# High IQ Exercise 26 proposes modeling IQ scores with N(100, 16). What IQ would you consider to be unusually high? Explain. Question
Modeling High IQ Exercise 26 proposes modeling IQ scores with N(100, 16). What IQ would you consider to be unusually high? Explain. 2021-02-26
Step 1 Any IQ more than 132 might be considered as unusually high. The given information is that the modeling IQ scores follow normal distribution with mean $$\mu=16\ \text{and standard deviation} \sigma = 16.$$ Let the random variable Y represents the IQ scores. The IQ more than three standard deviations below the mean is expected to see very rarely. The corresponding IQ can be obtained as follows: $$\mu\ +\ 3 \sigma = 100\ +\ 3(16) = 100\ +\ 48 = 148$$ Thus, there is very rare chance to find someone with an IQ over 148. Step 2 Generally any IQ scores more than two standard deviations above the mean is considered as unusually high. The corresponding IQ score can be obtained as follows: $$\mu\ +\ 2\ \sigma = 100\ +\ 2(16) = 100\ +\ 32 = 132$$ Thus, there are more than 132 are considered as unusually high IQ scores.

### Relevant Questions Consider that math modeling following initial valu problem
$$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{t}\right.}}={3}-{2}{t}-{0.5}{y},{y}{\left({0}\right)}={1}$$
We would liketo find an approximation solution with the step siza $$h = 0.05$$
What is the approximation of $$y(0.1)?$$ Illustrate and explain how you would introduce division of fractions by modeling $$\frac{3}{4} -: \frac{1}{6}$$ with Fraction Bars. Consider a capital budgeting problem with six projects represented by $$0-1\ \text{variables}\ x1,\ x2,\ x3,\ x4,\ x5,\ \text{and}\ x6.$$
a. Write a constraint modeling a situation in which two of the projects 1, 3, and 6 must be undertaken.
b. In which situation the constraint "$$x3\ -\ x5 = 0$$" is used, explain clearly:
c. Write a constraint modeling a situation in which roject 2 or 4 must be undertaken, but not both.
d. Write constraints modeling a situation where project 1 cannot be undertaken IF projects 3. also is NOT undertaken.
e. Explain clearly the situation in which the following 3 constraints are used simulataneously (together):
$$\displaystyle{x}{4}\le{x}{1}$$
$$\displaystyle{x}{4}\le{x}{3}$$
$$\displaystyle{x}{4}\ge{x}{1}+{x}{3}-{1}$$ Explain what changes would be required so that you could analyze the hypothesis using a chi-square test. For instance, rather than looking at test scores as a range from 0 to 100, you could change the variable to low, medium, or high. What advantages and disadvantages do you see in using this approach? Which is the better option for this hypothesis, the parametric approach or nonparametric approach? Mathematical modeling is about constructing one or two equations that represent real life situations. What are these math models used for? Provide at least two equations that can be used in the real world. For example: The equation $$s = 30\ h\ +\ 1000$$ can be used to find your salary given the fact you earn a fixed salary of $1000 per month, plus$30 per hours. Here s represents the total salary and h is the number of hours you worked. Consider a capital budgeting problem with seven projects represented by binary (0 or 1) variables $$X_{1},\ X_{2},\ X_{3},\ X_{4},\ X_{5}, X_{6}, X_{7}$$. Write a constraint modeling the situation in which only 2 of the projects from $$1,\ 2,\ 3\ and\ 4$$ must be selected. Write a constraint modeling the situation in which at least 2 of the project from $$1,\ 3,\ 4,\ and\ 7$$ must be selected. Write a constraint modeling the situation project 3 or 6 must be selected, but not both. Write a constraint modeling the situation in which at most 4 projects from the 7 can be selected. 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?  Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p $$200 53.00$$
$$250 52.50$$
$$300 52.00$$
$$35051.50$$ (a) Find a formula for pin terms of N modeling the data in the table. (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. $$R=$$ Is Ra linear function of N? (c) On the basis of the tables in this exercise and using cost, $$C= 35N + 900$$, use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month $$p=$$ (d) Is Pa linear function of N2 e. Explain how you would find breakeven. What does breakeven represent? 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? $$\begin{array}{|c|c|}Lemon imports &230&265&368&480&630\\ Crash FatalityRate&159&157&15.3&15.4&14.9\end{array}$$