# The article “Modeling of Urban Area Stop-andGo Traffic Noise” presents measurements of traffic noise, in dBA, from 10 locations in Bangkok, Thailand. Measurements, presented in the following table, were made at each location, in both the acceleration and deceleration lanes. begin{array}{|c|c|}hline text{Location} & text{Acceleration} & text{Decelaration} hline 1 & 78.1 & 78.6 hline 2 & 78.1 & 80.0 hline 3 & 79.6 & 79.3 hline 4 & 81.0 & 79.1 hline 5 & 78.7 & 78.2 hline 6 & 78.1 & 78.0 hline 7 & 78.6 & 78.6 hline 8 & 78.5 & 78.8 hline 9 & 78.4 & 78.0 hline 10 & 79.6 & 78.4 hline end{array} Can you conclude that there is a difference in the mean noise levels between acceleration and deceleration lanes? Question
Modeling The article “Modeling of Urban Area Stop-andGo Traffic Noise” presents measurements of traffic noise, in dBA, from 10 locations in Bangkok, Thailand. Measurements, presented in the following table, were made at each location, in both the acceleration and deceleration lanes.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Location}&\text{Acceleration}&\text{Decelaration}\backslash{h}{l}\in{e}{1}&{78.1}&{78.6}\backslash{h}{l}\in{e}{2}&{78.1}&{80.0}\backslash{h}{l}\in{e}{3}&{79.6}&{79.3}\backslash{h}{l}\in{e}{4}&{81.0}&{79.1}\backslash{h}{l}\in{e}{5}&{78.7}&{78.2}\backslash{h}{l}\in{e}{6}&{78.1}&{78.0}\backslash{h}{l}\in{e}{7}&{78.6}&{78.6}\backslash{h}{l}\in{e}{8}&{78.5}&{78.8}\backslash{h}{l}\in{e}{9}&{78.4}&{78.0}\backslash{h}{l}\in{e}{10}&{79.6}&{78.4}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Can you conclude that there is a difference in the mean noise levels between acceleration and deceleration lanes? 2021-01-23
Step 1
Given: $$\displaystyle{n}={10}$$
Let us assume: $$\displaystyle\alpha={5}\%={0.05}$$
Given claim: Difference $$\displaystyle\mu_{{{d}}}\ \ne{q}\ {0}$$
The claim is either the null hypothesis or the alternative hypothesis. The null hypothesisi and the alternative hypothesis state the opposite of each other.
$$\displaystyle{H}_{{{0}}}\ :\ \mu_{{{d}}}={0}$$
$$\displaystyle{H}_{{\alpha}}\ :\ \mu_{{{d}}}\ \ne{q}\ {0}$$
Determine the difference in value of each pair.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}\text{Sample 1}&\text{Sample 2}&\text{Difference D}\backslash{h}{l}\in{e}{78.1}&{78.6}&-{0.5}\backslash{h}{l}\in{e}{78.1}&{80}&-{1.9}\backslash{h}{l}\in{e}{79.6}&{79.3}&{0.3}\backslash{h}{l}\in{e}{81}&{79.1}&{1.9}\backslash{h}{l}\in{e}{78.7}&{78.2}&{0.5}\backslash{h}{l}\in{e}{78.1}&{78}&{0.1}\backslash{h}{l}\in{e}{78.6}&{78.6}&{0}\backslash{h}{l}\in{e}{78.5}&{78.8}&-{0.3}\backslash{h}{l}\in{e}{78.4}&{78}&{0.4}\backslash{h}{l}\in{e}{79.6}&{78.4}&{1.2}\backslash{h}{l}\in{e}\text{Mean}&&{0.17}\backslash{h}{l}\in{e}\text{sd}&&{1.0122}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Step 2
Determine the sample mean of the differences. The mean is sum of all values divided by number of values.
$$\displaystyle\overline{{{d}}}={\frac{{-{0.5}\ -\ {1.9}\ +\ {0.3}\ +\ \cdots\ -{0.3}\ +\ {0.4}\ +\ {1.2}}}{{{10}}}}\ \approx\ {0.17}$$
Determine the sample standard deviation of the differences:
$$\displaystyle{s}_{{{d}}}=\sqrt{{{\frac{{{\left(-{0.5}\ -\ {0.17}\right)}^{{{2}}}\ +\ \cdots\ +\ {\left({1.2}\ -\ {0.17}\right)}^{{{2}}}}}{{{10}\ -\ {1}}}}}}\ \approx\ {1.0122}$$
Determine the value of the test statistic:
$$\displaystyle{t}={\frac{{\overline{{{d}}}}}{{\frac{{s}_{{{d}}}}{\sqrt{{{n}}}}}}}={\frac{{{0.17}}}{{\frac{{1.0122}}{\sqrt{{{10}}}}}}}\ \approx\ {0.531}$$
The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. The P-value is the number (or interval) in the column title of the Students T distribution in the appendix containing the t-value in the row $$\displaystyle{d}{f}={n}\ -\ {1}={10}\ -\ {1}={9}$$ (Note: We double the boundaries, because the test is two-tailed).
$$\displaystyle{0.50}={2}\ \times\ {0.25}\ {<}\ {P}\ {<}\ {2}\ \times\ {0.40}={0.80}$$</span>
If the P-value is less than the significance level, reject the null hypothesis.
$$\displaystyle{P}\ {>}\ {0.10}={10}\%\ \Rightarrow\ {F}{a}{i}{l}\ \to\ {r}{e}{j}{e}{c}{t}\ {H}_{{{0}}}$$
There is sufficient evidence to support the claim that is a difference in the mean noise levels between accelaration and decelaration lanes.

### Relevant Questions The article “Anodic Fenton Treatment of Treflan MTF” describes a two-factor experiment designed to study the sorption of the herbicide trifluralin. The factors are the initial trifluralin concentration and the $$\displaystyle{F}{e}^{{{2}}}\ :\ {H}_{{{2}}}\ {O}_{{{2}}}$$ delivery ratio. There were three replications for each treatment. The results presented in the following table are consistent with the means and standard deviations reported in the article. $$\displaystyle{b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}\text{Initial Concentration (M)}&\text{Delivery Ratio}&\text{Sorption (%)}\ {15}&{1}:{0}&{10.90}\quad{8.47}\quad{12.43}\ {15}&{1}:{1}&{3.33}\quad{2.40}\quad{2.67}\ {15}&{1}:{5}&{0.79}\quad{0.76}\quad{0.84}\ {15}&{1}:{10}&{0.54}\quad{0.69}\quad{0.57}\ {40}&{1}:{0}&{6.84}\quad{7.68}\quad{6.79}\ {40}&{1}:{1}&{1.72}\quad{1.55}\quad{1.82}\ {40}&{1}:{5}&{0.68}\quad{0.83}\quad{0.89}\ {40}&{1}:{10}&{0.58}\quad{1.13}\quad{1.28}\ {100}&{1}:{0}&{6.61}\quad{6.66}\quad{7.43}\ {100}&{1}:{1}&{1.25}\quad{1.46}\quad{1.49}\ {100}&{1}:{5}&{1.17}\quad{1.27}\quad{1.16}\ {100}&{1}:{10}&{0.93}&{0.67}&{0.80}\ {e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}$$ a) Estimate all main effects and interactions. b) Construct an ANOVA table. You may give ranges for the P-values. c) Is the additive model plausible? Provide the value of the test statistic, its null distribution, and the P-value. 1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance. A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared. In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.
This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.
The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination. PSK\begin{array}{|c|c|} \hline Geographical\ regions & Before\ vaccination & After\ vaccination\\ \hline 1 & 85 & 11\\ \hline 2 & 77 & 5\\ \hline 3 & 110 & 14\\ \hline 4 & 65 & 12\\ \hline 5 & 81 & 10\\\hline 6 & 70 & 7\\ \hline 7 & 74 & 8\\ \hline 8 & 84 & 11\\ \hline 9 & 90 & 9\\ \hline 10 & 95 & 8\\ \hline \end{array}ZSK
Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following: Construct a one-sided $$\displaystyle{95}\%$$ confidence interval for the true difference in population means. Test the null hypothesis that the population means are identical at the 0.05 level of significance. Determine the algebraic modeling which of the following data sets are linear and which are exponential. For the linear sets, determine the slope. For the exponential sets, determine the growth factor or the decay factor
a) $$\begin{array}{|c|c|}\hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & \frac{1}{9} & \frac{1}{3} & 1 & 3 & 9 & 27 & 81 \\ \hline \end{array}$$ b) $$\begin{array}{|c|c|}\hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & 2 & 2.6 & 3.2 & 3.8 & 4.4 & 5.0 & 5.6 \\ \hline \end{array}$$
c) $$\begin{array}{|c|c|}\hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & 3.00 & 5.0 & 7 & 9 & 11 & 13 & 15 \\ \hline \end{array}$$
d) $$\begin{array}{|c|c|}\hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & 5.25 & 2.1 & 0.84 & 0.336 & 0.1344 & 0.5376 & 0.021504 \\ \hline \end{array}$$ Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of $$\alpha = 0.05$$. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.) Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities? $$\begin{matrix} \text{Lemon Imports} & 230 & 265 & 358 & 480 & 530\\ \text{Crashe Fatality Rate} & 15.9 & 15.7 & 15.4 & 15.3 & 14.9\\ \end{matrix}$$ The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations 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 654.1 vehicles per hour, with a standard deviation of 311.7 vehicles per hour.
a) Find a $$\displaystyle{95}\%$$ confidence interval for the improvement in traffic flow due to the new system.
b) Find a $$\displaystyle{98}\%$$ confidence interval for the improvement in traffic flow due to the new system.
c) A traffic engineer states that the mean improvement is between 581.6 and 726.6 vehicles per hour. With what level of confidence can this statement be made?
d) Approximately what sample size is needed so that a $$\displaystyle{95}\%$$
confidence interval will specify the mean to within $$\displaystyle\pm\ {50}$$ vehicles per hour?
e) Approximately what sample size is needed so that a $$\displaystyle{98}\%$$ confidence
interval will specify the mean to within $$\displaystyle\pm\ {50}$$ vehicles per hour? Iron is very important for babies' growth. A common belief is that breastfeeding will help the baby to get more iron than formula feeding. To justify the belief, a study followed 2 groups of babies from born to 6 months. With one group babies are breast fed, and the other group are formula fed without iron supplements. Data below shows iron levels of those two groups of babies. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{G}{r}{o}{u}{p}&{S}{a}\mp\le\ {s}{i}{z}{e}&{m}{e}{a}{n}&{S}{\tan{{d}}}{a}{r}{d}\ {d}{e}{v}{i}{a}{t}{i}{o}{n}\backslash{h}{l}\in{e}{B}{r}{e}\ast-{f}{e}{d}&{23}&{13.3}&{1.7}\backslash{h}{l}\in{e}{F}{\quad\text{or}\quad}\mu{l}{a}-{f}{e}{d}&{23}&{12.4}&{1.8}\backslash{h}{l}\in{e}{D}{I}{F}{F}={B}{r}{e}\ast-{F}{\quad\text{or}\quad}\mu{l}{a}&{23}&{0.9}&{1.4}\backslash{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ (1) There are two groups we need to compare for the study: Breast-Fed and Formula- Fed. Are those two groups dependent or independent? Based on your answer, what inference procedure should we apply for this research? (2) Please perform the inference you decided in (1), and make sure to follow the 5-step procedure for any hypothesis test. (3) Based on your conclusion in (2), what kind of error could you make? Explain the type of error using the context words for this research n an experiment designed to study the effects of illumination level on task performance (“Performance of Complex Tasks Under Different Levels of Illumination,” J. Illuminating Eng., 1976: 235–242), subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\mathcal}\right\rbrace}{h}{l}\in{e}&{a}\mp&{a}\mp&{a}\mp\ \text{Subject}\backslash{h}{l}\in{e}&{a}\mp\ {1}&{a}\mp\ {2}&{a}\mp\ {3}&{a}\mp\ {4}&{a}\mp\ {5}&{a}\mp\ {6}&{a}\mp\ {7}&{a}\mp\ {8}&{a}\mp\ {9}&{a}\mp\backslash{h}{l}\in{e}\text{Black}&{a}\mp\ {25.85}&{a}\mp\ {28.84}&{a}\mp\ {32.05}&{a}\mp\ {25.74}&{a}\mp\ {20.89}&{a}\mp\ {41.05}&{a}\mp\ {25.01}&{a}\mp\ {24.96}&{a}\mp\ {27.47}&{a}\mp\backslash{h}{l}\in{e}\text{White}&{a}\mp\ {18.23}&{a}\mp\ {20.84}&{a}\mp\ {22.96}&{a}\mp\ {19.68}&{a}\mp\ {19.509}&{a}\mp\ {24.98}&{a}\mp\ {16.61}&{a}\mp\ {16.07}&{a}\mp\ {24.59}&{a}\mp\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Does the data indicate that the higher level of illumination yields a decrease of more than 5 sec in true average task completion time? Test the appropriate hypotheses using the P-value approach. In an experiment designed to study the effects of illumination level on task performance (“Performance of Complex Tasks Under Different Levels of Illumination,” J. Illuminating Eng., 1976: 235–242), subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task.
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{S}{u}{b}{j}{e}{c}{t}&{\left({1}\right)}&{\left({2}\right)}&{\left({3}\right)}&{\left({4}\right)}&{\left({5}\right)}&{\left({6}\right)}&{\left({7}\right)}&{\left({8}\right)}&{\left({9}\right)}\backslash{h}{l}\in{e}{B}{l}{a}{c}{k}&{25.85}&{28.84}&{32.05}&{25.74}&{20.89}&{41.05}&{25.01}&{24.96}&{27.47}\backslash{h}{l}\in{e}{W}{h}{i}{t}{e}&{18.28}&{20.84}&{22.96}&{19.68}&{19.509}&{24.98}&{16.61}&{16.07}&{24.59}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
Does the data indicate that the higher level of illumination yields a decrease of more than 5 sec in true average task completion time? Test the appropriate hypotheses using the P-value approach. A random sample of $$\displaystyle{n}_{{1}}={16}$$ communities in western Kansas gave the following information for people under 25 years of age.
$$\displaystyle{X}_{{1}}:$$ Rate of hay fever per 1000 population for people under 25
$$\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}$$
A random sample of $$\displaystyle{n}_{{2}}={14}$$ regions in western Kansas gave the following information for people over 50 years old.
$$\displaystyle{X}_{{2}}:$$ Rate of hay fever per 1000 population for people over 50
$$\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}$$
(i) Use a calculator to calculate $$\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.$$ (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $$\displaystyle\alpha={0.05}.$$
(a) What is the level of significance?
State the null and alternate hypotheses.
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}$$
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value $$\displaystyle>{0.250}$$
$$\displaystyle{0.125}<{P}-\text{value}<{0},{250}$$
$$\displaystyle{0},{050}<{P}-\text{value}<{0},{125}$$
$$\displaystyle{0},{025}<{P}-\text{value}<{0},{050}$$
$$\displaystyle{0},{005}<{P}-\text{value}<{0},{025}$$
P-value $$\displaystyle<{0.005}$$
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value
...