# Using the health records of ever student at a high school, the school nurse created a scatterplot relating y= text{height (in centimeters) to} x= text{age (in years).} text{After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be} mu_{0}=105 + 4.2x text{with} sigma=7 cm. About what percent of 15-year-old students at this school are taller than 180 cm?

Question
Scatterplots
Using the health records of ever student at a high school, the school nurse created a scatterplot relating $$\displaystyle{y}=\ \text{height (in centimeters) to}\ {x}=\ \text{age (in years).}$$
$$\displaystyle\text{After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be}\ \mu_{{{0}}}={105}\ +\ {4.2}{x}\ \text{with}\ \sigma={7}\ {c}{m}.$$ About what percent of 15-year-old students at this school are taller than 180 cm?

2020-11-11
Given: (Equation population regression line): $$\displaystyle\mu_{{{y}}}={105}\ +\ {4.2}{x}$$
$$\displaystyle\sigma={7}$$ The average height of 15-year-old students at this high scool according to the population regression line can be found by replacing x in the regression line equation by 15 and evaluating. $$\displaystyle\mu_{{{y}}}={105}\ +\ {4.2}{\left({15}\right)}={105}\ +\ {63}={168}$$ Thus the mean is 168 and the standard deviation is7. Since the conditions are met, the response y varies according to a Normal distribution. The z-score is the value decreased by the mean, divided by the standard deviation. $$\displaystyle{z}=\ {\frac{{{x}\ -\ \mu}}{{\sigma}}}=\ {\frac{{{180}\ -\ {168}}}{{{7}}}}\ \approx\ {1.71}$$ Determine the corresponding probability using the normal probability table in the appendix. $$\displaystyle{P}{\left({Z}\ {<}\ {1.71}\right)}$$</span> is given in the row starding with 1.7 and in the column starting with .01 of the standard normal probability table in the appendix. $$\displaystyle{P}{\left({X}\ {>}\ {180}\right)}={P}{\left({Z}\ {>}\ {1.71}\right)}$$
$$\displaystyle={1}\ -\ {P}{\left({Z}\ {<}\ {1.71}\right)}$$</span>
$$\displaystyle={1}\ -\ {0.9564}$$
$$\displaystyle={0.0436}$$
$$\displaystyle={4.36}\%$$ Thus about $$\displaystyle{4.36}\%$$ of the 15-year-old students at this scool are expected to be taller than 180 cm.

### Relevant Questions

Using the health records of ever student at a high school, the school nurse created a scatterplot relating y = height (in centimeters) to x = age (in years).
After verifying that the conditions for the regression model were met, the nurse calculated the equation of the population regression line to be $$\displaystyle\mu_{{0}}={105}+{4.2}{x}\ \text{with}\ \sigma={7}{c}{m}$$.
About what percent of 15-year-old students at this school are taller than 180 cm?

Using the daily high and low temperature readings at Chicago's O'Hare International Airport for an entire year, a meteorologist made a scatterplot relating y = high temperature to x = low temperature, both in degrees Fahrenheit.
After verifying that the conditions for the regression model were met, the meteorologist calculated the equation of the population regression line to be $$\displaystyle{\left[\mu_{{y}}={16.6}+{1.02}\right]}{w}{i}{t}{h}{\left[\sigma={6.6}+^{\circ}{F}\right]}$$.
About what percent of days with a low temperature of $$\displaystyle{40}^{\circ}$$ F?
Using the daily high and low temperature readings at Chicago's O'Hare International Airport for an entire year, a meteorologist made a scatterplot relating y = high temperature to x = low temperature, both in degrees Fahrenheit.
After verifying that the conditions for the regression model were met, the meteorologist calculated the equation of the population regression line to be $$\displaystyle{\left[\mu_{{y}}={16.6}+{1.02}\right]}{w}{i}{t}{h}{\left[\sigma={6.6}+^{\circ}{F}\right]}$$.
About what percent of days with a low temperature of $$\displaystyle{40}^{\circ}$$ F have a high temperature greater than $$\displaystyle{70}^{\circ}$$ F?
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung.
1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest?
2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.