Improve Your Understanding of College Level Statistics Problems

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let e be the level of confidence used to construct a confidence interval from sample data. Let $\alpha$ be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters such as p, $m{u}_1-{\mu }_2$, or $p_1-p_2$, which we will study in Section 9.3, 10.2, and 10.3.) Whenever the value of k given in the null hypothesis falls outside the ${\displaystyle c=1-\alpha }$ confidence interval for the parameter, we reject $H_0$. For example, consider a two-tailed hypothesis test with $\alpha =0.01\text{and}{H}_0:\mu =20H_1:\mu 20$ sample mean ${\displaystyle x¯=22}$ from a population with standard deviation ${\displaystyle \sigma =4}$.
(a) What is the value of c${\displaystyle c=1-\alpha }$;. Using the methods, construct a ${\displaystyle c=1-\alpha }$; confidence interval for μ from the sample data. What is the value of $\mu$; given in the null hypothesis (i.e., what is k)? Is this value in the confidence interval? Do we reject or fail to reject $H_0$ based on this information?
(b) using methods, find the P-value for the hypothesis test. Do we reject or fail to reject $H_0$? Compare your result to that of part (a).

Random variables Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of: $X+Y$

Suppose that the random variables X and Y have joint p.d.f.
$(f(x,y)=\{\begin{array}{l}kx(x-y)\\ 0\end{array}$Find the marginal p.d.f. of the two random variables.

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires were returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the two-way table. $\begin{array}{lcc}& \text{Business size}\text{Response?}& \begin{array}{lcccc}& \text{ Small }& \text{ Medium }& \text{ Large }& \text{ Total }\\ \text{ Yes }& 125& 81& 40& 246\\ \text{ No }& 75& 119& 160& 354\\ \text{ Total }& 200& 200& 200& 600\end{array}\end{array}$

 Which variable should be used as the explanatory variable? Explain.

The accompanying data on y = normalized energy \(\displaystyle{\left(\frac{{J}}{{m}}{2}\frac{{J}}{{m}^{{2}}}\right)}\) and x = intraocular pressure (mmHg) appeared in a scatterplot in the article “Evaluating the Risk of Eye Injuries: Intraocular Pressure During High Speed Projectile Impacts” (Current Eye Research, 2012: 43–49); an estimated regression function was superimposed on the plot.
x2761197642571339801278219008 y155314999328131667874116526 x2078219028143979606390525731 y267701652698686640122030730
The standardized residuals resulting from fitting the simple linear regression model (in the same order as the observations) are .98, -1.57, 1.47, .50, -.76, -.84, 1.47, -.85, -1.03, -.20, .40, and .81. Construct a plot of e* versus x and comment. [Note: The model fit in the cited article was not linear.]

A random sample of 1200 U.S. college students was asked, "What is your perception of your own body? Do you feel that you are overweight, underweight, or about right?" The two-way table summarizes the data on perceived body image by gender. $\begin{array}{cc}& \text{Gender}\\ \text{Body image}& \begin{array}{lccr}& \text{ Female }& \text{ Male }& \text{ Total }\\ \text{ About right }& 560& 295& 855\\ \text{ Overwerght }& 163& 72& 235\\ \text{ Underweight }& 37& 73& 110\\ \text{ Total }& 760& 440& 1200\end{array}\end{array}$

What percent of respondents feel that their body weight is about right?

Driven by technological advances and financial pressures, the number of surgeries performed in physicians' offices nationwide has been increasing over the years. The function ${\displaystyle f\left(t\right)=-0.00447t^3+0.09864t^2+0.05192t+0.8(0\le t\le 15)}$ gives the number of surgeries (in millions) performed in physicians' offices in year t, with $t=0$ corresponding to the beginning of 1986.
a. Plot the graph of f in the viewing window $[0,15]\times [0,10]$.
b. Prove that f is increasing on the interval [0, 15].

Gender and housing 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. $\begin{array}{lrr}& \text{Gender}& \\ \text{Housingstatus Housing status }& \begin{array}{cccc}& \text{ Male }& \text{ Female }& \text{ Total }\\ \text{ Own }& 132& 122& 254\\ \text{ Rent }& 50& 44& 94\\ \text{ Total }& 182& 166& 348\end{array}\end{array}$ State the hypotheses for a test of the relationship between gender and housing status for U.S. residents.

People with high blood pressure suffer from hypertension. A study of the lipid profiles of hypertensive patients was carried out and the results published in Biology and Medicine (Vol. 2, 2010). Data on fasting blood sugar (milligrams/deciliter) and magnesium (milligrams/deciliter) in blood specimens collected from 50 patients diagnosed with hypertension were collected. The accompanying MlNlTAB printout gives 90% confidence intervals for the mean fasting blood sugar (FBS) and mean magnesium level (MAG). a. Locate and interpret the 90% confidence interval for mean fasting blood sugar on the printout. b. Locate and interpret the 90% confidence interval for mean magnesium level on the printout. c. If the confidence level is increased to 95%, what will happen to the width of the intervals? d. If the sample of hypertensive patients is increased from 50 to 100, what will likely happen to the width of the intervals?$\begin{array}{cccccc}\text{Variable}& \text{N}& \text{Mean}& \text{StDev}& \text{SE Mean}& \text{90\% CI}\\ \text{FBS}& \text{50}& \text{101.62}& \text{34.13}& \text{4.83}& \text{(93.53, 109.71)}\\ \text{MAG}& \text{50}& \text{1.94100}& \text{0.05607}& \text{0.00793}& \text{(1.92771, 1.95429)}\end{array}$

Construct the indicated confidence intervals for (a) the population variance 
${\displaystyle {\sigma }^2}$ 
and (b) the population standard deviation 
${\displaystyle \sigma }$ 
Assume that the population in the sample has a normal distribution.
${\displaystyle c=0.95,s^2=11.56,n=30}$