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STatiana [176]
3 years ago
9

9.50 One way to evaluate the effectiveness of a teaching assistant is to examine the scores achieved by his or her students on a

n examination at the end of the course. Obviously, the mean score is of interest. However, the variance also contains useful information— some teachers have a style that works very well with more-able students but is unsuccessful with less-able or poorly motivated students. A professor sets a standard examination at the end of each semester for all sections of a course. The variance of the scores on this test is typically very close to 300. A new teaching assistant has a class of 30 students whose test scores had a variance of 480. Regarding these students’ test scores as a random sample from a normal population, test, against a two-sided alternative, the null hypothesis that the population variance of their scores is 300.
Mathematics
1 answer:
EleoNora [17]3 years ago
5 0

Answer:

Step-by-step explanation:

Given that a professor sets a standard examination at the end of each semester for all sections of a course. The variance of the scores on this test is typically very close to 300.

H_0: s^2 = 300\\H_a: s^2 \neq 300

(Two tailed test for variance )

Sample variance =480

We can use chi square test for testing of hypothesis

Test statistic = \frac{(n-1)s^2}{\sigma^2} \\=62.4

p value = 0.0100

Since p <0.05 our significance level, we reject H0.

The sample variance cannot be claimed as equal to 300.

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A metal cylinder can with an open top and closed bottom is to have volume 4 cubic feet. Approximate the dimensions that require
Aleksandr-060686 [28]

Answer:

r\approx 1.084\ feet

h\approx 1.084\ feet

\displaystyle A=11.07\ ft^2

Step-by-step explanation:

<u>Optimizing With Derivatives </u>

The procedure to optimize a function (find its maximum or minimum) consists in :

  •  Produce a function which depends on only one variable
  •  Compute the first derivative and set it equal to 0
  •  Find the values for the variable, called critical points
  •  Compute the second derivative
  •  Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum

We know a cylinder has a volume of 4 ft^3. The volume of a cylinder is given by

\displaystyle V=\pi r^2h

Equating it to 4

\displaystyle \pi r^2h=4

Let's solve for h

\displaystyle h=\frac{4}{\pi r^2}

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

\displaystyle A=\pi r^2+2\pi rh

Replacing the formula of h

\displaystyle A=\pi r^2+2\pi r \left (\frac{4}{\pi r^2}\right )

Simplifying

\displaystyle A=\pi r^2+\frac{8}{r}

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

\displaystyle A'=2\pi r-\frac{8}{r^2}=0

Rearranging

\displaystyle 2\pi r=\frac{8}{r^2}

Solving for r

\displaystyle r^3=\frac{4}{\pi }

\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet

Computing h

\displaystyle h=\frac{4}{\pi \ r^2}\approx 1.084\ feet

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

\displaystyle A''=2\pi+\frac{16}{r^3}

We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.

The minimum area is

\displaystyle A=\pi(1.084)^2+\frac{8}{1.084}

\boxed{ A=11.07\ ft^2}

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Answer:

A. 45°

Step-by-step explanation:

The smallest interior angle measure of any regular polygon is that of an equilateral triangle: 60°.

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