a) Let X be the number of passengers in a car (excluding the driver) in rush hour traffic in Sydney. The random variable X is likely to assume the values 0, 1, 2, 3, and 4, with probabilities given as follows:

x01234 Probability 0.56 0.34 0.07 0.02 0.01

Calculate the mean number of passengers in a car in rush hour traffic in Sydney.

b) The lifetime of a certain brand of tyres is normally distributed, with a mean of 65,000 km and a standard deviation of 2,800 km.

i) If the tyres carry a warranty for 60,000 km, what proportion of the tyres will fail before th warranty expires?

The tyre manufacturer wants to advertise that the tyres will last a specific amount of kilometres. If the tyre manufacturer wants to be correct 99% of the time, what should the amount be advertised as?

c)A firm specializing in agricultural products wants to conduct a market trial for one of its new products. A random sample of 600 potential customers is chosen to receive advertising material describing the new product.

It is decided that additional advertising and promotion will occur only if the sample results provide strong evidence that the actual (population) response rate will exceed 7%.

What decision will be made if 50 out of the 600 people make a purchase? Test using = 0.05

A life insurance company is interested to find the importance of phone calls on prospective clients and ultimately on policy sales. The company carried out a study where ten randomly selected lif insurance salespeople were surveyed to determine the number of weekly sales calls they made and the number of policy sales they made. The following data were recorded.

No.ofcalls(x) 66 43 57 37 32 48 39 58 54 62 No.ofpoliciessold (y) 20 15 18 9 12 14 13 17 16 18

You may use the following Excel print-outs or summary statistics provided to answer the questions

SUMMARY OUTPUT

Regression Statistics

SUMMARY STATISTICS

Multiple R R Square Adjusted R Square Standard Error Observations

Intercept No. of calls

0.919 0.845 0.825 1.376 10

Coefficients

2.169 0.263

Standard Error

2.023 0.040

n = = y = s2x = s2y = sxy =

t Stat

1.072 6.597

10 49.6 15.2 132.711 10.844 34.867

P-value

0.315 0.000

a) Determine the linear regression equation to predict the weekly policy sales from the number of calls made.

b) Predict the approximate number of policies sold if 50 calls were made. How confident are you in this prediction?

e

c) Use an appropriate statistical test to determine, at the 5% level of significance, whether there is any evidence of a significant linear relationship between the number of calls and the number of policies sold.