Reference no: EM132232476

Calculus and Statistics Assignment -

Question 1 -

(a) An experiment involves the moving of an object whose position s at time t is given by s(t) = 3t⁴ - 40t³ +126t² - 9. At the start of the experiment, the object is given a push to its right.

(i) Determine the velocity of the object at any time t.

(ii) Does the object ever stop moving? If so, solve for the value(s) of t when this occurs.

(iii) When is the object moving to its right, and when is the object moving to its left?

(b) Solve for the values of x where the tangent line to f(x) = x3 - 5x2 + x is parallel to the line y = 4x + 23.

Question 2 -

Identify the following integrals:

(a) ₀∫¹1 - ((5x+3)/(x²+5x+6))dx

(b) ∫(tanx + cotx)² dx

(c) ∫(sinx+sec³x-secx)/tanx dx

Question 3 -

(a) It is known that 3% of products on a production line are defective. Products are inspected until the first defective product is encountered. Solve for the probability, giving your answers correct to 4 decimal places, that

(i) The first defective occurs in the fifth item inspected,

(ii) The first defective occurs in the first five items inspected.

(b) A standard pack of 52 playing cards is used for a game. Solve for the probability that a randomly selected card is an ace or a spade.

Question 4 -

Machine tools from two companies, Company X and Company Y, were sampled and tracked for the time that they lasted.

For Company X, 40 tools were sampled. The sample mean and sample standard deviation were 16.2 weeks and 0.2 weeks respectively.

For Company Y, 40 tools were sampled. The sample mean and sample standard deviation were 15.9 weeks and 0.2 weeks respectively.

(a) Carry out a two-sample z-test to investigate whether, in general, there is a difference between the mean lasting time of tools from Company X and the mean lasting time of tools from Company Y.

(b) Calculate a 95 % confidence interval for the mean lasting time of tools from Company X, giving your answer to 2 decimal places.

Question 5 -

An IT company has two stores that sell computers. The company recorded the value of sales (in thousands of dollars) each store made for the past 12 months. The following are the results:

Store 1: 350, 460, 20, 160, 580, 250, 210, 120, 200, 510, 290, 380.

Store 2: 520, 180, 260, 380, 80, 500, 630, 420, 210, 70, 440, 140.

Sketch two box and whisker plots, one for Store 1 and one for Store 2.

Compare the two stores for the minimum sales value, the maximum sales value and the median sales value.

Which store seems to be doing better in terms of sales?