The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations presents a new method for timing traffic signals in he

Step 1
Given:
${\displaystyle n=Samplesize=50}$
${\displaystyle \bar{x}=Samplemean=654.1}$
${\displaystyle s=Samplestandarddeviation=311.7}$
a) ${\displaystyle c=Confidencecoefficient=95\mathrm{\%}=0.95}$
Since the sample size n is large (at least 30), it is appropriate to estimate the population standard deviation by the sample standard deviation.
${\displaystyle \sigma \approx s}$
For confidence level ${\displaystyle 1-\alpha 0.95,}$ determine
${\displaystyle z_{\frac{\alpha }{2}}=z_{0.025}}$ using the normal probability table in the appendix (look up 0.025 in the table, the z-score is then the found z-score with opposite sign):
${\displaystyle z_{\frac{\alpha }{2}}=1.96}$
The margine of error is then:
${\displaystyle E=z_{\frac{\alpha }{2}}\times \frac{\sigma }{\sqrt{n}}\approx z_{\frac{\alpha }{2}}\times \frac{s}{\sqrt{n}}=1.96\times \frac{311.7}{\sqrt{30}}\approx 111.5404}$
The boundaries of the confidence interval then become:
${\displaystyle \bar{x}-E=654.1-111.5404=542.5596}$
${\displaystyle \bar{x}+E=645.1+111.5404=765.6404}$
We are ${\displaystyle 95\mathrm{\%}}$ confident that the improvement in traffic flow due to the new system is between 542.5596 vehicles per hour and 765.6404 vehicles per hour.
Step 2
b) ${\displaystyle c=Confidencecoefficient=98\mathrm{\%}=0.98}$
Since the sample size n is large (at leasr 30), it is appropriate to estimate the population standard deviation by the sample standard deviation.
${\displaystyle \sigma \approx s}$
For confidence level ${\displaystyle 1-\alpha =0.98,determine{z}_{\frac{\alpha }{2}}=z_{0.01}}$ using the normal probability table in the appendix (look up 0.01 in the table, the z-score is then the found z-score with opposite sing):
${\displaystyle z_{\frac{\alpha }{2}}=2.33}$
The margin of error is then:
${\displaystyle E=z_{\frac{\alpha }{2}}\times \frac{\sigma }{\sqrt{n}}\approx z_{\frac{\alpha }{2}}\times \frac{s}{\sqrt{n}}=2.33\times \frac{311.7}{\sqrt{30}}\approx 132.5965}$
The boudaries of the confidenceinterval then become:
${\displaystyle \bar{x}-E=654.1-132.5965=521.5035}$
${\displaystyle \bar{x}+E=654.1+132.5965=786.6965}$
We are ${\displaystyle 98\mathrm{\%}}$ confident that the improvement in traffic flow due to the new system is between 521.5035 vehicles per hour and 786.6965 vehicles per hour.
Step 3
c) Given confidence interval: ${\displaystyle (581.6,726.6)}$
The sampling distribution of the sample mean ${\displaystyle \bar{x}}$ has mean
${\displaystyle \mu }$ and standard deviation
$\frac{\sigma }{\sqrt{n}}.$