Does increasing the amount of calcium in our diet reduce blood pressure? Researchers designed a randomized comparative experiment to find out. The subjects were 21 healthy black men who volunteered to take part in the experiment. They were randomly assigned to two groups: 10 of the men received a calcium supplement for 12 weeks, while the control group of 11 men received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic (top number) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative number. Here are the data.

$$\begin{array}{|l|r|r|r|r|r|r|r|r|r|r|r|} \hline \begin{array}{l} \text { Group 1 } \\ \text { (calcium) } \end{array} & 7 & -4 & 18 & 17 & -3 & -5 & 1 & 10 & 11 & -2 & \\ \hline \begin{array}{l} \text { Group 2 } \\ \text { (placebo) } \end{array} & -1 & 12 & -1 & -3 & 3 & -5 & 5 & 2 & -11 & -1 & -3 \\ \hline \end{array}$$

Researchers want to know if a calcium supplement reduces blood pressure more than a placebo, on average, for healthy black men like these. State appropriate hypotheses for performing a significance test. Be sure to define the parameters of interest.

Do fast-paced shows affect a child's ability to delay gratification? A study randomly assigned twenty 4 -year-olds to view a fast-paced children's show about "an animated sponge that lives under the sea." Another twenty 4-year-olds were randomly assigned to watch a slower-paced PBS children's show about "a typical American preschool-aged boy." In the fast-paced show, there was a scene change every 11 seconds, on average. In the slower-paced show, there was a scene change every 34 seconds, on average. After watching 9 minutes of the assigned program, children were asked if they preferred mini-marshmallows or goldfish crackers. Once the children had chosen their snack, two plates were prepared: one with 10 pieces and one with 2 pieces. A bell was placed between them and children were told that they could have the plate with 10 pieces if they waited for the researcher to return. Otherwise, they could ring the bell at any time and get the plate with 2 pieces. The amount of time (in seconds) it took for each child to ring the bell was recorded (330 seconds if the child waited the full time for the researcher to return). The table summarizes the data.

$$\begin{array}{|c|c|c|c|} \hline \text { Group } & n & \text { Mean } & \text { SD } \\ \hline \text { Fast-paced } & 20 & 146.15 & 151.29 \\ \hline \text { Slower-paced } & 20 & 257.20 & 132.16 \\ \hline \end{array}$$

Graphs of the data for both groups revealed no strong skewness or outliers. Calculate and interpret a 95% confidence interval for the difference in the true mean amount of time 4-year-old kids like these can delay gratification when watching fast-paced and slower-paced children's shows (Slow - Fast).

What goes wrong if you try to fit an exponential curve to data and one of the points has a y-coordinate that is 0 or a negative number?

Aviation and high-altitude physiology is a specialty in the study of medicine. Let x = partial pressure of oxygen in the alveoli (air cells in the lungs) when breathing naturally available air. Let y = partial pressure when breathing pure oxygen. The (x, y) data pairs correspond to elevations from 10,000 feet to 30,000 feet in 5000-foot intervals for a random sample of volunteers. Although the medical data were collected using airplanes, they apply equally well to Mt. Everest climbers (summit 29,028 feet).

$$\begin{array}{c|ccccc} \hline x & 6.7 & 5.1 & 4.2 & 3.3 & 2.1 \text { (units: mm Hg/10) } \\ \hline y & 43.6 & 32.9 & 26.2 & 6.2 & 13.9 \text { (units: mm Hg/10) } \\ \hline \end{array}$$

Find the predicted pressure when breathing pure oxygen if the pressure from breathing available air is x = 4.0..

Construct and interpret a 95% confidence interval for the difference (men minus women).

The formula for the area of a circle is $A=\pi r^{2}$, where r is the radius of the circle and $\pi$ is a constant. a. Show that the derivative of the area formula is $2 \pi r$, the formula for the circumference of a circle. b. Give an explanation for this in terms of rates of change.

Consider the following two events for an individual: A=owns a cell phone. B=owns a laptop computer. Translate each event into words. $A^{c}$

Solve each first-order linear differential equation. $y^{\prime}=x+y$

In many parts of the northern United States, two color variants of the Eastern Gray Squirrel-gray and black-are found in the same habitats. A scientist studying squirrels in a large forest wonders if there is a difference in the sizes of the two color variants. He collects random samples of 40 squirrels of each color from a large forest and weighs them. The 40 black squirrels have a mean weight of $\bar{x}_{\mathrm{B}}$=20.3 ounces and standard deviation of $s_{\mathrm{B}}$=2.1 ounces. The 40 gray squirrels have a mean weight of $\bar{x}_{\mathrm{G}}$=19.2 ounces and standard deviation of $s_{\mathrm{G}}$=1.9 ounces. Construct and interpret a 95% confidence interval for the difference in mean weight of all black and gray squirrels in this forest.

In many parts of the northern United States, two color variants of the Eastern Gray Squirrel-gray and black-are found in the same habitats. A scientist studying squirrels in a large forest wonders if there is a difference in the sizes of the two color variants. He collects random samples of 40 squirrels of each color from a large forest and weighs them. The 40 black squirrels have a mean weight of $\bar{x}_{1}=20.3$ ounces and standard deviation of $s_{1}=2.1$ ounces. The 40 gray squirrels have a mean weight of $\bar{x}_{2}=19.2$ ounces and standard deviation of $s_{2}=1.9$ ounces. Construct and interpret a 95% confidence interval for the difference in mean weight of black and gray squirrels in this forest.

In many parts of the northern United States, two color variants of the Eastern Gray Squirrel-gray and black-are found in the same habitats. A scientist studying squirrels in a large forest wonders if there is a difference in the sizes of the two color variants. He collects random samples of 40 squirrels of each color from a large forest and weighs them. The 40 black squirrels have a mean weight of 20.3 ounces and a standard deviation of 2.1 ounces. The 40 gray squirrels have a mean weight of 19.2 ounces and a standard deviation of 1.9 ounces. Do these data provide convincing evidence at the $\alpha=0.01$ significance level of a difference in the mean weights of all gray and black Eastern Gray Squirrels in this forest?

Refer to the previous exercise. Here are the summary statistics for the two samples.

Calculate a 99% confidence interval for the difference (Katy Perry - Justin Timberlake) in the true mean number of likes for all their Twitter posts.

Find the sum of each finite geometric series. $3-3 \cdot 2+3 \cdot 2^{2}-3 \cdot 2^{3}+\cdots+3 \cdot 2^{6}$

You have two cards. One is red on both sides, and the other is red on one side and black on the other. After shuffling the cards behind your back, you select one of them at random and place it on your desk with your hand covering it. Upon lifting your hand, you observe that the face showing is red. a. What is the probability that the other side is red? b. Provide an intuitive explanation for the result in part (a).