Improve Your Understanding of College Level Statistics Problems

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 ${\displaystyle Pr\left(X>2000\right)\ge 0.95}$.

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.

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.

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.

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.

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.

To explain:
When the relative frequencies are useful in comparing two data sets.

To find: The ${\displaystyle 95\mathrm{\%}}$ confidence interval for the comparison.

Let ${\displaystyle X_1,X_2,\dots ,X_6}$ be an i.i.d. random sample where each ${\displaystyle X_i}$ is a continuous random variable with probability density function
${\displaystyle f\left(x\right)=e^{-(x-0)},x>0}$
Find the probability density function for ${\displaystyle X_6}$.

Two random variables X and Y with joint density function given by:
$f(x,y)=\{\begin{array}{ll}\frac{1}{3}(2x+4y)& 0\le x,\le 1\\ 0& elsewhere\end{array}$
Find the marginal density of Y.

A researcher always need population to collect sample for research.Identify the relevant population for the following research studies and suggest the appropriate sampling design to investigate the issues, explaining why they are appropriate.
1. A hospital administrator wants to find out if the single parents working in the hospital have ahigher rate of absenteeism than parents who are not single.
b. A marketer wants to generate some ideas on how women differ from men in acquiring productknowledge about cars

Random variables ${\displaystyle X_1,X_2,\dots ,X_n}$ are independent and identically distributed; 0 is a parameter of their distribution.
If ${\displaystyle X_1,X_2,\dots ,X_n}$ are Normally distributed with unknown mean 0 and standard deviation 1, then ${\displaystyle \bar{X}\sim N\left(\frac{0,1}{n}\right)}$. Use this result to obtain a pivotal function of X and 0.

The population mean may be reliably estimated using a statistic calculated from data recorded from randomly selected individuals.’ Explain this statement with reference to sampling distributions and the central limit theorem.

Range=max value- min value
h=r/m m=no of classes
The weights shown in data given to the nearest tenth of a pound, were obtained from a sample of 18- to 24-year-old males. Organize these data into frequency and relative-frequency distributions. Use a class width of 20 and a first cutpoint of 120.
129.2 185.3 218.1 182.5 142.8 155.2 170.0 151.3 187.5 145.6 167.3 161.0 178.7 165.0 172.5 191.1 150.7 187.0 173.7 178.2 161.7 170.1 165.8 214.6 136.7 278.8 175.6 188.7 132.1 158.5 146.4 209.1 175.4 182.0 173.6 149.9 158.6
Also calculate cumulative frequency and midpoint.

List 7 ways of representing data

Random variables ${\displaystyle X_1,X_2,\dots ,X_n}$ are independent and identically distributed; 0 is a parameter of their distribution.
If ${\displaystyle q(X,0)\sim N(0,1)}$ is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

Critical Thinking Suppose you take a random sample from a normal population and you want to determine whether there is sufficient statistical evidence to claim that the population variance differs from a corresponding variance specified in a goverment contract. Which type of test is appropriate, a test of one variance or a test of two variances?

What are important features of hypothesis testing?

Suppose that ${\displaystyle X_1,X_2\text{and}{X}_3}$ are three independent random variables with the same distribution as X.
What is the ecpected value of the sum ${\displaystyle X_1+X_2+X_3}$? The product ${\displaystyle X_1{X}_2{X}_3}$?
Suppose a discrete random variable X assumes the value ${\displaystyle \frac{3}{2}}$ with probability 0.5 and assumes the value ${\displaystyle \frac{1}{2}}$ with probability 0.5.