## Find out the average and probability, Basic Statistics

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Question 1:

(a)  The time it takes to deliver a pizza after an order is placed is uniformly distributed between 20 and 50 minutes. What is the probability that a pizza will be delivered within half-an-hour after an order is placed?

(b)  After completing a study, the Chicago O'Hare Airport managers have concluded that the time needed to get passengers loaded onto an airplane is normally distributed with a mean equal to 15 minutes and a standard deviation equal to 3.5 minutes.  Recently an Airbus A380 airplane required 22 minutes to get passengers on board and ready for take off. What is the probability that a flight will take 22 minutes or more to get passengers loaded?

(c) The Haines Internet Services Company has determined that the number of customers who attempt to connect to the Internet per hour is Poisson distributed with a mean of 30 per hour.  The time between connect requests is exponentially distributed. Because of the system that Haines uses, if customer requests are too close together - 45 seconds or less - the connection will fail. Find the probability that a customer will fail to connect.

Question 2:

(a) The Commonwealth Bank Property Guide shows that the median value of houses in the New South Wales Hunter region in August 2008 was \$261,000.  Suppose that the mean price was \$276,000 and standard deviation of prices was \$80,000. If you take a random sample of 100 houses, what is the probability that the sample mean will be less than \$270,000? What assumptions did you have to make?

(b) Skilled migrants accounted for 43% of all permanent arrivals to Australia in 2006-07. In comparison, Family Stream migrants accounted for 26% and Humanitarian Program migrants contributed 9%. Suppose a government department is conducting a follow-up study and randomly selects 60 migrants who arrived in the above period. What is the probability that more than 45% of the people in the sample are skilled migrants?

(c) Suppose that the population size is 500 and sample size is 50. Do you apply the finite population correction factor in calculating the standard error if the sampling is done with replacement?

Question 3:

(a) The data below represent the lowest daily rate in NZ\$ for a double room in a sample of motels and motor lodges in 20 New Zealand cities and towns in January 2008.

 City/Town Room cost City/Town Room cost Lake Taupo 130 Hamilton 140 Whitianga 110 Waitomo 80 Auckland 120 Whangarei 115 Paihia 90 Russell 115 Wellington 110 Kerikeri 85 Tauranga 79 Havelock North 115 New Plymouth 115 Thames 115 Hastings 115 Palmerston North 130 Napier 99 Wanganui 95 Gisborne 130 Rotorua 100

Construct a 95% confidence interval for the population mean lowest room rate.

(b)  The Parental Leave Survey, carried out by University of Queensland researchers, collected information from 3,500 families with young children. It found that 26% of the mothers used some paid maternity leave, of which 18% used it at half-pay to double the duration. Construct a 95% confidence interval for the proportion of mothers taking paid maternity leave who wish to receive it at half-pay so as to lengthen the duration.

(c)  Explain (including functions and arguments) what the following command in Microsoft Excel will find:

=AVERAGE(A2:A20) + TINV(0.05, ROWS(A2:A20-1))*STDEV(A2:A20)/SQRT(ROWS(A2:A20))

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