What is the probability that a random sample of 49 different one-hour shopping periods will yield a sample mean between 441 and 446 shoppers?

Question 1

A national survey found that 17% Of Australians consume milk with their breakfast. However, in Victoria, a large milk producer believes that more than 17% of Victorians consume milk with their breakfast. To test this idea, a marketing organisation randomly selected 550 Victorians and asked if they consume milk with their breakfast. It was found that 115 did. Using a 0.05 level of significance, test the idea that more than 17% of Victorians consume milk with their breakfast.

(5 marks)

Question 2

A rental car company promotes cheap, affordable, old cars to backpackers. The company is interested in estimating the average number of days its cars have been rented for over past few years. The company has the records for all its cars, but to use this data would be very expensive.  To make a quick estimate, a random sample of 32 hire cars is selected from the company records. The number of days each car was hired is given below (in days).

2 3 7 5 8 2 1 5
9 6 3 2 5 10 1 2
4 8 4 3 7 5 8 2
5 3 7 4 9 11 4 3

 

Using these data, construct a 95% confidence interval to estimate the average number of days a car is rented out. Assume that the number of days a car is rented is normally distributed in the population. What can you conclude from your calculations?

(5 marks)

 

Question 3

Using a people-counting device at the entry to a particular department store in Sydney, the average number of shoppers visiting the store during any one-hour period was determined as 448 shoppers with a standard deviation of 21 shoppers.

(a)    What is the probability that a random sample of 49 different one-hour shopping periods will yield a sample mean between 441 and 446 shoppers?

(2 marks)

(b)   Interpret your answer in words.

(2 marks)

(c)    If the random sample size increased to 196 what effect will this have on the standard error of the mean?

(1 mark)

Question 4

The Bureau of Labor Statistics provides compensation information on and services for various positions. As of May 2008, the national average salary for an RN (registered nurse) was $65,130. Suppose the standard deviation is $9385. For a random sample of 100 such nurses find the following:

(a)    The probability that the mean of the sample is less than $62,500.

(1 mark)

(b)   The probability that the sample mean is between $64,000 and $67,500

(2 marks)

(c)    The probability that the sample mean is greater than $66,000

(1 mark)

(d)   Explain why the assumption of normality about the distribution of wages was not involved in the solutions parts to a, b, and c.

(1 mark)

 

Question 5

Sara is a stockbroker and psychic.  She claims that she provides clients with stock market advice which is in good harmony with a client’s “aura” and “karma” and which is attuned to the client’s own personal astrological chart.  Daniela is a bit wary of these claims, but decides to invest with Sara for a trial period of one year.  She purchases five stocks, on Sara’s advice.  At the end of the year Daniela checks to see whether she could on an average depend on Sarah’s psychic abilities for investment decisions.

 

(a)    State the null and alternative hypotheses, in words and in symbols.

(1 mark)

(b)   The five stocks that Sara recommended to Daniela had returns of +10%, -15%, +5%, +30%, and –5% over the one-year trial period.  (The overall stock market, as measured by the S&P 500, lost 6% over the course of that year.)  Compute the test statistic. Explain the choice of your test statistic. Use α = 0.05. Give the p-value.

(2 marks)

(c)    Based on your calculations do you think Sara has investing abilities based on her being a psychic?

(1 mark)

(d)   What would be a Type I error in this scenario

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