Quadratic equation-logarithm equation-probability

Question

Question 1:

(a) without the use of a calculator, find the value of the following, showing all steps in your calculations:

i) 6(1/2) + 4(1/2)

ii) (2/3)² x √(36/9)

iii) Convert each decimal value to a fraction and then evaluate: (0.01 ÷ 0.1) . ((0.1)² ÷ 0.001)

(b) Express 598.77468:

i) Correct to 1 decimal place; 
ii) To the nearest whole number.

(c) Use a calculator to evaluate the following correct to 1 decimal place:

i) (2.25 . log 4.4) ÷ (3.84 . ln 1.21),
 
(d) Use a calculator to find the value, correct to 2 decimal places, of:

i) 5^-1.33  
ii) e^3.4             
iii) ln 4.2 
 
Question 2:

(a) Solve the following equations:

i) 7x – 3 = 15 + x 
ii) x² – 8x + 12 = 0, using factorisation; 
iii) 3x² + 6x – 2 = 0, using the quadratic formula. (Give your answer correct to 2 decimal places.) 
iv) e^2x = 10,
v) log 3x = 3.

(b) Simplify the following logarithm equation to a single log term: log (x – 2) + log (x).
 
Question 3:

(a) Explain the term “expected monetary value” when applied to a business project with an uncertain outcome.
 
(b) Two alternatives business projects. I and II, have the following probability distributions of profits:

Project I:

Profit                      Probability
£10,000                  0.6
£5,000                    0.3
-£5,000                   0.1
 
Project II:
Profit                     Probability
£10,000                  0.4
£6,000                    0.6
 
i) Find the expected profit for each project and state which project you would recommend the business to pursue.
ii) What other factors should a business take into account when deciding between alternative projects?
 
(c) A producer has developed a new product and now has to decide whether to manufacture and market the new product or simply sell the product design for £5000. If marketed successfully, the product will yield and expected profit of £10,000. If not successful, a profit of just £2000 is expected.
 
The producer also has the option of conducting a market research survey, which would cost £500. The estimated probabilities associated with the possible outcomes are summarised below:

• Probability of success with no market research = 0.4
• Probability of success following a favourable market research report = 0.8
• Probability of success following an unfavourable market research report = 0.2
• Probability of the market research report being favourable = 0.5

i) Calculate a decision tree diagram to summarise the decisions that the producer may make.
ii) Calculate the expected monetary values for the various options and make a recommendation to the producer.

Question 4: A manufacturer of computers claims that his computers are operational for more than 80% of the time. During the course of a year one computer was operational for 270 days. Test, at the 1% level, whether the manufacturer's claim was justified.
 
Question 5: The following set of represents the annual expenditure on home insurance by 150 households in 2005.

Expenditure (£)                        Frequency

Less than 50                                20
50 but less than 100                      45
100 but less than 150                    45
150 but less than 200                    35
200 but less than 250                     3
250 but less than 300                     2

(a) Find the arithmetic mean and the median for this distribution. From your results, would you say that the distribution was positively or negatively skewed?

(b) Calculate the standard deviation.

(c) An insurance trade association claims that average expenditure on home insurance in the country as a whole is £100. You are asked to test the association’s claim. State the null and alternative hypotheses, identify the critical region (using a 5% significant level), calculate the test statistic and draw an appropriate conclusion.
 
Question 6:

(a) Distinguish between simple random sampling and quota sampling
(b) Discuss one advantages and one disadvantage of quota sampling
(c) Define the standard error of the mean
(d) To investigate the amount spent each year on car repairs by the average households in a
country, a random sample of 1000 households is estimated to be £250.
 
i) Calculate 90%, 95% and 99% confidence intervals for the population mean.
ii) What sample size would be requiring to estimate the population mean to within £10 with %95 confidence.
 
Question 7:

The following data shows the number of daily deliveries made by a delivery firm over a 30 day period:

19 32 21 28 49 44 38 12 33 51
26 10 38 32 12 45 21 33 24 54
41 31 23 28 35 30 34 53 24 17

(a) Tabulate the data above in the form of a grouped frequency distribution table (using class intervals of: ‘10 to 19’; ‘20 to 29’; ‘30 to 39’; ‘40 to 49’ and ‘50 to 59’) and, giving your answers correct to 2 decimal places, calculate the:

i) Mean
ii) Standard deviation

(b) Using the tabulated grouped frequency distribution data, draw a fully labelled histogram of the number of daily deliveries. Using the histogram:

i) Comment on the shape of the distribution;
ii) Determine the most likely value of the mode.