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Modeling data distributions

### a) To calculate: The least squares regression line for the data points using the table given below. \begin{array}{|c|c|} \hline Fertilizer & x & 100 & 150 & 200 & 250 \\ \hline Yield & y & 35 & 44 & 50 & 56 \\ \hline \end{array} b)To calculate: The approximate yield when 175 pounds of fertizers were used per acre of land.

Modeling data distributions

### An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by $$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$ $$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$ ​(A) Complete the table below. $$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$ ​(Round to one decimal place as​ needed.) $$A. 20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate. $$B. 20602060xf(x)$$ Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate. $$C. 20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate. $$D.20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate. ​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29 $$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35 $$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is (Round to two decimal places as​ needed.) (D) Write a brief description of the relationship between tire pressure and mileage. A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase. B. As tire pressure​ increases, mileage decreases. C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease. D. As tire pressure​ increases, mileage increases.

Modeling data distributions

### 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?

Modeling data distributions

### Define the term Mean.

Modeling data distributions

### It is estimated that aproximately $$\displaystyle{8.36}\%$$ Americans are afflicted with Diabetes . Suppose that a ceratin diagnostic evaluation for diabetes will correctly diagnose $$\displaystyle{94.5}\%$$ of all adults over 40 with diabetes as having the disease and incorrectly diagnoses $$\displaystyle{2}\%$$ of all adults over 40 without diabetes as having the disease . 1) Find the probability that a randamly selected adult over 40 doesn't have diabetes and is diagnosed as having diabetes ( such diagnoses are called "false positives"). 2) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes. 3) Find the probability that a randomly selected adult over 40 actually has diabetes , given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives"). Note: It will be helpful to first draw an appropriate tree diagram modeling the situation.

Modeling data distributions

### Decide which of the following statements are true. Answer: square Normal distributions are bell-shaped, but they do not have to be symmetric. square The line of symmetry for all normal distributions is x square On any normal distribution curve, you can find data values more than 5 standard v square deviations above the mean. square The x-axis is a horizontal asymptote for all normal distributions.

Modeling data distributions

### M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing Theory to Reservation Networks” (TIMS, Vol. 22, No. 5, pp. 540–546). They defined a Type 1 call to be a call from a motel’s computer terminal to the national reservation center. For a certain motel, the number, X, of Type 1 calls per hour has a Poisson distribution with parameter $$\displaystyle\lambda={1.7}$$. Determine the probability that the number of Type 1 calls made from this motel during a period of 1 hour will be: a) exactly one. b) at most two. c) at least two. (Hint: Use the complementation rule.) d. Find and interpret the mean of the random variable X. e. Determine the standard deviation of X.

Modeling data distributions

### A parks and recreation department is constructing a new bike path. The path will be parallel to the railroad tracks shown and pass through the parking area al the point $$\displaystyle{\left({4},\ {5}\right)}.$$ Write an equation that represents the path.

Modeling data distributions

### 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.

Modeling data distributions

### To determine: The number of luxury home sales S(t) in a major Canadian urban area over a period of 12 year is given by: $$\displaystyle\Rightarrow\ {S}{\left({t}\right)}={5.8}\ {t}^{{{2}}}\ -\ {81.2}\ {t}\ +\ {1200}$$

Modeling data distributions

### Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. $$\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}$$ (a) Find a formula for p in terms of N modeling the data in the table. $$\displaystyle{p}=$$ (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month. $$\displaystyle{R}=$$ Is R a linear function of N? (c) On the basis of the tables in this exercise and using cost, $$\displaystyle{C}={35}{N}+{900}$$, use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month. $$\displaystyle{P}=$$ (d) Is P a linear function of N?

Modeling data distributions

### 1)What factors influence the correspondence between the binomial and normal distributions? 1.Twenty percent of individuals who seek psychotherapy will recover from their symptoms irrespective of whether they receive treatment. A research finds that a particular type of psychotherapy is successful with 30 out of 100 clients. Using an alpha level of 0.05 as a criterion, what should she conclude about the effectiveness of this psychotherapeutic approach? 2.How does the size of the data set help cut down on the size of the error terms in the approximation process?

Modeling data distributions

### An experiment designed to study the relationship between hypertension and cigarette smoking yielded the following data. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{T}{e}{n}{s}{i}{o}{n}\ \le{v}{e}{l}&{N}{o}{n}-{s}{m}{o}{k}{e}{r}&{M}{o}{d}{e}{r}{a}{t}{e}\ {s}{m}{o}{k}{e}{r}&{H}{e}{a}{v}{y}\ {s}{m}{o}{k}{e}{r}\backslash{h}{l}\in{e}{H}{y}{p}{e}{r}{t}{e}{n}{s}{i}{o}{n}&{20}&{38}&{28}\backslash{h}{l}\in{e}{N}{o}\ {h}{y}{p}{e}{r}{t}{e}{n}{s}{i}{o}{n}&{50}&{27}&{18}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Test the hypothesis that whether or not an individual has hypertension is independent of how much that person smokes.

Modeling data distributions

### The following observations are lifetimes (days) subsequent to diagnosis for individuals suffering from blood cancer ("A Goodness of Fit Approach to the Class of Life Distributions with Unknown Age," Quality and Reliability Engr. Intl., $$2012: 761-766): 115, 181, 255, 418, 441, 461, 516, 739, 743, 789, 807, 865, 924, 983, 1025, 1062, 1063, 1165, 1191, 1222, 1222, 1251, 1277, 1290, 1357, 1369, 1408, 1455, 1278, 1519, 1578, 1578, 1599, 1603, 1605, 1696, 1735, 1799, 1815, 1852, 1899, 1925, 1965.$$ a) can a confidence interval for true average lifetime be calculated without assuming anything about the nature of the lifetime distribution? Explain your reasoning. [Note: A normal probability plot of data exhibits a reasonably linear pattern.] b) Calculate and interpret a confidence interval with a 99% confidence level for true average lifetime. [Hint: mean = 1191.6, s = 506.6.]

Modeling data distributions

### Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. $$\displaystyle{\left({1},{3}\right)},{\left({2},{6}\right)},{\left({3},{12}\right)},{\left({4},{24}\right)}$$ Part A: Is this data modeling an algebraic sequence or a geometric sequence? Explain your answer. Part B: Use a recursive formula to determine the time she will complete station 5. Part C: Use an explicit formula to find the time she will complete the 9th station.

Modeling data distributions

### In general, the highest price p per unit of an item at which a manufacturer can sell N items is not constant but is, rather, a function of N. Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. $$\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 250 & 52.50\\ \hline300 & 52.00\\\hline 350 & 51.50\\ \hline 400 & 51.00\\ \hline \end{array}$$ (a) Find a formula for p in terms of N modeling the data in the table. $$\displaystyle{p}=$$ (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month. $$\displaystyle{R}=$$

Modeling data distributions

### Means $$=59.47444, SDS = 12.91711, Min = 23.599, Max = 82.603$$ AND Means $$= 67.00742, SDS = 12.7302, Min = 39.613, Max = 82.603$$ Means Based on your findings write a preliminary statistical report that comprises of the Methods, Results/Analysis and Conclusions sections. Include a comparison of the two distributions in (1) and (2) in terms of their central tendencies and variability. While your audience is one that lacks statistical expertise you are still expected to correctly interpret the data and statistical analyses, in a manner that is understandable to your audience. Be mindful to present an impartial report that distinguishes conclusive and inferential statements for the audience.

Modeling data distributions

### The volume of a sphere is given bby the equation $$\displaystyle{V}\ =\ {\frac{{{1}}}{{{6}\sqrt{{\pi}}}}}\ {S}^{{\frac{{3}}{{2}}}}$$, where S is the surface area of the sphere. Find the volume of a sphere, to the nearest cubic meter, that has a surface area of 60 square meter. Use 3.14 for $$\displaystyle\pi$$.

Modeling data distributions