Improve Your Understanding of College Level Statistics Problems

In government data, a household consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States.
$\begin{array}{|cccccccc|}\hline \text{Number of people}& 1& 2& 3& 4& 5& 6& 7\\ \begin{array}{l}\text{ Household }\text{ probability }\end{array}& 0.25& 0.32& 0.17& 0.15& 0.07& 0.03& 0.01\\ \begin{array}{l}\text{ Family }\text{ probability }\end{array}& 0& 0.42& 0.23& 0.21& 0.09& 0.03& 0.02\\ \hline\end{array}$
Let H = the number of people in a randomly selected U.S. household and F = the number of people in a randomly chosen U.S. family. Find the expected value of each random variable. Explain why this difference makes sense.

In government data, a house-hold consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States.
$\begin{array}{|cccccccc|}\hline \text{Number of people}& 1& 2& 3& 4& 5& 6& 7\\ \begin{array}{l}\text{ Household }\text{ probability }\end{array}& 0.25& 0.32& 0.17& 0.15& 0.07& 0.03& 0.01\\ \begin{array}{l}\text{ Family }\text{ probability }\end{array}& 0& 0.42& 0.23& 0.21& 0.09& 0.03& 0.02\\ \hline\end{array}$
Let H = the number of people in a randomly selected U.S. household and F = the number of people in a randomly chosen U.S. family. The standard deviations of the 2 random variables are σH=1.421 and σF=1.249.. Explain why this difference makes sense.

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 24 subjects had a mean wake time of 104.0 min. After treatment, the 24 subjects had a mean wake time of 94.5 min and a standard deviation of 23.2 min. Assume that the 24 sample values appear to be from a normally distributed population and construct a ${\displaystyle 99\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 104.0 min before the treatment? Does the drug appear to be effective?
Construct the ${\displaystyle 99\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with the treatment.
${\displaystyle ?min<\mu <?min}$

Do male and female students have different favorite seasons? The two-way table shows the favorite season and gender for a simple random sample of 89 high school juniors and seniors in the United States from the Census At School database. Is there convincing evidence of an association between gender and favorite season for students like those who participated in the Census At School survey?

Presenting data in the form of table. For the data set shown by the table, Solve,
a) Create a scatter plot for the data.
b) Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, you will use your graphing utility to obtain these functions.)
$\begin{array}{|cc|}\hline \text{Intensity (wattd per}mete{r}^2)& \text{Loudness Level (decibels)}\\ 0.1\text{(loud thunder)}& 110\\ 1\text{(rock concert, 2 yd from speakers)}& 120\\ 10\text{(jackhammer)}& 130\\ 100\text{(jet take off, 40 yd away)}& 140\\ \hline\end{array}$

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-50s.
$\begin{array}{lcc}& \text{Soccer experience}\text{Arthritis}& \begin{array}{lcccc}& & & \text{ Did not }& \\ & \text{ Elite }& \text{ Non-elite }& \text{ play }& \text{ Total }\\ \text{ Yes }& 10& 9& 24& 43\\ \text{ No }& 61& 206& 548& 815\\ \text{ Total }& 71& 215& 572& 858\end{array}\end{array}$
Suppose we choose one of these players at random. What is the probability that the player has arthritis?

The accompanying data on y = normalized energy \(\displaystyle{\left(\frac{{J}}{{{m}}^{2}}\right)}\) 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.
\(\begin{array}{} x&2761&19764&25713&3980&12782&19008\\ y&1553&14999&32813&1667&8741&16526 \\ x&20782&19028&14397&9606&3905&25731\\ y&26770&16526&9868&6640&1220&30730 \\ \end{array}\)
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.]

MODELING REAL LIFE. To make 18 glasses of juice, combine 0.25 cups of juice concentrate with 2 cups of water. Use how much juice concentrate? How much water do you use? You use ___ cups of juice concentrate and ___ cups of water.