Exam 11: Two Quantitative Variables

If you fit a least squares line to two quantitative variables x and y, and the slope of the line differs from zero, then you know the correlation coefficient also differs from zero.

Social warmth is a term referring to the feeling of being connected to others. A study published in PLoS One in 2016 looked at a potential relationship between physical warmth (body temperature) and social warmth among a group of 54 volunteers (Inagki et al.). These volunteers had their oral temperature taken by a registered nurse and then assessed themselves using a scale of 1 to 5 on twelve items related to a feeling of social connection for which the average was recorded. Higher average scores indicated higher levels of social warmth. The theory was that the thermoregulatory system, which helps maintain a relatively warm internal body temperature, may also help people assess feelings of social connection. Below is a scatterplot and least-squares regression line of the data. -Below is a picture of a simulated null distribution of slopes created using the Corr/Regression applet. How is this distribution used calculate the p-value?

For a given dataset, a test of association based on a slope is equivalent to a test of association based on a correlation coefficient. Being equivalent means which of the following is true?

Social warmth is a term referring to the feeling of being connected to others. A study published in PLoS One in 2016 looked at a potential relationship between physical warmth (body temperature) and social warmth among a group of 54 volunteers (Inagki et al.). These volunteers had their oral temperature taken by a registered nurse and then assessed themselves using a scale of 1 to 5 on twelve items related to a feeling of social connection for which the average was recorded. Higher average scores indicated higher levels of social warmth. The theory was that the thermoregulatory system, which helps maintain a relatively warm internal body temperature, may also help people assess feelings of social connection. Below is a scatterplot and least-squares regression line of the data. -Based off of the simulated null distribution in question 51, what is the strength of evidence against the null hypothesis?

It is commonly expected that as a person ages, their muscle mass decreases. To further examine this relationship in women, a nutritionist randomly selected 60 female patients from her clinic, 15 women from each 10-year age group beginning with age 40 and ending with age 80. For each patient, her age and current muscle mass was recorded. A scatterplot, least squares regression line, and coefficient of determination are as follows. -How would you simulate one sample, assuming the null hypothesis is true?

The correlation coefficient is the proportion of total variation in the response variable that is accounted for by changes in the explanatory variable.

Are people with bigger brains more intelligent? Forty college students volunteered to participate in a study which examined brain size (measured as 1000's of pixels counted in a brain scan), and IQ scores (measured in points). A scatterplot of the data is shown below. -The p-value for this test is 0.008. What can we conclude?

It is commonly expected that as a person ages, their muscle mass decreases. To further examine this relationship in women, a nutritionist randomly selected 60 female patients from her clinic, 15 women from each 10-year age group beginning with age 40 and ending with age 80. For each patient, her age and current muscle mass was recorded. A scatterplot, least squares regression line, and coefficient of determination are as follows. -Can these results be generalized to the population of all patients at this clinic?

Data from gapminder.org on 184 countries was used to examine if there is an association between (average) female life expectancy (that is, the average lifespan of women in the country) and the average number of children women give birth to for the year 2019. A scatterplot of the data and a regression table from the Corr/Regression applet follows. -Is there significant evidence of an association between female life expectancy and the average number of children women give birth to for the year 2019?

Data from gapminder.org on 184 countries was used to examine if there is an association between (average) female life expectancy (that is, the average lifespan of women in the country) and the average number of children women give birth to for the year 2019. A scatterplot of the data and a regression table from the Corr/Regression applet follows. -How would you interpret the standardized statistic for a test of the regression slope?

It is commonly expected that as a person ages, their muscle mass decreases. To further examine this relationship in women, a nutritionist randomly selected 60 female patients from her clinic, 15 women from each 10-year age group beginning with age 40 and ending with age 80. For each patient, her age and current muscle mass was recorded. A scatterplot, least squares regression line, and coefficient of determination are as follows. -The p-value for these data was less than 0.0001. Write a conclusion of the test in the context of the study.

Data from gapminder.org on 184 countries was used to examine if there is an association between (average) female life expectancy (that is, the average lifespan of women in the country) and the average number of children women give birth to for the year 2019. A scatterplot of the data and a regression table from the Corr/Regression applet follows. -Using the regression table output, state the equation of the regression line.

Data from the World Bank for 25 Western Hemisphere countries was used to examine the association between (average) female life expectancy (that is, the average lifespan of women in the country) and the average number of children women give birth to. Given below is the scatterplot for the data. The regression equation for this context is found to be: y= 84.5 - 4.4x where Y is female life expectancy in years, and is the average number of births per woman. -Interpret the slope in the context of the study.

The following scatterplot displays the finish time (in minutes) and age (in years) for the male racers at the 2018 Strawberry Stampede (a 10k race through Arroyo Grande).  Below are the same data for the female racers in this year's race. -Do you think the correlation coefficient for the females will be larger, smaller, or remain the same as the male's?

It is commonly expected that as a person ages, their muscle mass decreases. To further examine this relationship in women, a nutritionist randomly selected 60 female patients from her clinic, 15 women from each 10-year age group beginning with age 40 and ending with age 80. For each patient, her age and current muscle mass was recorded. A scatterplot, least squares regression line, and coefficient of determination are as follows. -Write the null and alternative hypotheses of interest for testing if there is a negative linear relationship between age and muscle mass using proper notation for a test of slope.

A student in an AP Statistics class decided to conduct a study to determine whether you could predict the number of followers a teen has on Instagram based on the number of people he or she is following. To do this, she randomly selected fifty students from her high school that had Instagram accounts and for each student recorded the number of people they were following and the number of followers they had. A scatterplot of the data is shown. The regression line is . followers=-45.3+1.21(following) -A 95% confidence interval for the population slope is (1.07, 1.35). How would you interpret this in the context of the study?

The following scatterplot displays the finish time (in minutes) and age (in years) for the male racers at the 2018 Strawberry Stampede (a 10k race through Arroyo Grande).  Below are the same data for the female racers in this year's race. -What is the null hypothesis for a simulation-based test of the correlation coefficient for the females.

Which of the following plots has the strongest correlation between the two variables plotted?

Annual measurements of the number of powerboat registrations (in thousands) and the number of manatees killed by powerboats in Florida were collected over the 14 years 1977-1990. A scatterplot of the data, least squares regression line, and correlation coefficient follow. -Which of the following is a correct interpretation of the coefficient of determination?

The following scatterplot displays the finish time (in minutes) and age (in years) for the male racers at the 2018 Strawberry Stampede (a 10k race through Arroyo Grande). -What is the form of this scatterplot?