Improve Your Understanding of College Level Statistics Problems

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance ${\displaystyle {\sigma }^2}$ and (b) the population standard deviation ${\displaystyle \sigma }$. Interpret the results. The maximum wind speeds (in knots) of 13 randomly selected hurricanes that have hit the U.S. mainland are listed. Use a 95% level of confidence. $\begin{array}{ccccccccccccc}70& 85& 70& 75& 100& 100& 110& 105& 130& 75& 85& 75& 70\end{array}$

A random variable X has the discrete uniform distribution
$f(x;k)=\frac{1}{k},x=1,2...,k$
$f(x;k)=0,$ elsewhere.
Show that the moment-generating function of X is
${\displaystyle M_x\left(t\right)=\frac{e^t(1-e^{kt})}{k(1-e^t)}}$.

Prove this version of the Bonferroni inequality: ${\displaystyle P\left({\cap }_{i=1}^n{A}_i\right)\ge 1-\sum _{i=1}^{n}P(A_i^c)}$(Use Venn diagrams if you wish.) In the context of simultaneous confidence intervals, what is Ai and what is ${\displaystyle A_i^c}$?

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $\sigma$;2 and (b) the population standard deviation$\sigma$;. Interpret the results. The acceleration times (in seconds) from 0 to 60 miles per hour for 33 randomly selected sedans are listed. Use a 98% level of confidence.
6.5 5.0 5.2 3.3 6.6 6.3 5.1 5.3 5.4 9.5 7.5

4.5 5.8 8.6 6.9 8.1 6.0 6.7 7.9 8.8 7.1 7.9

7.2 18.4 9.1 6.8 12.5 4.2 7.1 9.9 9.5 2.8 4.9

In the article "On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals," by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement:
"Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute." Based on the preceding results, what should you conclude about the equality of $p_1$ and $p_2$?
Which of the three preceding methods is least effective in testing for the equality of $p_1$ and $p_2$?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties:

$$\begin{gathered} \text{Soccer level} \\ \begin{array}{llccc}& & \text{ Ellte }& \text{ Non-elite }& \text{ Did not play }\\ \text{ Whether person }& \text{ Yes }& 10& 9& 24\\ \text{ developed arthritis }& \text{ No }& 61& 206& 548\end{array} \end{gathered}$$

What percent of the elite soccer players developed arthritis? What percent of those who got arthritis were elite soccer players?

Take into account two mound-shaped independent distributions. A random sample of size $n_1=36$ from the first distribution showed $x_1=15$, and a random sample of size $n_2=40$ from the second distribution showed $x_2=14$. Based on the calculated confidence intervals, can you be 95% confident that ${\mu }_1$ is larger than ${\mu }_2$? Explain.

Find the levels of the confidence intervals that have the following values of ${\displaystyle z_{\frac{\alpha }{2}}}$:
a. ${\displaystyle z_{\frac{\alpha }{2}}=1.96}$
b. ${\displaystyle z_{\frac{\alpha }{2}}=2.17}$
c. ${\displaystyle z_{\frac{\alpha }{2}}=1.28}$
d. ${\displaystyle z_{\frac{\alpha }{2}}=3.28}$

Mutually exclusive versus independent. The two-way table summarizes data on the gender and eye color of students in a college statistics class. Imagine choosing a student from the class at random. Define event A: student is male, and event B: student has blue eyes.

$\text{Gender}\text{Eye color}\begin{array}{lccc}& \text{ Male }& \text{ Female }& \text{ Total }\\ \text{ Blue }& & & 10\\ \text{ Brown }& & & 40\\ \text{ Total }& 20& 30& 50\end{array}$

Copy and complete the two-way table so that events A and B are mutually exclusive.

Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article ''Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis'' (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)). If $\overline{X}$ is the sample mean Young’s modulus for a random sample of $n=16$ sheets, where is the sampling distribution of $\overline{X}$ centered, and what is the standard deviation of the $\overline{X}$ distribution?

In the treatment of critically ill patients, teamwork among health care professionals is essential. What is the level of collaboration between nurses and resident doctors working in the intensive care unit (ICU)? This was the question of interest in an article published in the Journal of Advanced Nursing (Vol. 67, 2011). Independent samples of 31 nurses and 46 resident doctors, all working in the ICU, completed the Baggs Collaboration and Satisfaction about Care Decisions survey. Responses to all questions were measured on a 7-point scale, where $1=$ never and $7=$ always. The data for the following two questions (simulated from information provided in the article) are listed above. a. Conduct a nonparametric test (at $\alpha =05\alpha =05$) to compare the response distributions for nurses and doctors on Question 4. Practically interpret the result. b. Conduct a nonparametric test (at $\alpha =05\alpha =05$) to compare the response distributions for nurses and doctors on Question 5. Practically interpret the result. Question 4: Physicians and nurses cooperate in making decisions.
Nurses:1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 5 5 5 5 5 5 5 7 7 Doctors:1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7
Question 5: In making decisions, both nursing and medical concerns about patients' needs are considered.
Nurses:1 1 2 2 2 2 3 3 3 3 33 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 7 Doctors:2 2 2 2 2 3 3 3 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7

Another type of confidence interval is called a one-sided confidence interval. A one-sided confidence interval provides either a lower confidence bound or an upper confidence bound for the parameter in question. You are asked to examine one-sided confidence intervals. Presuming that the assumptions for a one-mean z-interval are satisfied, we have the following formulas for (1−α)-level confidence bounds for a population mean ${\displaystyle \mu }$: Lower confidence bound: $\overline{x}-z_{\alpha }\cdot \sigma /\sqrt{n}$, Upper confidence bound: $\overline{x}+z_{\alpha }\cdot \sigma /\sqrt{n}$. Interpret the preceding formulas for lower and upper confidence bounds in words.

Use the one-standard-deviation ${\displaystyle x^2}$-test and the one-standard-deviation ${\displaystyle x^2}$-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval. $s=2$ and $n=10$
a. ${\displaystyle H_0:\sigma =4,H_{\left\{a\right\}}:\sigma <4,\alpha =0.05}$.
b. $90\mathrm{\%}$ confidence interval