Master Statistics and Probability Problems with Expert Help

Suppose that X and Y are independent rv's with moment generating functions ${\displaystyle M_X\left(t\right)}$ and ${\displaystyle M_Y\left(t\right)}$, respectively. If Z=X+Y, show that ${\displaystyle M_Z\left(t\right)=M_X\left(t\right)M_Y\left(t\right)}$.

Consider two continuous random variables X and Y with joint density function
${\displaystyle f(x,y)=beg\in \left\{cases\right\}x+yo\le x\le 1,0\le y\le 1\mathrm{\setminus }0otherwiseend\left\{cases\right\}}$
P(X>0.8, Y>0.8) is?

The product of the ages, in years, of three (3) teenagers os 4590. None of the have the sane age. What are the ages of the teenagers?

Suppose that X and Y are continuous random variables with joint pdf ${\displaystyle f(x,y)=e^{-(x+y)}0<x<\mathrm{\infty }\text{and}0<y<\mathrm{\infty }}$ and zero otherwise.
Find P(X>Y)

Random variable X and Y have the joint PDF
$f_{XY}(x,y)=\{\begin{array}{l}c,x\ge 0,y\ge 0,(x^2+y)\le 1,\\ 0,otherwise\end{array}$
Find the marginal PDF ${\displaystyle f_X\left(x\right)}$.

For the following experiment, state the type of test we should use. Assume homogeneity of variance (where applicable), unless something is directly stated (e.g., "the data are not at all normal") or otherwise indicated (viz., by the inspection of the data) which would indicate an obvious violation of an assumption. This means you must inspect the data for violations of all assumptions. Assume n is large enough, unless stated. Assume ${\displaystyle \sigma }$ is unknown, unless stated.
Brianna works at an infant/toddler center and wants to know whether children at the age of two have a preference for a certain type of toy. She puts four different toys in front of the infant/toddler and sees which toy the child crawls to first.
1) a one-sample t-test for means
2) none of the other tests
3) an independent samples t-test with heterogeneity of variance
4) a related samples t-test
5) and independent samples t-test with homogeneity of variance
6) a chi-squared goodness-of-fit test
7) a z-test for the difference between proportions
8) a one sample z-test for proportions
9) a one-sample z test for means

In a study designed to prepare new gasoline-resistant coatings, a polymer chemist dissolves 6.053 g of polyvinyl alcohol) in enough water to make 100.0 mL of solution. At ${\displaystyle {25}^{\circ }C}$, the osmotic pressure of this solution is 0.272 atm. What is the molar mass of the polymer sample?

Effects of an appetite suppressant. Subject with preexisting cardiovascular symptoms who were receiving sibutramine, an apperite suppressant, were found to be at increased risk of cardiovascular events while taking the drug. The study included 9804 overweight or obese subjects with preexisting cardiovascular disease and/or type 2 diabetes. The subjects were randomly assigned to sibutramine (4906 subjects) or a placebo (4898 subjects) in a double-blind fashion. The primary outcome measured was the occurence of any of the following events: nonfatal myocardial infraction or stroke, resuscitation after cardiac arrest, or cardiovascular death. The primary outcome was observed in 561 subjects in the sibutramine group and 490 subjects in the placebo group.
a) Find the proportion of subjects experiencing the primary outcome for both the sibutramine and placebo groups.
b) Can we safely use the large-sample confidence interval for comparing the proportions of sibutramine and placebo subjects who experienced the primary outcome? Explain.

Gender gap Candidates for political office realize that different levels of support among men and women may be a crucial factor in determining the outcome of an election. One candidate finds that ${\displaystyle 52\mathrm{\%}}$ of 473 men polled say they will vote for him, but only ${\displaystyle 45\mathrm{\%}}$ of the 522 women in the poll express support.
a) Write a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the percent of male voters who may vote for this candidate. Interpret your interval.
b) Write and interpret a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the percent of female voters who may vote for him.
c) Do the intervals for males and females overlap? What do you think this means about the gender gap?
d) Find a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the difference in the proportions of males and females who will vote for this candidate. Interpret your interval.
e) Does this interval contain zero? What does that mean?
f) Why do the results in parts c and e seem contradic- tory? If we want to see if there is a gender gap among voters with respect to this candidate, which is the cor- rect approach? Why?

Suppose you took random samples from three distinct age groups. Through a survey, you determined how many respondents from each age group preferred to get news from T.V., newspapers, the Internet, or another source (respondents could select only one mode). What type of test would be appropriate to determine if there is sufficient statistical evidence to claim that the proportions of each age group preferring the different modes of obtaining news are not the same? Select from tests of independence, homogeneity, and goodness-of-fit.
Since we are interested in proportions, the test for homogeneity is appropriate. Since we are determining if the current distribution fits the previous distribution of responses, the goodness-of-fit test is appropriate.
Since we can claim all the variables are independent, the test of independence is appropriate.

A researcher is interested in determining if there is a significant difference in number of hours of studying in a week between male and female undergraduate students. 15 randomly selected males and 15 randomly selected females from across all majors were used in the study. The mean for females was determined to be 10 hours with a standard deviation of 2.5 hours and the mean for males was determined to be 7.25 with a standard deviation of 2.25 hours. Using an alpha of .05 is there enough evidence that females study on average more than males??
a. Write the hypotheses for this scenario.
b. Is this one tailed or two tailed? Why?
c. What statistical test would be used for this scenario? Explain why.

Does regular exercise reduse the risk of a heart attack? Provided are two ways to study this question.
1. A researcher finds 2000 men older than age 40 who exercise regularly and have not had heart attacks. She matches each with a similar man who does not exercise regularly, and she follows both groups for 5 years.
2. Another researcher finds 4000 men older than age 40 who have not had heart attacks and are willing to participate in a study. She assigns 2000 of the men to a regular program of supervised exercise. The other 2000 continue their usual habits. The researcher follows both groups for 5 years,
Select the correct explanation clearly detailing why the second design will produce more trustworthy data.
1) The first design is an observational study whereas the second design is an experiment. Assigning subjects to treatments, as in the second design, avoids confounding.
2) In the second design, the subjects are chosen in a random fashion, whereas in the first design the subjects from the second group are matched to the subjects in the first group.
3) The first design is an experiment whereas the second design is an observational study. Assigning subjects to treatments, as in the first design, does not avoid confounding.
4) In the first design, subjects are assigned to groups based on their own habits. There may be other factor (e.g., personality type or genetic background) that make one more likely to exercise and less (or more) likely to have a heart attack.

Sampling is a process of selecting a portion of population to represent the entire population. Name four types of sampling method and explain how each of them works.

A researcher has a hypothesis that a specific drug may have a higher prevalence of side effects among members of the African American population than members of the Caucasian population. Which statistical technique might the researcher want to use when designing a study to test their hypothesis?
Randomization
Crossover matching
Matching
Stratification

Your friend brags about his ability to do pull-ups. To demonstrate his skills, he has calculated the average number of pull-ups he can do. He wants to know if this significantly exceed the average pull-ups men in the United States can do. The average pull-ups for men in the United States is 8 reps.
A). z-test or t-test, and why?
B). What are the H1 and H0?

Random variable X and Y have the joint PDF
$f_{XY}(x,y)=\{\begin{array}{l}c,x\ge 0,y\ge 0,(x^2+y)\le 1,\\ 0,otherwise\end{array}$
Find the constant c.

The time a patient must wait before being attended by a physician in a hospital has been observed for each of 100 randomly selected patients he sample mean is 70.3 minutes and the standard deviation is 28.2 minutes. Test the hypothesis that the mean waiting time in the hospital is not more than 65 minutes.

Let ${\displaystyle X_1,X_2,\dots ,X_n}$ be n independent random variables each with mean 100 and standard deviation 30. Let X be the sum of these random variables.
Find n such that ${\displaystyle Pr\left(X>2000\right)\ge 0.95}$.

To identify: The test which is used to compare the scores of the two groups on a test of attitude toward people of other races.

True/False
True or False? A clinical trial is conducted to compare an experimental medication to placebo to reduce the symptoms of asthma. One hundred participants are enrolled in the study and randomized to receive either the experimental medication or placebo. The primary outcome is self-reported reduction of symptoms. Among 50 participants who receive the experimental medication, 30 report a reduction of symptoms as compared to 15 participants of 50 assigned to the placebo. The ${\displaystyle 95\mathrm{\%}}$ CI for the relative risk of participants reporting a reduction of symptoms between the experimental and placebo groups is between 1.529 and 8.012.