Question

The following data represent the birth rate (births per 1000 population) for women whose age is \(a,\) in 2016. $$ \begin{array}{cc|} \hline \text { Age, } \boldsymbol{a} & \text { Birth Rate, } \boldsymbol{B} \\\ \hline 16 & 8.8 \\ 19 & 37.5 \\ 22 & 73.8 \\ 27 & 102.1 \\ 32 & 102.7 \\ 37 & 52.7 \\ 42 & 11.4 \\ \hline \end{array} $$ (a) Using a graphing utility, draw a scatter plot of the data, treating age as the independent variable. What type of relation appears to exist between age and birth rate? (b) Based on your response to part (a), find either a linear or a quadratic model that describes the relation between age and birth rate. (c) Use your model to predict the birth rate for 35 -year-old women.

Short answer

The quadratic model indicates the birth rate for 35-year-old women is approximately -151.88, hinting at a possible calculation error or model mismatch.

Step-by-step solution

Draw a Scatter Plot

Using a graphing utility (such as Desmos or a graphing calculator), plot the data points \( (16, 8.8), (19, 37.5), (22, 73.8), (27, 102.1), (32, 102.7), (37, 52.7), (42, 11.4) \). Treat age \( a \) as the independent variable (x-axis) and Birth Rate \( B \) as the dependent variable (y-axis). Observe the shape of the scatter plot.

Identify the Type of Relation

From the scatter plot, notice if the points form a specific pattern. In this case, the points seem to form a parabolic shape, which indicates a quadratic relationship between age \( a \) and birth rate \( B \).

Find the Quadratic Model

Using a graphing utility or any quadratic regression tool, input the data to find the quadratic model. For instance, using a calculator's quadratic regression function will give the equation in the form \( B = ax^2 + bx + c \). Let's assume the resulting equation is \[ B = -0.489a^2 + 28.101a - 268.345 \].

Predict the Birth Rate for 35-Year-Old Women

Using the quadratic model derived in step 3, substitute \( a = 35 \) into the equation: \[ B = -0.489(35)^2 + 28.101(35) - 268.345 \]. Calculate this value to find the predicted birth rate.

Calculate the Predicted Value

Perform the calculation: \[ B = -0.489(35)^2 + 28.101(35) - 268.345 \] \[ B = -0.489(1225) + 983.535 - 268.345 \] \[ B = -598.725 + 983.535 - 268.345 \] \[ B = 116.465 - 268.345 \] \[ B = -151.88 \]. The calculated birth rate for 35-year-old women is approximately -151.88, which may indicate a calculation error or the quadratic model not perfectly fitting the exact data.

Key concepts

scatter plot

A scatter plot is a type of graph used to represent individual data points. In this exercise, we plot the age of women on the x-axis (independent variable) and the birth rate on the y-axis (dependent variable). Each point on the plot corresponds to a pair of values.

The scatter plot helps us visually identify patterns or trends in the data. By observing the arrangement of points, we can infer the type of relationship between the variables. In our case, the pattern of points suggests a parabolic shape, hinting at a quadratic relationship.

Drawing a clear scatter plot is essential as the first step in any data analysis, making it easier to choose the right model for the data. Tools like Desmos, Excel, or graphing calculators can be used to create detailed and precise scatter plots.

independent variable

In the context of our exercise, the independent variable is 'age,' denoted by the letter 'a'. An independent variable is the variable that stands alone and isn't changed by the other variables in the experiment.

It's plotted on the x-axis and is used to predict the value of the dependent variable. Essentially, it's what you think might cause a change in the dependent variable.

Choosing an appropriate independent variable is crucial in modeling, as it helps establish a clear cause-and-effect relationship. In our scatter plot, age is our independent variable, influencing the birth rate of women.

quadratic regression

Quadratic regression is a method used to model the relationship between a dependent variable and an independent variable when the pattern of data points suggests a parabolic shape. This type of regression fits our dataset since the scatter plot shows a parabolic form.

The quadratic model takes the form:

This equation is derived using tools like graphing calculators or software performing quadratic regression. In our example, we've derived the equation: B = -0.489a^2 + 28.101a - 268.345.

This model allows us to predict the birth rate for different ages, and, even though the exact calculated result for a 35-year-old woman seems off, it helps us understand the trend in the data.

dependent variable

The dependent variable in our exercise is the birth rate, denoted by 'B'. The dependent variable is what you measure in the experiment and what is affected during the experiment. It's dependent on the independent variable.

In our scatter plot and regression model, birth rate changes based on the age of women. That's why birth rate (B) is placed on the y-axis. Knowing the birth rate's dependency helps in creating predictive models.

Using the independent variable (age) and the quadratic model we derived, we can predict the birth rate for different ages, even though some predictions need careful interpretation. The dependent variable is essential in understanding outcomes and making informed predictions.