Homework for Week 7

Those who favor gun control often point to a positive correlation between the availability of handguns and murder rates to support their position that gun control would save lives. Does this​ correlation, by​ itself, indicate that handgun availability causes a higher murder​ rate? Suggest some other factors that might support or weaken this conclusion.

Availability is not itself a cause.​ Social, economic, or personal conditions cause individuals to use the available handguns.

What is a best fit​ line? How is a best fit line​ useful? How is a best fit line​ useful?

It is a line that lies closer to the data points than any other possible line. It is useful to make predictions within the bounds of the data points.

The figure on the right shows the birth and death rates for different​ countries, measured in births and deaths per 1000 people. Complete parts​ (a) through​ (c) below.

The correlation coefficient is r≈0.8 which indicates a strong positive correlation. The points toward the left correspond to relatively wealthy countries, which have low birth rates and low death rates. The points toward the right correspond to relatively poor countries, which tend to have high birth rates and high death rates. Wealthier countries have a negative correlation, so higher birth rates are associated with lower death rates. Poorer countries have a positive correlation, so higher birth rates are associated with higher death rates.

Determine whether the statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain clearly. The scatterplot showed all the data points following a nearly straight diagonal​ line, but only a weak correlation between the two variables being plotted.

The statement does not make sense. The data points following a nearly straight diagonal line would indicate a very strong correlation between the two variables.

Determine whether the statement below makes sense or does not make sense. Explain clearly. I created a scatterplot of CEO salaries and corporate revenue for 10 companies and found a negative​ correlation, but when I left out a data point for a company whose CEO took no​ salary, there was no correlation for the remaining data.

The statement makes sense. A CEO taking no salary is an​ outlier, and an outlier can make a correlation appear where there otherwise is none.

Determine whether the following statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). I used a​ best-fit line for data showing the ages and arm spans of thousands of boys of various ages to predict the mean arm span of 17​-year-old boys.

The statement makes sense. Assuming the data were collected in a reasonable way and all ages were​ sampled, a scatterplot for thousands of boys should produce a​ best fit line that makes reasonable predictions of mean arm spans at different ages.

Determine whether the statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Explain clearly Researchers conducted animal experiments to study smoking and lung cancer because it would have been unethical to conduct these experiments on humans.

The statement makes sense. Researchers cannot randomly assign people to treatment and control groups and ask subjects in the treatment group to smoke.

Several things besides smoking have been shown to be probabilistic causal factors in lung cancer. For​ example, exposure to asbestos and exposure to radon​ gas, both of which are found in many​ homes, can cause lung cancer. Suppose that you meet a person who lives in a home that has a high radon level and insulation that contains asbestos. The person tells​ you, "I​ smoke, too, because I figure​ I'm doomed to lung cancer​ anyway." What would you say in​ response? Explain.

This person may or may not be doomed to lung​ cancer, but smoking will only increase the risk of getting lung cancer.

What is a​ correlation? Give three examples of pairs of variables that are correlated. Give three examples of pairs of variables that are correlated.

A correlation exists between two variables when higher values of one variable consistently go with higher or lower values of another variable. Amount of smoking and lung​ cancer, height and weight of​ people, price of a good and demand of the good

What does the square of the correlation​ coefficient, r2​, tell us about a​ best-fit line?

It tells us the proportion of the variation that is accounted for by the​ best-fit line. For​ example, if r2=​0.9, or​ 90%, then​ 90% of the variability is accounted for by the​ best-fit line, but​ 10% is not.

Determine whether the stated causal connection is valid. If the causal connection appears to be​ valid, provide an explanation. Drinking greater amounts of alcohol slows a​ person's reaction time.

The causal connection is valid. Alcohol is a depressant to the central nervous​ system, which leads to slower reaction time.

Determine whether the statement makes sense​ (or is clearly​ true) or does not make sense​ (or is clearly​ false). Using sample data on footprint lengths and heights from​ men, the equation of the​ best-fit line is​ obtained, and it is used to find that a man with a footprint length of 35 inches is predicted to have a height of 148 ​inches, or 12 feet, 4 inches.

The statement does not make sense since a prediction is being made regarding a value that is beyond the bounds of the data points.

For the following pair of​ variables, state whether you believe the two variables are correlated. If you believe they are​ correlated, state whether the correlation is positive or negative. Explain your reasoning. The IQ scores and hat sizes of randomly selected adults.

The variables are not correlated.

For the description​ below, state the correlation clearly.​ (For example, state that​ "there is a positive correlation between variable A and variable​ B.") Then state whether the correlation is most likely due to​ coincidence, a common underlying​ cause, or a direct cause. Explain your answer. It has been found that as gas prices​ increase, the distances vehicles are driven tend to get shorter.

There is negative correlation between gas prices and the distances vehicles are driven. The correlation is most likely due to a direct cause. As gas prices increase by large​ amounts, people​ can't afford to drive as​ much, so they cut costs by driving less.

For the description​ below, state the correlation clearly.​ (For example, state that​ "there is a positive correlation between variable A and variable​ B.") Then state whether the correlation is most likely due to​ coincidence, a common underlying​ cause, or a direct cause. Explain your answer. It has been found that as the number of traffic lights​ increases, the number of car crashes also increases.

There is positive correlation between the number of traffic lights and the number of car crashes. The correlation is most likely due to a common underlying​ cause, such as the general increase in the number of cars and traffic.

For the description​ below, state the correlation clearly.​ (For example, state that​ "there is a positive correlation between variable A and variable​ B.") Then state whether the correlation is most likely due to​ coincidence, a common underlying​ cause, or a direct cause. Explain your answer. In one​ state, the number of unregistered handguns steadily increased over the past several​ years, and the crime rate increased as well.

There is positive correlation between the number of unregistered handguns and an increase in crime rate. The correlation is most likely due to a common underlying cause. Many crimes are committed with handguns that are not registered.

A study reported in Nature claims that women who give birth later in life tend to live longer. Of the 78 women who were at least 100 years old at the time of the​ study, 19% had given birth after their 40th birthday. Of the 54 women who were 73 years old at the time of the​ study, only​ 5.5% had given birth after their 40th birthday. A researcher stated that​ "if your reproductive system is aging slowly enough that you can have a child in your​ 40s, it probably bodes well for the fact that the rest of you is aging slowly​ too." Was this an observational study or an​ experiment? Does the study suggest that later child bearing causes longer lifetimes or that later child bearing reflects an underlying​ cause? Comment on how persuasive you find the conclusions of the report.

This was an observational study. The study suggests that later child bearing reflects an underlying cause. There are other possible explanations for the findings. For​ example, it's also possible that the younger women lived during a time when having babies after age 40 was less likely​ (by choice). It is still possible for them to live to be 100 years old.

Listed below are altitudes​ (in thousands of​ feet) and outside air temperatures​ (in °​F) recorded during a flight between two cities. Altitude Temperature 4 57 12 37 13 25 23 -4 27 -29 32 -41 34 -53 Construct a scatterplot. Choose the correct answer below.

strong negative correlation Estimate the correlation coefficient​. -1.0 From the​ data, it seems that as the aircraft gains​ altitude, the outside temperature appears to drop in a strong and consistent pattern.