The average weight of a male child's brain is 970 grams at...

The average weight of a male child's brain is 970 grams at age 1 and 1270 grams at age 3 (Source: American Neurological Association) (a) Assuming that the relationship between brain weight $y$ and age $t$ is linear, write a linear model for the data. (b) What is the slope and what does it tell you about brain weight? (c) Use your model to estimate the average brain weight at age 2 (d) Use your school's library, the Internet, or some other reference source to find the actual average brain weight at age $2 .$ How close was your estimate? (e) Do you think your model could be used to determine the average brain weight of an adult? Explain.

The average weight of a male child's brain is 970 grams at age 1 and 1270 grams at age 3 (Source: American Neurological Association)
(a) Assuming that the relationship between brain weight $y$ and age $t$ is linear, write a linear model for the data.
(b) What is the slope and what does it tell you about brain weight?
(c) Use your model to estimate the average brain weight at age 2
(d) Use your school's library, the Internet, or some other reference source to find the actual average brain weight at age $2 .$ How close was your estimate?
(e) Do you think your model could be used to determine the average brain weight of an adult? Explain.

Key Concepts

Linear Model

This concept involves creating an equation that relates two variables in a straight-line pattern, typically expressed as y = mx + b, where m is the slope and b is the y-intercept. It is a fundamental approach in statistics and mathematics for modelling relationships between variables under the assumption that changes in one correspond to proportional changes in the other.

Slope

The slope in a linear equation defines the rate of change between the dependent and independent variables; it shows how much the dependent variable changes when the independent variable increases by one unit. Understanding the slope is crucial for interpreting the direction and strength of the relationship between the variables.

Y-intercept

The y-intercept is the value of the dependent variable when the independent variable is zero. While it mathematically anchors the linear equation, its practical meaning depends on the context and whether zero is a realistic or meaningful value for the independent variable.

Interpolation

Interpolation refers to the estimation of values within the range of the known data points. This technique assumes that the relationship modeled between the known data points holds true within that interval, and it is generally more reliable than extrapolation.

Extrapolation

Extrapolation is the process of estimating values outside the range of the available data. This method carries more risk as the linear relationship may not continue beyond the observed range, making predictions less reliable when applied to data beyond the original scope.

Model Limitations

Every mathematical model is a simplified representation of reality and has limitations. When using a model like a linear equation, it is important to consider the assumptions made and recognize situations where the model may not adequately capture the complexity of real-world phenomena, such as biological growth trends that could be non-linear over larger ranges.

Transcript

00:01 Okay, so we are discussing linear functions.

00:05 And we are told that the average weight of a male child's brain, so it's the average mass, not the weight.

00:11 So the average mass of a male's child's brain is 970 grams at age 1 and 1270 grams at age 3.

00:20 So assuming the relationship between the brain mass and age is linear, write a linear model for the data.

00:30 What is the slope and what is it? tell you about brain weight, use your model to estimate the brain weight at age 2, look it up on the internet and how much does it actually, what is its actual mass at age 2, and do you think of kimi used to find the average mass of an adult's brain? okay, so first we're told that our points are t in years and let's just go wait.

01:04 In grams even though weight is a force so it's technically a mass but we'll go with it so we are given two points we're told that at age one it is 1270 and at age 3 excuse me 970 at age 1 and 1270 at age 3 okay so recall that a linear equation is in the form y equals mx plus b where m is the slope.

01:51 So if m is the slope, m is equal to y2 minus y1 over x2 minus x1.

02:02 So this is equal to 1270 minus 970, which is 300.

02:16 And 3 minus 2, sorry 3 minus 1 is 2.

02:22 Is 150.

02:24 So then y is equal to 150x plus b.

02:29 Now we need to know what b is the y intercept.

02:33 So to do that we plug in one of our points.

02:36 Let's plug in the point 970.

02:40 So you have 970 is equal to 150 times 1 plus b.

02:50 So 970 minus 150...