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Modeling data distributions

Random variables

Two-way tables

### Baseball star David Ortiz-nicknamed "Big Papi"-is known for his ability to deliver hits in high-pressure situations. Here is a two-way table of his hits, walks, and outs in all of his regular-season and post-season plate appearances from 1997 through 2014. Choose a plate appearance at random. Are the events "Hit" and "Post-season" independent? Justify your answer.

Confidence intervals

### Do piano lessons improve the spatial-temporal reasoning of preschool children? A study designed to investigate this question measured the spatial-temporal reasoning of a random sample of 34 preschool children before and after 6 months of piano lessons. The differences (After - Before) in the reasoning scores have mean 3.618 and standard deviation 3.055.

Comparing two groups

### The mean + 1 sd of In [calcium intake (mg)] among 25 females, 12 to 14 years of age, below the poverty level is $$6.56 + 0.64.$$ Similarly, the mean + 1 sd of In [calcium intake (mg)] among 40 females, 12 to 14 years of age, above the poverty level is $$6.80 + 0.76.$$ 8.2 Test for a significant difference between the variances of the two groups. 8.3 What is the appropriate procedure to test for a signifi- cant difference in means between the two groups? 8.4 Implement the procedure in Problem 8.3 using the critical-value method. 8.5 What is the p-value corresponding to your answer to Problem 8.4? 8.6 Compute a $$95\%$$ Cl for the difference in means between the two groups.

Analyzing categorical data

### Is the mode the measure of central tendency that is best used with a skewed distribution?

Analyzing categorical data

### Describe the summary of the perfomance of eight graders nationwide.

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.

Confidence intervals

### The local energy company claims the average annual electricity bill for its subscribers is just $600. A consumer watchdog group wants to dispute this claim. All agree that the standard deviation sigma of annual electricity bills is$150. Some time later, a wealthy activist provides funding for a simple random sample of 250 households. The average annual electricity bill for this sample is \$622. Find a $$95\%$$ confidence interval for the true mean annual electric bill, based on this sample.

Alternate coordinate systems

### Convert the point from spherical coordinates to rectangular coordinates. $$(9, \pi, \frac{\pi}{2})$$ $$(x, y, z) = ?$$

Confidence intervals

### Let $$f(x)=4−\frac{2}{x}+\frac{6}{x^{2}}$$. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). f is increasing on the intervals f is decreasing on the intervals The relative maxima of f occur at x = The relative minima of f occur at x = Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none". In the last two, your answer should be a comma separated list of x values or the word "none".

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?

Sampling distributions

### 1) Describe sampling distributions and sampling variavility 2) Explain The Central Limit Theorem 3) Explain how confidence intervals are created and what can they tell us about population parameters

Confidence intervals

### Calculate confidence intervals for ratio of two population variances and ratio of standard deviations. Assume that samples are simple random samples and taken from normal populations. a) $$\displaystyle\alpha={0.05},\ {n}_{{{1}}}={30},\ {s}_{{{1}}}={16.37},\ {n}_{{{2}}}={39},\ {s}_{{{2}}}={9.88}$$ b) $$\displaystyle\alpha={0.01},\ {n}_{{{1}}}={25},\ {s}_{{{1}}}={5.2},\ {n}_{{{2}}}={20},\ {s}_{{{2}}}={6.8}$$

Confidence intervals

Random variables

### A distribution of values is normal with a mean of 203.8 and a standard deviation of 50.2. Find the probability that a randomly selected value is less than 113.4. $$\displaystyle{P}{\left({X}{<}{113.4}\right)}=$$? Write your answers as numbers accurate to 4 decimal places.

Comparing two groups

### Decide equation using only factors to solve this problem $$12x^{2}\ +\ 5x\ -\ 2 = 0$$

Modeling data distributions