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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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- The U.S. government wants to determine whether immigrants should be tested for a contagious disease, and it is planning to base this decision on financial considerations. Assume that each immigrant who is allowed to enter the United States and has the disease costs the country $100,000. Also, assume that each immigrant who is allowed to enter the United States and does not have the disease will contribute $10,000 to the national economy. Finally, assume that x percent of all potential immigrants have the disease. The U.S. government can choose to admit all immigrants, admit no immigrants, or test immigrants for the disease before determining whether they should be admitted. It costs T dollars to test a person for the disease, and the test result is either positive or negative. A person who does not have the disease always tests negative. However, 10% of all people who do have the disease test negative. The government’s goal is to maximize the expected net financial benefits per…Ten years ago, the town of Easton decided to increase its annual spending on education so that its high school graduates would be able to earn higher wages. Now Easton has asked you to evaluate the effectiveness of the spending increase. Their data show that before the spending increase, the average annual salary of recent high school graduates was $25,000 and that now that average salary has risen to $28,500. Fortunately for your analysis, a neighboring community (Allentown) did not change its annual spending on education. Ten years ago, recent Allentown high school graduates earned an average of $22,500, and now that average is $23,750. (b) What underlying assumption do you have to make in order for your estimate to be valid? What might cause your underlying assumption to be invalid?Ten years ago, the town of Easton decided to increase its annual spending on education so that its high school graduates would be able to earn higher wages. Now Easton has asked you to evaluate the effectiveness of the spending increase. Their data show that before the spending increase, the average annual salary of recent high school graduates was $25,000 and that now that average salary has risen to $28,500. Fortunately for your analysis, a neighboring community (Allentown) did not change its annual spending on education. Ten years ago, recent Allentown high school graduates earned an average of $22,500, and now that average is $23,750. (a) Use a difference-in-differences estimator to determine whether Easton’s spending increase caused the wages of their high school graduates to increase. please provide an accurate answer.
- A manufacturing company has been experimenting with a new additive that is used to reduce polluting effluents from its production facility. Test results suggest that if no additive is used, pollution levels are 3000 parts per million (ppm). For each pound of additive used per week, pollution levels are reduced by 0.49 ppm. The additive costs $30 per pound. The reason for experimenting with the additive is that a government agency is conducting weekly air quality tests in the vicinity of the production facility. Regulations state that if pollution levels exceed 2000 ppm, a fine is imposed. The fine is $1000 for each ppm by which the pollution level exceeds the 2000 ppm limit. If the company has budgeted $80,000 per week to spend on the additive and any pollution fines, how much additive should be purchased each week for use by the company? What is the resulting amount that the company is willing to pay in terms of weekly fines?A group of scientists studied the effect of a chemical on various strains of bacteria. Bacteria A started with 12000 cells and decreased at a constant rate of 3000 cells per hour after the chemical was applied. Strain B started with 4000 cells and decreased at a constant rate of 2000 cells per hour after the chemical was applied. When will the strains have the same number of cells?The U.S. Bureau of the Census prediction for the percentage of the population 65 years and older can be modeled as p(x) = −0.00022x3 + 0.014x2 − 0.0033x + 12.236 percent where x is the number of years since 2000, data from 0 ≤ x ≤ 50. 1. Determine the value of x in the domain 0 ≤ x ≤ 50 for which the percentage is predicted to be increasing most rapidly (Round your answer to three decimal places). 2. Thus, in which year is the percentage is predicted to be increasing most rapidly? 3. Calculate the percentage at that time. (Round your answer to two decimal places.) 4. Calculate the rate of change of the percentage at that time. (Round your answer to three decimal places.)
- John Swanson, president of Market Research Inc., has asked you to estimate the coefficients of the modelY = β0 + β1X1 + β2X21 + b3X2where Y is the expected sales of office supplies for a large retail distributor of office supplies, X1 is the total disposable income of residents within 5 miles of the store, and X2 is the total number of persons employed in information-based businesses within 5 miles of the store. Recent work by a national consulting firm has concluded that the coefficients in the model must have the following restriction:b1 + b2 = 2Describe how you would estimate the model coefficients using least squares.In a hypothetical city, there is a population of 200,000 people. During the year 2000, there were 1,000 deaths from all causes. All new cases of disease “X” have been accounted for and they total 600 (400 males and 200 females). During the year 2000 there were 120 deaths from disease “X”, 100 of them in males. What is the crude death rate for all causes per 1,000 population?The population, P, of the city of Hazelton has grown according to the mathematical model P = 65,000(1.075)*, where t is the number of years since 2005. a) At what percentage is the population growing? b) If this trend continues, during which year will the population reach approx. 100,000?
- The car manufacturer Edison is the sole producer of electric cars in the market, selling to two different types of customers. All else equal, travelers (type t) prefer batteries that go longer distances between charges and are willing to pay for this luxury. Urban drivers (type u) do not require as great a battery. Edison knows the willingness-to-pay of each type, and that there are q percent of type ts in the market. The willingness-to-pay for each type for different battery sizes are, Vt(k = 100) = $160, 000, Vt(k = 60) = $80, 000, Vu(k = 100) = $100, 000, Vu(k = 60) = $60, 000, where k is the battery life in kilowatt hours (kWh). Producing 60 kWh battery costs $30,000, which is half the cost of a 100 kWh battery. (a) Edison showroom salesmen believe they can perfectly identify (1st/3rd-degree) which type of buyer walks through the door. Find the profit-maximizing prices (pi) and battery sizes (ki). What is Edison’s average profit per customer? (b) It turns out Edison salesmen are…Data were collected from a group of senior citizens, ages 65 and older, about their age and pet ownership. The results were used to calculate the model ŷ = –0.24x + 18.48, where x represents the age in years and ŷ represents the predicted number of pets owned. If a 45-year-old is predicted to have more than 7 pets, would that be a reasonable use of the model? This is not reasonable because the prediction is based on someone outside the original population. This is reasonable because the model is based on real data. This is not reasonable because it is impossible to own more than 7 pets. This is reasonable because the prediction is based on someone within the original population.A clinical trial is being conducted in order to determine the efficacy of a new drug that will be used to treat rheumatoid arthritis. The efficacy of the medication will not only be determined by the physical improvement of symptoms but will also be determined by using a blood test to examine the concentration of c-reactive protein in an individual's blood. If the researcher wants a margin of error for the level of c-reactive protein to be less than or equal to 2.0 mg/d and if the standard deviation for c-reactive protein concentrations among arthritis patients was previously documented as 7 mg/dl, how many patients should be recruited for each group of individuals in the study, assuming a 95% confidence interval will be used to quantify the mean differences between the control group and the treatment group? A. n for treatment group = 95 and n for control group = 95 B. n treatment group = 48 and control group = 48 C. n treatment group = 47 and control = 48 D. n treatment group = 48…