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A is the set which itdicates the odd numbers of throwing a fair die. Find the power set of
A. Also find A°
The aluminum bottle, first introduced in 1991 by CCL Container for mainly personal and household items such as lotions, has become popular with beverage manufacturers. Besides being lightweight and requiring less packaging, the aluminum bottle is reported to cool faster and stay cold longer than typical glass bottles. A small brewery tests this claim and obtains the following information regarding the time (in minutes) required to chill a bottle of beer from room temperature (75ºF) to serving temperature (45ºF). At α = 0.10, can it be concluded that aluminum bottles chills at a lesser time than glass bottles? (12 points)
Aluminum Glass
Sample Size 35 42
Mean time to chill 92.4 133.8
Sample standard deviation 7.3 9.9
a. State the hypothesis and identify the claim of the researcher.
b. Find the critical value(s).
c. Compute the test value.
d. Make a decision on the null hypothesis.
e. Make a decision on the claim of the researcher.
(1) If the total cost of a firm is c(x) = 1.5x^3 - 5x^2 + 20x + 5 . Find the marginal cost (in dollars) when 25 units are produced and sold. (2) The number q of roller blades a firm is willing to sell per week at a price of $p is given by q = 60 square root of p + 25 + 30 for 20 < p < 100 . (i) Find dq/dp . (ii) Find the amount supplied when the price is $56. (ii) Find the instantaneous rate of change of supply with respect to price when the price is $56.
Bluereef real estate agent wants to form a relationship between the prices of houses, how many bedrooms, House size in sq ft and Lot Size in sq ft. The data pertaining to 100 houses were processed using MINITAB and the following is an extract of the output obtained: The regression equation is 𝑃𝑟𝑖𝑐𝑒 = 𝛽 + 𝜙𝐵𝑒𝑑𝑟𝑜𝑜𝑚 + 𝛾𝐻𝑜𝑢𝑠𝑒 𝑆𝑖𝑧𝑒 + 𝜆𝐿𝑜𝑡 𝑆𝑖𝑧𝑒 Predictor Coef SE Coef T P Constant 37718 14177 2.66 ** Bedrooms 2306 6994 0.33 0.742 House Size 74.3 52.98 * 0.164 Lot Size -4.36 17.02 -0.26 0.798 S= 25023 R-Sq=56.0% R-Sq(adj)=54.6% Source DF SS MS F P Regression 3 76501718347 25500572782 *** **** Residual Error 96 60109046053 626135896 Total 99 a) Write out the regression equation. [1] b) Fill in the missing values *, **, *** and ****. [6]
a) Persons who visit the restroom of a certain fast-food outlet were asked to state their opinion of the quality of the restroom facilities, The following tables show the responses from a sample of 100 persons. Gender of Respondent Totals Male Female Quality of Facilities Above Average 8 7 15 Average 26 24 50 Below Average 7 28 35 Totals 41 59 100 A 𝜒 2 test is carried out to determine whether there is an association between the gender of persons and their opinion. i) State appropriate null and alternative hypotheses [2] ii) Determine the critical region of the test at the 1% level of significance [1] iii) Calculate the expected value for Male and below average [2] iv) For a test statistic of 9.825, explain with reason, the conclusion of your test. [3]
a) A machine is set to produce disc plates with a mean diameter of 14 mm. A sample of 8 discs gave a mean diameter, 𝑥̅ = 14.9 mm and a standard deviation, s = 1.33 mm. A test was carried out at the 5% level of significance to determine whether the machine is in good working order. Assume that the diameter of the disc follows a normal distribution. i. State, in symbols, the null and alternate hypotheses for this test. [2] ii. State, with reasons, whether a t-test or a z-test will be appropriate. [3] iii. (Determine the rejection region(s) of the test. [3] iv. Calculate the value of the test statistic. [3] v. State, with reason, a valid conclusion for the test. [2]
a) On average 2.5 faulty reports are made to a company’s switchboard per day. i. Name the random variable present in this problem and state its distribution. [2] Calculate the probability that ii. FOUR faulty reports will be made on Monday [2] iii. Less than 3 faulty reports in a 5-day work week [4] b) The number of attempts at shooting goals made by a netballer in a tournament can be modelled by a binomial distribution with a probability of success equal to 0.35. (i) In a sample of 12 attempts at shooting goals, calculate the probability that EXACTLY 4 were successful. [4] (ii) Given that the netballer made a total of 120 attempts at shooting goals in a tournament, calculate the expected number of successful shoots. [2]
The human resource manager at a car dealership wants to know if the ages of its employees are related to the department that they work in. Data was compiled and tabulated in a 2-way contingency table. The employees were classified according to their age and department. Expected counts are printed below observed counts Sales Accounts Marketing Repairs Total 20-29 8 10 27 43 88 17.74 *** 26.20 23.62 30-39 29 26 38 22 * 23.18 26.72 34.24 30.86 40-49 33 32 72 82 219 44.15 50.88 65.20 58.77 50-59 81 106 86 54 327 65.92 75.97 97.36 87.75 Total 151 ** 223 201 749 Chi-Sq = 5.348 + **** + 0.024 + 15.912 + 1.459 + 0.019 + 0.413 + 2.544 + 2.816 + 7.003 + 0.709 + 9.182 + 3.448 + 11.875 + 1.325 + 12.983 = 80.395 DF = *****, P-Value = ****** No cells with expected counts less than 5.
f) Do the records suggest a relationship or independence between age and department for the employees of this car dealership at the 5% significance level? Give reason(s) for your answer. [2]
The human resource manager at a car dealership wants to know if the ages of its employees are related to the department that they work in. Data was compiled and tabulated in a 2-way contingency table. The employees were classified according to their age and department. Expected counts are printed below observed counts Sales Accounts Marketing Repairs Total 20-29 8 10 27 43 88 17.74 *** 26.20 23.62 30-39 29 26 38 22 * 23.18 26.72 34.24 30.86 40-49 33 32 72 82 219 44.15 50.88 65.20 58.77 50-59 81 106 86 54 327 65.92 75.97 97.36 87.75 Total 151 ** 223 201 749 Chi-Sq = 5.348 + **** + 0.024 + 15.912 + 1.459 + 0.019 + 0.413 + 2.544 + 2.816 + 7.003 + 0.709 + 9.182 + 3.448 + 11.875 + 1.325 + 12.983 = 80.395 DF = *****, P-Value = ****** No cells with expected counts less than 5. e) What is relevance of the statement” No cells with expected counts less than 5.” [1]
_____________________________________________________________________________ The human resource manager at a car dealership wants to know if the ages of its employees are related to the department that they work in. Data was compiled and tabulated in a 2-way contingency table. The employees were classified according to their age and department. Expected counts are printed below observed counts Sales Accounts Marketing Repairs Total 20-29 8 10 27 43 88 17.74 *** 26.20 23.62 30-39 29 26 38 22 * 23.18 26.72 34.24 30.86 40-49 33 32 72 82 219 44.15 50.88 65.20 58.77 50-59 81 106 86 54 327 65.92 75.97 97.36 87.75 Total 151 ** 223 201 749 Chi-Sq = 5.348 + **** + 0.024 + 15.912 + 1.459 + 0.019 + 0.413 + 2.544 + 2.816 + 7.003 + 0.709 + 9.182 + 3.448 + 11.875 + 1.325 + 12.983 = 80.395 DF = *****, P-Value = ****** No cells with expected counts less than 5.
c) Fill in the gaps marked by ‘*’, ‘**’, ‘***’, ‘****’ , ‘*****’ and ‘******’ [8] d) What is the assumption on which these calculations are based? [1]
