For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population maybe modeled by the following function: y = 12.000 + 8.000 sin(0.628x), where the domain is the years since 1980 and the range is the population of the city. Graph the function on the domain of [0,40]

Question
Comparing two groups
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population maybe modeled by the following function: $$y = 12.000 + 8.000 \sin(0.628x)$$, where the domain is the years since 1980 and the range is the population of the city.
Graph the function on the domain of [0,40]

2021-02-21
Given:
The population of a city risen and fallen over a 20-year interval. Its populations may be modeled by the function $$y = 12000 + 8000 \sin (0.628x)$$, where domain is the year since 1980 and the range is the population of the city.
Calculation:
The given function is $$y = 12000 + 8000 \sin (0.628x).$$
Comparing, the function with the standard form of the sinusoidal function $$y = A \sin (Bx - C) + D$$
$$A = 8000, B = 0.628, C = 0, D = 12000$$
The amplitude is $$A = 8000$$
The periodis given by
$$\frac{2 \pi}{B}$$
$$\frac{2 \pi}{0.628}= \frac{3.14}{0.314}$$
$$= 10$$
And the phase shift is $$C = 0$$
Using these properties, the graph of the function in the domain [0,40] has been shown below

Relevant Questions

For Questions 1 — 2, use the following. Scooters are often used in European and Asian cities because of their ability to negotiate crowded city streets. The number of scooters (in thousands) sold each year in India can be approximated by the function $$N = 61.86t^2 — 237.43t + 943.51$$ where f is the number of years since 1990. 1. Find the vertical intercept. What is the practical meaning of the vertical intercept in this situation? 2. Use a numerical method to find the year when the number of scooters sold reaches 1 million. (Note that 1 million is 1,000 thousand, so N = 1000) Show three rows of the table you used to support your answer and write a clear answer to the problem.
The population P (in thousands) of Tallahassee, Florida, from 2000 through 2014 can be modeled by $$P = 150.9e^{kt},$$ where t represents the year, with $$t = 0$$ corresponding to 2000. In 2005, the population of Tallahassee was about 163,075.
(a) Find the value of k. Is the population increasing or decreasing? Explain.
(b) Use the model to predict the populations of Tallahassee in 2020 and 2025. Are the results reasonable? Explain.
(c) According to the model, during what year will the — populates reach 200,000?
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.
Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13,$$ where $$t = 0$$ corresponds to 2000.
(Source: U.S. Census Bureau)
Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013.
Analytically find the maximum and minimum populations over the interval.
The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
We need to calculate: The absolute extrema of the popullation $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13$$ over the closed interval [0, 13].
Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13,$$ where $$t = 0$$ corresponds to 2000.
(Source: U.S. Census Bureau)
Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013.
Analytically find the maximum and minimum populations over the interval.
The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
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.
True // False: comparing means. Determine if the following statements are true or false, and explain your reasoning for statements you identify as false.
(a) When comparing means of two samples where $$n_{1} = 20$$
and $$n_{2} = 40$$,
we can use the normal model for the difference in means since $$n_{2} \geq 30.$$
(b) As the degrees of freedom increases, the t-distribution approaches normality.
(c) We use a pooled standard error for calculating the standard error of the difference between means when sample sizes of groups are equal to each other.
Use a calculator with a $$y^x$$ key or a key to solve: India is currently one of the world’s fastest-growing countries. By 2040, the population of India will be larger than the population of China, by 2050, nearly one-third of the world’s population will live in these two countries alone. The exponential function $$f(x)=574(1.026)^x$$ models the population of India, f(x), in millions, x years after 1974.
The following quadratic function in general form, $$\displaystyle{S}{\left({t}\right)}={5.8}{t}^{2}—{81.2}{t}+{1200}$$ models the number of luxury home sales, S(t), in a major Canadian urban area, according to statistical data gathered over a 12 year period. Luxury home sales are defined in this market as sales of properties worth over \$3 Million (inflation adjusted). In this case, $$\displaystyle{\left\lbrace{t}\right\rbrace}={\left\lbrace{0}\right\rbrace}{Z}{S}{K}\ \text{represents}\ {2000}{\quad\text{and}\quad}{\left\lbrace{t}\right\rbrace}={\left\lbrace{11}\right\rbrace}$$represents 2011. Use a calculator to find the year when the smallest number of luxury home sales occurred. Without sketching the function, interpret the meaning of this function, on the given practical domain, in one well-expressed sentence.