Statistics

A lifeguard counts the number of swimmers in the pool when she starts her shift and notes the temperature. The equation of the regression line for the data in the table is y^ = 1.62x - 105.5, where x represents the temperature and y^ is the number of swimmers. Day | Temperature (°F) | Number of swimmers Monday | 85 | 32 Tuesday | 90 | 40 Wednesday | 92 | 41 Thursday | 88 | 39 Friday | 95 | 50 What is the meaning of 1.62 in the equation?

Step-by-step solution with explanation

Final Answer

The 1.62 is the slope. It means that for every 1°F increase in temperature, the predicted number of swimmers in the pool increases by about 1.62 swimmers.

Step-by-step solution

1

Identify the regression line equation

We are given a regression line equation. In this equation, x is the temperature in °F and ŷ is the predicted number of swimmers.
2

Recall the slope-intercept form meaning

In any linear equation, m is the slope and b is the y-intercept. Here, the slope is 1.62 and the y-intercept is -105.5.
3

Interpret the slope in context

The slope tells us how much the predicted number of swimmers changes for each 1-degree increase in temperature. A slope of 1.62 means for every 1°F increase in temperature, the number of swimmers is predicted to increase by about 1.62.

Understanding this problem

Learning Insight

The slope in a regression equation always describes the rate of change between the two variables. It answers the question: 'If x goes up by 1, how much does y change?' The sign tells you direction (positive = both go up together), and the number tells you how fast.

Quick Tip

To interpret any slope in context, just fill in this sentence: 'For every 1 [x-unit] increase, y is predicted to [increase/decrease] by [slope value] [y-unit].'

Common Mistake

Students often confuse the slope (1.62) with the y-intercept (-105.5) and try to interpret -105.5 as the rate of change, or they forget to include the units (°F and number of swimmers) when explaining the slope's meaning.