Question

You work on a traffic management team for the city. There has been a proposal to add a new lane to a road near a busy intersection. As part of the research into whether or not to build the new lane, the team would like to know how many cars, on average, pass through this intersection at peak hour each day (5 pm to 6 pm). Your colleague has developed the hypothesis that the population mean is 886. You intend to test this hypothesis. For the purposes of this research, the team is assuming that the standard deviation in the number of cars passing through each hour is 38.

You randomly select 49 days on which to monitor the traffic going through this intersection.

a)Complete the following statement by filling in the correct numbers. Give your answers to the nearest whole number of cars.

If the true population mean really is 886, then for 90% of all samples of size n = 49, the sample mean will be somewhere between  and  cars.

b)Over the 49 days, you find that an average of 891 cars pass through the intersection. Therefore with 90% confidence you

can rule out the possibility that your colleague's hypothesis was correct
cannot rule out the possibility that your colleague's hypothesis was correct

Answer 1

Let x be the number of cars pass through this intersection at peak hour each day.

x has mean mean ( µ ) = 886 and standard deviation (σ) = 38

be the average number of cars pass through this intersection at peak hour in 49 days.

follows approximately normal distribution with mean ( µ ) = 886 and standard deviation (σ) =   = 5.4286

a) We are asked to find interval such that 90% of average cars lies between them.

The total area under the curve is 1 , then middle shaded area is 0.90 therefore left tail area is  (1 - 0.90)/2 = 0.05 and right tail area is also 0.05

So we have to find the z score corresponding to area 0.05 on z score table

So corresponding z scores are -1.645 and 1.645

  ₁ = z*σ + µ = ( -1.645* 5.4286 + 886 )

₁ = 877

₂ = z*σ + µ = ( 1.645* 5.4286 + 886 )

₂ = 895

If the true population mean really is 886, then for 90% of all samples of size n = 49, the sample mean will be somewhere between 877 and 895 cars.

b) Since 891 lies in the interval ( 877,895)

Over the 49 days, you find that an average of 891 cars pass through the intersection. Therefore with 90% confidence you can rule out the possibility that your colleague's hypothesis was correct.