Question 1
Daily production of gadgets is normally distributed with mean 1000 (μ) and standard deviation 100 (o).
a) What is the probability that today’s production will be more than 950?
b) If there are 22 working days in June, how many could be expected to have production being more than 950? (Give your answer to the nearest whole number.)
c) What is the probability that today’s production will be between 950 and 1050?
d) The company has a policy that if daily production exceeds a certain level, X, all the workers will get a bonus. Assuming that the company does not want to give the workers a bonus more often that 5% of the time, what value should X have?
e) Find the interquartile range for daily production.
(Hint: It would be helpful if you draw the Z curve and shade the required region to see the required probability).
Question 2(A)
A survey of 20 randomly selected residents in the city of Hot Town found their mean annual income to be $39,580 with standard deviation $7,490.
a) Find a 95% confidence interval on the mean income for all residents in Hot
Town, stating clearly the formula you would use.
b) If it is now given that the population standard deviation is known to be $8,000, compute the 99% confidence interval using the same sample mean from part (a) above.
Question 2(B)
The supervisor of a Muesli Bar producing factory noted that the weight of each 30g bar is a normally distributed random variable with a mean of 30.25g and a standard deviation of 0.2g.
a) Find the probability that the mean weight of 5 Muesli bars is less than 30g?
b) Find the probability that the mean weight of 7 Muesli bars is less than 30g?
c) What can you conclude about the probability when the sample size increases? Use your answers to part (a) and (b).
Question 3
A study of the operation of a multi-storey car park in Chadstone plaza shows a historical mean parking time of 220 minutes per car. The car park area has been remodeled and the parking charges have increased. The manager believes that these changes have had an adverse effect on the mean parking time of the customers.
a) Specify the null and alternative hypotheses to test the manager’s belief.
b) Is this a one tail or two tailed test?
c) If a sample of 40 cars yields a mean of 208 minutes and a standard deviation of 80 minutes, what conclusion can be drawn at 5% level of significance about the mean parking time by the customers? Show all relevant details and workings to support your conclusion.
d) Explain a type I error in the context of this problem.
Question 4
In a study of shipping costs incurred by a firm, a company controller has randomly selected 15 air freight invoices from current shippers in
order to estimate the relationship between shipping costs and distance travelled for a given volume of goods. The results of this sample follow.
A summary output of the regression analysis of these data using Excel is given below.
Answer the following questions based on the output above.
a) Describe the type of relationship that exists between the Invoice Charges and the Distance travelled.
b) Report the estimated linear regression equation that related Invoice charges to Distance travelled.
c) Interpret the slope coefficient of the regression line.
d) Interpret the intercept of the regression line.
e) Report the value of the coefficient of determination. What information is contained in this value?
f) On the basis of the given regression output, can we say that the linear relationship between Invoice Charges and the Distance travelled is statistically significant at 5% level of significance? Show all relevant details and use the p value approach to support your answer.
g) Predict the Invoice charge if the distance travelled is 2000 Km.
Question 5
A survey of fruit preference of northern and southern Australians was carried out.
Researchers were interested to know whether Australians living in northern areas were more interested in tropical fruit because it was plentiful, as opposed to citrus and other fruit. A survey of 500 persons was carried out and the results are given below. Perform a χ2 test to
see whether fruit preference and location in Australia are independent. Use a significance level of α= 0.05.
a. State the hypotheses that you are testing
b. Determine the critical value of χ2 and your decision rule
c. Calculate the χ2 test statistic for this data. (Show all workings)
d. State your decision and conclusion.