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Chapter 6: Problem 2

introduces a data set on birth weight of babies. Another variable we consider is parity, which is 0 if the child is the first born, and 1 otherwise. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, from parity. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 120.07 & 0.60 & 199.94 & 0.0000 \\ \text { parity } & -1.93 & 1.19 & -1.62 & 0.1052 \\ \hline \end{array} $$ (a) Write the equation of the regression line. (b) Interpret the slope in this context, and calculate the predicted birth weight of first borns and others. (c) Is there a statistically significant relationship between the average birth weight and parity?

Short Answer

Expert verified
(a) \( \text{Birth weight} = 120.07 - 1.93 \times \text{parity} \). (b) Slope interpretation: others weigh 1.93 oz less. Firstborns: 120.07 oz, others: 118.14 oz. (c) No, the relationship is not statistically significant (p=0.1052).

Step by step solution

01

Writing the Regression Equation

The linear regression model relates birth weight (dependent variable) to parity (independent variable). The formula for a regression line is given by \( y = \beta_0 + \beta_1x \), where \( \beta_0 \) is the intercept and \( \beta_1 \) is the slope. Using the estimates from the summary table, the regression equation is: \[ \text{Birth weight} = 120.07 - 1.93 \times \text{parity} \]
02

Interpret the Slope

The slope of -1.93 indicates that, on average, if the baby is not the first born (parity = 1), then the birth weight is predicted to decrease by 1.93 ounces compared to the firstborn babies (parity = 0). This is the effect of parity on birth weight.
03

Calculate Predicted Birth Weight

To calculate the predicted birth weight: For first-born babies (parity = 0), substitute 0 for parity in the equation: \( \text{Birth weight} = 120.07 - 1.93 \times 0 = 120.07 \) ounces. For other babies (parity = 1), substitute 1 for parity: \( \text{Birth weight} = 120.07 - 1.93 \times 1 = 118.14 \) ounces.
04

Evaluate Statistical Significance

To determine if there is a statistically significant relationship, we look at the p-value for parity. The provided p-value for parity is 0.1052. A common significance level is 0.05. Since 0.1052 > 0.05, we conclude that parity does not have a statistically significant relationship with birth weight at the 0.05 level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Significance
Statistical significance helps us understand whether the result of a statistical test is likely due to chance or if there is a genuine effect. In the context of linear regression, it tells us whether we can confidently say that a relationship exists between the independent variable and the dependent variable.

We use the p-value from statistical tests to determine significance. A p-value is the probability of observing the data, or something more extreme, if there is no actual effect. In this model:
  • If the p-value is less than the common significance level of 0.05, the relationship is considered statistically significant.
  • The given p-value for parity is 0.1052, which is greater than 0.05.

    This means that the effect of parity on birth weight is not statistically significant, suggesting that any observed difference in birth weight between firstborn and later-born babies may be due to random variation rather than parity.
Birth Weight
Birth weight, in this context, is the dependent variable in the regression analysis. It represents the weight of newborn babies in ounces. The goal of performing the regression analysis is to understand how different factors, like parity, affect this weight. In this data set, birth weight helps us explore and quantify differences between firstborn babies and their siblings.

Key points about birth weight as a variable:
  • The intercept of the regression equation shows us the average birth weight when the independent variable parity is zero (firstborn). In this case, the intercept is 120.07 ounces, which represents the average weight of firstborn babies.
  • This quantitative measure allows researchers to assess public health needs and understand growth patterns in different populations.

    Understanding birth weight is critical in medical research, as it has associations with health outcomes later in life.
Slope Interpretation
The slope in a regression equation gives insight into how much the dependent variable changes for a one-unit change in the independent variable. In this exercise, the slope is -1.93, which we apply to understand the relationship between parity and birth weight.

Understanding slope:
  • The negative value reflects that as parity changes from 0 to 1, the birth weight decreases by 1.93 ounces on average. In simple terms, firstborn babies tend to weigh 1.93 ounces more than their later-born siblings.
  • This effect size gives researchers an idea of how much parity (the variable for whether a baby is the firstborn or not) affects the weight of the baby.
  • It's valuable in understanding specific behavioral or health interventions needed for different birth orders.

    While statistically, the slope shows an effect, we previously noted that it's not statistically significant, meaning this effect might not hold true beyond the sample studied.
Regression Equation
The regression equation is a mathematical representation of how the dependent variable relates to independent variables. It helps make predictions or identify patterns in the data. For this problem, the regression equation illustrating the relationship between birth weight and parity is: \[ \text{Birth weight} = 120.07 - 1.93 \times \text{parity} \]

Features of the regression equation:
  • The intercept (120.07) is the expected birth weight of a firstborn baby, where parity equals zero.
  • The slope (-1.93) captures the mean change in birth weight when the parity changes by one unit. In this instance, this represents the difference in birth weight between firstborn and subsequently born children, with firstborns weighing about 1.93 ounces more.
  • This equation provides a simple tool for estimating birth weights based on parity, though we must be cautious since the relationship isn't statistically significant.

    Equations like this guide researchers in hypothesis testing and establishing related cause-effect links in studies.

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Most popular questions from this chapter

Researchers interested in the relationship between absenteeism from school and certain demographic characteristics of children collected data from 146 randomly sampled students in rural New South Wales, Australia, in a particular school year. Below are three observations from this data set. $$ \begin{array}{rcccc} \hline & \text { eth } & \text { sex } & \text { lrn } & \text { days } \\ \hline 1 & 0 & 1 & 1 & 2 \\ 2 & 0 & 1 & 1 & 11 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 146 & 1 & 0 & 0 & 37 \\ \hline \end{array} $$ The summary table below shows the results of a linear regression model for predicting the average number of days absent based on ethnic background (eth: 0 - aboriginal, 1 - not aboriginal), sex (sex: 0 - female, 1 - male), and learner status (lrn: 0 - average learner, 1 - slow learner). $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 18.93 & 2.57 & 7.37 & 0.0000 \\ \text { eth } & -9.11 & 2.60 & -3.51 & 0.0000 \\ \text { sex } & 3.10 & 2.64 & 1.18 & 0.2411 \\ \text { lrn } & 2.15 & 2.65 & 0.81 & 0.4177 \\ \hline \end{array} $$ (a) Write the equation of the regression line. (b) Interpret each one of the slopes in this context. (c) Calculate the residual for the first observation in the data set: a student who is aboriginal, male, a slow learner, and missed 2 days of school. (d) The variance of the residuals is 240.57 , and the variance of the number of absent days for all students in the data set is 264.17. Calculate the \(R^{2}\) and the adjusted \(R^{2} .\) Note that there are 146 observations in the data set.

Considers a model that predicts a newborn's weight using several predictors. Use the regression table below, which summarizes the model, to answer the following questions. If necessary, refer back to Exercise 6.3 for a reminder about the meaning of each variable. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.41 & 14.35 & -5.60 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.26 & 0.0000 \\ \text { parity } & -3.33 & 1.13 & -2.95 & 0.0033 \\ \text { age } & -0.01 & 0.09 & -0.10 & 0.9170 \\ \text { height } & 1.15 & 0.21 & 5.63 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 1.99 & 0.0471 \\ \text { smoke } & -8.40 & 0.95 & -8.81 & 0.0000 \\ \hline \end{array} $$ (a) Determine which variables, if any, do not have a significant linear relationship with the outcome and should be candidates for removal from the model. If there is more than one such variable, indicate which one should be removed first. (b) The summary table below shows the results of the model with the age variable removed. Determine if any other variable(s) should be removed from the model. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.64 & 14.04 & -5.74 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.28 & 0.0000 \\ \text { parity } & -3.29 & 1.06 & -3.10 & 0.0020 \\ \text { height } & 1.15 & 0.20 & 5.64 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 2.00 & 0.0459 \\ \text { smoke } & -8.38 & 0.95 & -8.82 & 0.0000 \\ \hline \end{array} $$

The Child Health and Development Studies investigate a range of topics. One study considered all pregnancies between 1960 and 1967 among women in the Kaiser Foundation Health Plan in the San Francisco East Bay area. Here, we study the relationship between smoking and weight of the baby. The variable smoke is coded 1 if the mother is a smoker, and 0 if not. The summary table below shows the results of a linear regression model for predicting the average birth weight of babies, measured in ounces, based on the smoking status of the mother. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 123.05 & 0.65 & 189.60 & 0.0000 \\ \text { smoke } & -8.94 & 1.03 & -8.65 & 0.0000 \\ \hline \end{array} $$ The variability within the smokers and non-smokers are about equal and the distributions are symmetric. With these conditions satisfied, it is reasonable to apply the model. (Note that we don't need to check linearity since the predictor has only two levels.) (a) Write the equation of the regression line. (b) Interpret the slope in this context, and calculate the predicted birth weight of babies born to smoker and non-smoker mothers. (c) Is there a statistically significant relationship between the average birth weight and smoking?

A survey of 55 Duke University students asked about their GPA, number of hours they study at night, number of nights they go out, and their gender. Summary output of the regression model is shown below. Note that male is coded as \(1 .\) $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & 3.45 & 0.35 & 9.85 & 0.00 \\ \text { studyweek } & 0.00 & 0.00 & 0.27 & 0.79 \\ \text { sleepnight } & 0.01 & 0.05 & 0.11 & 0.91 \\ \text { outnight } & 0.05 & 0.05 & 1.01 & 0.32 \\ \text { gender } & -0.08 & 0.12 & -0.68 & 0.50 \\ \hline \end{array} $$ (a) Calculate a \(95 \%\) confidence interval for the coefficient of gender in the model, and interpret it in the context of the data. (b) Would you expect a \(95 \%\) confidence interval for the slope of the remaining variables to include 0? Explain

We considered the variables smoke and parity, one at a time, in modeling birth weights of babies in Exercises 6.1 and \(6.2 .\) A more realistic approach to modeling infant weights is to consider all possibly related variables at once. Other variables of interest include length of pregnancy in days (gestation), mother's age in years (age), mother's height in inches (height), and mother's pregnancy weight in pounds (weight). Below are three observations from this data set. $$ \begin{array}{rccccccc} \hline & \text { bwt } & \text { gestation } & \text { parity } & \text { age } & \text { height } & \text { weight } & \text { smoke } \\ \hline 1 & 120 & 284 & 0 & 27 & 62 & 100 & 0 \\ 2 & 113 & 282 & 0 & 33 & 64 & 135 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1236 & 117 & 297 & 0 & 38 & 65 & 129 & 0 \\ \hline \end{array} $$ The summary table below shows the results of a regression model for predicting the average birth weight of babies based on all of the variables included in the data set. $$ \begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -80.41 & 14.35 & -5.60 & 0.0000 \\ \text { gestation } & 0.44 & 0.03 & 15.26 & 0.0000 \\ \text { parity } & -3.33 & 1.13 & -2.95 & 0.0033 \\ \text { age } & -0.01 & 0.09 & -0.10 & 0.9170 \\ \text { height } & 1.15 & 0.21 & 5.63 & 0.0000 \\ \text { weight } & 0.05 & 0.03 & 1.99 & 0.0471 \\ \text { smoke } & -8.40 & 0.95 & -8.81 & 0.0000 \\ \hline \end{array} $$ (a) Write the equation of the regression line that includes all of the variables. (b) Interpret the slopes of gestation and age in this context. (c) The coefficient for parity is different than in the linear model shown in Exercise 6.2 . Why might there be a difference? (d) Calculate the residual for the first observation in the data set. (e) The variance of the residuals is \(249.28,\) and the variance of the birth weights of all babies in the data set is 332.57. Calculate the \(R^{2}\) and the adjusted \(R^{2}\). Note that there are 1,236 observations in the data set.
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