Improve Your Understanding of College Level Statistics Problems

The Great White Shark. In an article titled “Great White, Deep Trouble” (National Geographic, Vol. 197(4), pp. 2–29), Peter Benchley—the author of JAWS—discussed various aspects of the Great White Shark (Carcharodon carcharias). Data on the number of pups borne in a lifetime by each of 80 Great White Shark females are provided on the WeissStats site. Use the technology of your choice to
a. obtain frequency and relative-frequency distributions, using single-value grouping.
b. construct and interpret either a frequency histogram or a relativefrequency histogram.

Construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.
The table li sts intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity.
$$\begin{array}{|cccccc|}\hline \text{Sound Intensity}& 316& 500& 750& 2000& 5000\\ \text{Scale Value}& 25.0& 27.0& 28.75& 33.0& 37.0\\ \hline\end{array}$$

Suppose X and Y are random variables with joint density function ${\displaystyle f(x,y)=\{0.1e^{-(0.5x+0.2y)}}$ if ${\displaystyle \ge 0,y\ge 0,0}$ otherwise. Find the following probabilities. ${\displaystyle P(Y\ge 1)}$

Using data from the 2000 census, a random sample of 348 U.S. residents aged 18 and older was selected. The two-way table summarizes the relationship between gender and housing status for these residents.
$$\begin{array}{|cccc|}\hline & \text{Male}& \text{Female}& \text{Total}\\ \text{Own}& 132& 122& 254\\ \text{Rent}& 50& 44& 94\\ \text{Total}& 182& 166& 348\\ \hline\end{array}$$
Make a graph that displays the relationship between gender and housing status for the members of the sample. Briefly describe the association.

Make a scatterplot for the data below on the number of people working on farms in various years, and draw a line of best fit. Describe the correlation as strong positive, strong negative, or little to none.
$$\begin{array}{|ccccccc|}\hline Year& 1940& 1950& 1960& 1970& 1980& 1990\\ \text{Number of farm workers in thousands}& 8995& 6858& 4132& 2881& 2818& 2864\\ \hline\end{array}$$

The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl. a) How much more cereal do you expect to be in the large bowl? b) What 's the standard deviation of this difference? c) If the difference follows a Normal model, what's the probability the small bowl contains more cereal than the large one? d) What are the mean and standard deviation of the total amount of cereal in the two bowls? e) If the total follows a Normal model, what's the probability you poured out more than 4.5 ounces of cereal in the two bowls together? f) The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of 16.3 ounces and a standard deviation of 0.2 ounces. Find the expected amount of cereal left in the box and the standard deviation.

The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320,000, what fraction earn (a) below 250,000; (b) between 260,00 and 300,000?

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of
${\displaystyle 1-{(1-p)}^2}$ if it does rain
${\displaystyle 1-p^2}$ if it does not rain
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability ${\displaystyle p\cdot }$, what value of p should he or she assert so as to maximize the expected score?

The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is the probability that there will be (a) more than 2 such accidents in the next month? (b) more than 4 such accidents in the next 2 months? (c) more than 5 such accidents in the next 3 more than 5 such accidents in the next 3 months? Explain your reasoning!

The annual rainfall (in inches) in a certain region is normally distributed with ${\displaystyle \mu =40}$ and ${\displaystyle \sigma =4}$
What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?

Suppose that X is a normal random variable with mean 5. If P{X>9}=.2, approximately what is Var(X)?

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Are X and Y independent?

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find Var(Y).

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X<20]

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X<8]

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find E[Y].

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Find E[X].

If X is a normal random variable with parameters
${\displaystyle \mu =10}$
and
${\displaystyle {\sigma }^2=36}$
, compute P[X>16]