M_{night} $$ The test statistic for t = $$ \frac{d}{(s_{1}^{2} / n_{1} + s_{2}^{2} / n_{2})} $$ we'll put in our data for this problem into this formula t = $$ \frac{6}{(3^{2} / 20 + 4^{2} / 20)} = \frac{6}{(3^{2} / 20 + 4^{2} / 20)} = \frac{6}{1.25} = 4.8 $$ Using a t-critical value table with (20-1) = 19 degrees of freedom, we find a p-value that is less than 0.0005, which means there is quite a significant difference between the means. Therefore, we can conclude that students in the morning underwater-basketweaving class scored higher than night class!

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