- A 95% confidence interval for the mean income of shop assistants in a certain city is found to be ($12,000, $15,000). Explain briefly what this means. Would a 99% confidence interval be better than a 95% one? Justify your answer.
- A charity believes that when it puts out an appeal for charitable donations the donations it receives will be normally distributed with a mean of $50 and a standard deviation of $
- Find the probability that the first donation it receives will be less than $
- Find the value x such that 5% of donations are more than $x.
- A consultant for Dell was investigating computer usage among students at a particular university. 200 undergraduates and 100 postgraduates were chosen at random and asked if they owned a laptop. It was found that 81 of the undergraduates and 63 of the postgraduates owned a laptop. The consultant calculated that 48% (144 out of 300) of the students interviewed owned a laptop.
Explain, with reasons, whether the figure of 48% will be a good estimate of the proportion of all students who own a laptop.
- For a certain variable, the standard deviation in a large population is equal to 12.5. How big a sample is needed to be 95% sure that the sample mean is within 1.5 units of the population mean?
- (a) What conclusions would you draw from a test which is significant at the 1% level?
(b) What conclusions would you draw from a test which is significant at the 10% level, but not the 5% level?
(c) An accounting firm wishes to test the claim that no more than 5% of a large number of transactions contains errors. In order to test this claim, they examine a random sample of 225 transactions and find that exactly 20 of these are in error. What conclusion should the firm draw? Use a 5% significance level.
- A profit-maximising retailer can obtain cameras from the manufacturer at a cost of $50 per camera. The retailer has been selling the cameras at a price of $80, and at this price consumers have been buying 40 cameras per month. The retailer is planning to lower the price to stimulate sales and knows that for each $5 reduction in the price, 10 more cameras will be sold each month. Assuming price is a multiple of $5, what price should the retailer charge and what will the monthly profits be?
- Explain briefly the purpose of
- Hypothesis testing
- The prospective operator of a shoe store has the opportunity to locate in an established and successful shopping centre. Alternatively, at lower cost, he can locate in a new centre, whose development has recently been completed. If the new centre turns out to be very successful, it is expected that annual store profits from location in it would be $130,000. If the centre is only moderately successful, annual profits would be $60,000. If the new centre is unsuccessful, an annual loss of $10,000 would be expected. The profits to be expected from location in the established centre will also depend to some extent on the degree of success of the new centre, as potential customers may be drawn to it. If the new centre was unsuccessful, annual profits for the shoe store located in the established centre would be expected to be $90,000. However, if the new centre was moderately successful, the expected profits would be $70,000, while they would be only $30,000 if the new centre turned out to be very successful. All profits are inclusive of location cost. The probability that the new shopping centre will be very successful is 0.4 and the probability it will be moderately successful is also 0.4.
- Draw the decision tree for this problem.
- According to the expected monetary value criterion, where should the shoe store be located? Assume a risk-neutral decision-maker.
- Explain briefly how a perfect forecast of shopping centre success changes the order of the decision tree in ‘(a)’.
- The vice president of purchasing for a large national retailer has asked you to prepare an analysis of retail sales by state. Data are available for the following variables:
- Y (retsal) = Per capita retail sales in $
- X1 (perinc) = Per capita personal income in $
- X2 (unempl) = Unemployment rate in %
- X3 (totpop) = State population in 000s
The excel regression output of a potential model is:
- Comment on the effects of unemployment and per capita personal income.
- You think the prediction equation can be improved by adding state population as an additional explanatory variable. You obtained the following output:
- Is this model better? Why/why not?
- For this model, write out an expression for sales.
- For this model, calculate a 95% confidence interval for predicted sales, if unemployment is 8.1%, per capita income is $15,000 and the state’s population is 6 million. Use a z-value of 1.96.
- Write down two additional explanatory variables which you think could help to explain sales. Give a brief justification for each.
- (a) Time series are usually considered to have a combination of four components. What are these components? For each of them, give one example of data for which you would expect that component to be present.
(b) The following table gives average UK household electricity demand in kilowatt-hours (kWh) over the last five years. Quarter 1 represents Spring.
- State two features about household electricity demand that are apparent from these data.
- Show that the 4-point centred moving average for Quarter 3 in 2007 is 5.025.
- Calculate the ratio-to-moving-average (R2MA) for Quarter 3 in 2007.
- Compute the four seasonal indices using the following table of R2MA values. Replace `?’ with your answer to part `iii.’
- The estimated trend line is found to be:
= 4:461 + 0:050x;
where x is the Quarter number (Q1 of 2005 corresponds to x = 1). Provide a forecast, to three decimal places, for average UK household electricity demand for the summer of 2015. Do you have any comment to make about this forecast?
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