Learn Analyzing Functions with These Examples

Regarding analysis of residuals, decide in each case which assumption for regression inferences may be violated. a. A residual plot-that is, a plot of the residuals against the observed values of the predictor variable-shows curvature. b. A residual plot becomes wider with increasing values of the predictor variable. c. A normal probability plot of the residuals shows extreme curvature. d. A normal probability plot of the residuals shows outliers but is otherwise roughly linear.

Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.
${\displaystyle f(x,y)=y{e}^x-e^y}$

Error Analysis A student evaluated $-4+xforx=-9$ and got an answer of 5. What might the student have done wrong?

If the annual percentage yield is 5%, what will $10,000 be worth 4 years form now?

Let S be an ordered set and A is a nonempty subset such that sup A exists. Suppose there is a ${\displaystyle B\subset A}$ such that whenever ${\displaystyle x\in A}$ there is a ${\displaystyle y\in B}$ such that ${\displaystyle x\le y}$. Show that ${\displaystyle \supset B}$ exists and ${\displaystyle \supset B=\supset A}$.

Find the critical points of the following functions. Use the Second Derivative Test to determine whether each critical point corresponds to a loal maximum, local minimum, or saddle point.
${\displaystyle f(x,y)=x^4+2y^2-4xy}$

Some prime numbers are odd. 2 is a prine number. Therefore, 2 is an odd number.

Geographical Analysis (Oct. 2006) published a study of a new method for analyzing remote-sensing data from satellite pixels in order to identify urban land cover. The method uses a numerical measure of the distribution of gaps, or the sizes of holes, in the pixel, called lacunarity. Summary statistics for the lacunarity measurements in a sample of 100 grassland pixels are $\overline{x}=225ands=20s=20$. It is known that the mean lacunarity measurement for all grassland pixels is 220. The method will be effective in identifying land cover if the standard deviation of the measurements is 10% (or less) of the true mean (i.e., if the standard deviation is less than 22).

a. Give the null and alternative hypotheses for a test to determine whether, in fact, the standard deviation of all grassland pixels is less than 22.

b. A MINITAB analysis of the data is provided below. Locate and interpret the p-value of the test. Use $\alpha =0.10$. Test for One Standard Deviation Method Null hypothesis $\mathrm{\Sigma }=22$ Method Alternative hypothesis $\mathrm{\Sigma }\le 22$ The standard method is only for the normal distribution. Statistics NStDevVariance 10020.0400 Tests

For the graphs ${\displaystyle x=\left|y\right|\text{and}2x=-y^2+2}$
(a) Sketch the enclosed region, showing all the intersection and boundary points.
(b) Find the area of the enclosed region by integrating along the y-axis

$3450 shares were bought and than sold for $6100 after a while. What is the capital gain as a percent of the original purchace price? What is the gross capital gain?

gestationperiod mean birthweight standard deviation
under 28weeks 1.88lbs 1.20lbs
28-31weeks 4.08lbs 1.86lbs
32-35weeks 5.71 1.47lbs
36weeks 6.44lbs 1.19lbs
37-39weeks 7.31lbs 1.09lbs
40weeks 7.69 1.04
41weeks 7.79 1.08lbs
42 weeks andover 7.61 1.11lbs
1. what percent of the babies born within each gestationperiod have a low birth weight (under 5.5 lbs)? Explain yourreasoning.
a. under 28 weeks
b.32-35 weeks
c. 37to 39 weeks
d. 42 weeks and over
2. Describe the weights of the top 10% of the babies born witheach gestation period. Expalin your reasoning.
a. 37 to 39 weeks
b. 42 weeks and over
3. For each gestation period, what is the probability that ababy will weigh between 6 and 9 lbs at birth?
a. 32 to 35 weeks
b. 37to 39 weeks
c. 42 weeks and over
4. A birth weight of less than 3.3 lbs is classified by theNCHS as a " very low birth weight." What is the probability that ababy has a very low birth weight for each gestation period?
a. under 28 weeks
b. 32 to 35 weeks
c. 37 to 39 weeks

An analysis of variance produces ${\displaystyle S{S}_{total}=40}$ and ${\displaystyle S{S}_{within}=10}$. For rhis analysis, what is ${\displaystyle S{S}_{between}}$?
a) 30
b) 400
c) Cannot be determined grom the information given
d) 50

Which of the following stetements bset descibes correlation analysis in a simple linear regression
a. Correlation analysis measures the strenght of relationship between two categorical variables.
b. Correlation analysis measures the direction of relationship between two numerical variables.
c. Correlation analysis measures the strenght and direction of relationship between two numerical variables.
d. Correlation analysis measures the strenght of relationship between two numerical variables."