Question 1. Use stokes' theorem to compute the line integral

∫_c-yx²dx -zy²dy + e^z₂dz

Where C is the curve of intersection of the plane 2x + y +2z = 5 and the cylinder x² + y² = 4, oriented counterclockwise as viewed from above.

Question 2. A data set of 4 observations for concession sales per person at a theatre and minutes before the movie begins results in the estimated regression model sales = 4.4 + 0.266 minutes, the standard error of the regression slope is 0.0452.

(a) Compute the value of the t-statistic to test if there is a significant relationship between sales and minutes. What is the hypothesis for this test?

Question 3.

The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week?

Question 4. Find the area of the region bound by y = x/2 and y = √x

Question 5. Scores on an English test are normally distributed with a mean of 32.8 and a standard deviation of 8.7. Find the score that separates the top 59 % from the bottom 41%

Question 6. A quality control expert wants to estimate the proportion of defective components that are being manufactured by his company. A sample of 300 components showed that 20 were defective. How large a sample is needed to estimate the true proportion of defective components to within 2.5 percentage points with 99 % confidence?

Question 7. Find the limit of the sequence if it converges; otherwise indicate divergence.

1) a_n = ln(6n -1) -ln(5n +6)

A) ln[5/6]  B) ln(6/5)  C) ln 1  D) Diverges

Question 8. Find the length of the curve.
x = 4sint + 4t, y = 4cos t, 0 ≤ t ≤ Π

Question 9. Five males with a particular genetic disorder have one child each. The random variable x is the number of children among the five who inherit the genetic disorder. Determine whether the table describes a probability distribution. If it does, find the mean and standard deviation.

Question 10. A company sells sets of kitchen knives. A Basic set consists of 4 utility knives and 1 chef's knife. A regular set consists of 2 utility knives, 1 chef's knifes, and 1 slicer. A deluxe set consists of 3 utility knives, 1 chef's knife, and 1 slicer. The profit is $20 on a basic set, $60 on a regular set, and $70 on a deluxe set. The factory has on hand 1200 utility knives, 600 chef's knives, and 300 slices. If all set will be sold, how many of each type should be made up in order to maximize profit ? What is the maximum profit ?

Question 11. Consider the following three points in R³:

P(-2, 5, 1), Q(0, 3, 1), R(3, 3, 5)

And let a = PQ^→, b = PR^→, c= QR^→

a) calculate the following if possible, or explain why it is not possible to do so:

(i) a x b (ii) c x (a.b) (iii) (b x a).c (iv) (a.b).c

(b) Use the cross product to find the area of the triangle with vertices P, Q and R.

Question 12.

Practitioners measured spiritual well - being (SWB) in a sample of 16 adults who were alcoholic before and following treatment for alcoholism.

Change in SWB following treatment

Use the normal approximation for the Wilcoxon singled-ranks T test to analyze the data above. (Round your answer to two decimal places)

Question 13.

State whether to retain or reject the null hypothesis. (Assume alpha equal to 0.05.)

Question 14.

The manufacturer of a new antidepressant claims that, among all people with depression who use the drug, the proportion p of people who find relief from depression is at least 65%. A random sample of 170 patients who use this new drug is selected, and 95 of them find relief from depression. Based on these data, can we reject the manufacturer's claim at the 0.05 level of significance?
Perform a one-tailed test.

Then fill in the table below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table.

1. The null hypothesis:
2. The alternative hypothesis:
3. The type of test statistic: z
4. The value of the test statistic:
(Round to at least three decimal places)
5. The p-value:
(Round to at least three decimal places.)
6. Can we reject the claim that the proportion of people who find relief from depression using the new antidepressant is at least 65%?

Question 15.

Perform Gaussian Elimination on the matrix

Question 16.

Find the inverse of the matrix

Question 17.

(a)verify that ∂²z/∂y∂x = ∂²z/∂x∂y if z = arcsin (x/y).

Question 18.

(b) use the chain rule to find ∂z/ ∂u and ∂z/∂v, where z is from 1(a), u = 2x + 3y, and v = 3x - 2y.

Question 19.

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y . Then find the area of the region.
2y = 4√x, y = 4 , and 2y + 4x = 8

Question 20. Find the dimensions of a rectangle with perimeter 200 feet whose area is as large as possible.

Question 21.

Professor Jennings claims that only 35% of the students at Flora College work while attending school. Dean Renata thinks that the professor has underestimated the number of students with part-time or full-time jobs. A random sample of 84 students shows that 37 have jobs. Do the data indicate that more than 35% of the students have jobs? Use a 5% level of significance. What are we testing in this problem? Single mean single proportion

(a) What is the level of significance? State the null and alternate hypotheses.

(b) What sampling distribution will you use? What assumptions are you making?

(c) Find the P-value. (Round your answer to four decimal places.)

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

(e) Interpret your conclusion in the context of the application.