An article presents a new method for timing traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in a simulation study. In 50 simulations, the mean improvement in traffic flow in a particular intersection was 653.5 vehicles per hour, with a standard deviation of 311.7 vehicles per hour. Find a 95% confidence interval for the improvement in traffic flow due to new system. Round the to three decimal places.

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X=653.5[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

s=311.7 represent the sample standard deviation

n=50 represent the sample size

Calculate the confidence interval

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=50-1=49[/tex]

Since the confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,2.01)".And we see that [tex]t_{\alpha/2}=2.01[/tex].

Now we have everything in order to replace into formula (1):

[tex]653.5-2.01\frac{311.7}{\sqrt{50}}=564.897[/tex]

[tex]653.5+2.01\frac{311.7}{\sqrt{50}}=742.103[/tex]

So on this case the 95% confidence interval would be given by (564.897;742.103)