Mathematics
Statistics

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Q1 A large hamburger franchise wants to estimate the average monthly sales level per outlet for the month of March. The March sales values were obtained (in $000) from a random sample of 40 outlets. The sample provided a mean of 36.2 and a standard deviation of 7.8. (a) Is the sampling distribution a normal distribution or a t distribution? Justify your choice. (b) Calculate a 95% confidence interval for the mean value of sales for March. (c) Interpret the confidence interval limits obtained in (b). Q2 A road safety group undertook a random survey of 16 vehicles on a particular stretch of road and found that their sample mean speed measured in kilometres per hour (kph) was 107 with a standard deviation of 7.13. (a) Explain why it is necessary to assume that the vehicle speeds are normally distributed before a confidence interval can be calculated. (b) Calculate a 90% confidence interval for the true mean vehicle speed on this stretch of road. (c) If the speed limit on this stretch of road is 100kph, would the road safety group be justified in lobbying for more police speed checks? Q3 An airline wants to determine the proportion of passengers who exceed their carryon luggage allowance. In a random sample of 200 passengers, 64% exceeded their allowance. (a) Is the sample large enough to calculate a confidence interval for the proportion of passengers exceeding the carry-on luggage rules? Explain. (b) Calculate the 99% confidence interval for the proportion of passengers exceeding the carry-on luggage rules. (c) Interpret the confidence interval limits obtained in (b). Q4 A popular weekly magazine wants to estimate the average age of its readership. A pilot study indicated that the standard deviation is 10 years. How large a sample must be selected if the company wants to be 90% confident that the maximum error of estimation is 2 years? Q5 A student organisation wants to estimate the proportion of students who support a proposed change to the academic calendar. As the true proportion of students who support the change is unknown, it is appropriate to use π = 0.5. How many students should be randomly selected to ensure there is 95% confidence that the true proportion can be estimated within an error of 0.03? Q6 A call centre claims that the mean time a customer’s call is “on hold” is normally distributed with a mean of 55 seconds and a standard deviation of 25 seconds. A consumer organisation suspects that the mean time customers are on hold is in fact longer than 55 seconds. A random sample of 40 customers indicated a sample mean time of 63 seconds. (a) Is there evidence to support the consumer organisation’s suspicions at the 5% significance level? (b) Calculate the p-value for this test. Q7 A sample of 20 customers were asked to rate the new Super-Box computer game. It was found that the average satisfaction rating was 43 with a standard deviation of 2.7. Use α = 0.1 to see if the actual satisfaction rating is different from 42. What assumption needs to be made in order to undertake this hypothesis test? Q8 A market research company wants to check the effectiveness of a television advertisement for a new car brand. To be considered a success, the day-after recall of the brand has to greater than 20%. Out of a random sample of 200 respondents who claimed to have watched the television show in which the advertisement was aired, 46 were able to recall the advertisement. Do the data support the claim that the advertisement was successful? (Use α = 0.01) ...
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