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A study conducted at the University of Denver investigated whether babies take longer to learn to crawl in cold months, when they are often bundled in clothes that restrict their movement, than in warmer months. \(^{17}\) Infants born during the study year were split into twelve groups, one for each birth month. We consider the average crawling age of babies in each group against the average temperature when the babies are six months old (that's when babies often begin trying to crawl). Temperature is measured in degrees Fahrenheit ( \({ }^{\circ} \mathrm{F}\) ) and age is measured in weeks. (a) Describe the relationship between temperature and crawling age. (b) How would the relationship change if temperature was measured in degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and age was measured in months? (c) The correlation between temperature in \({ }^{\circ} \mathrm{F}\) and age in weeks was \(R=-0.70 .\) If we converted the temperature to \({ }^{\circ} \mathrm{C}\) and age to months, what would the correlation be?

What would be the correlation between the annual salaries of males and females at a company if for a certain type of position men always made (a) \(\$ 5,000\) more than women? (b) \(25 \%\) more than women? (c) \(15 \%\) less than women?

The scatterplot shows the relationship between socioeconomic status measured as the percentage of children in a neighborhood receiving reduced-fee lunches at school (lunch) and the percentage of bike riders in the neighborhood wearing helmets (helmet). The average percentage of children receiving reduced-fee lunches is \(30.8 \%\) with a standard deviation of \(26.7 \%\) and the average percentage of bike riders wearing helmets is \(38.8 \%\) with a standard deviation of \(16.9 \%\). (a) If the \(R^{2}\) for the least-squares regression line for these data is \(72 \%\), what is the correlation between lunch and helmet? (b) Calculate the slope and intercept for the leastsquares regression line for these data. (c) Interpret the intercept of the least-squares regression line in the context of the application. (d) Interpret the slope of the least-squares regression line in the context of the application. (e) What would the value of the residual be for a neighborhood where \(40 \%\) of the children receive reduced-fee lunches and \(40 \%\) of the bike riders wear helmets? Interpret the meaning of this residual in the context of the application.

Is the gestational age (time between conception and birth) of a low birth- weight baby useful in predicting head circumference at birth? Twenty-five low birth-weight babies were studied at a Harvard teaching hospital; the investigators calculated the regression of head circumference (measured in centimeters) against gestational age (measured in weeks). The estimated regression line is head_circumference \(=3.91+0.78 \times\) gestational_age (a) What is the predicted head circumference for a baby whose gestational age is 28 weeks? (b) The standard error for the coefficient of gestational age is \(0.35,\) which is associated with \(d f=23\). Does the model provide strong evidence that gestational age is significantly associated with head circumference?