High School Common Core Mathematics

Front 1

This scatterplot shows the finishing times for the Olympic gold medalist in the men's 100-meter dash for many previous Olympic games. The data shows a linear trend, and the least squares regression line (best fit line) is also shown. A. Describe the relationship shown by the linear model. Be sure to explain the strength and direction of the association.

Back 1

The linear model shows that there are no obvious outliers, and that there is a strong negative association between the two variables because the points are tightly clustered in linear form.

Front 2

This scatterplot shows the finishing times for the Olympic gold medalist in the men's 100-meter dash for many previous Olympic games. The data shows a linear trend, and the least squares regression line (best fit line) is also shown. B. Let f represent finishing time and Y represent the number of years after 1900. The equation of the best fit line that is shown is f=10.878-0.0106Y. Given that the summer Olympic games only take place every 4 years, how should we expect the gold medalist's finishing time to change from one Olympic games to the next?

Back 2

The slope of the linear model is -0.0106. This means for every 1 year that passes, we would predict that the finishing time for the 100-meter dash decreases by 0.0106 seconds. Since the Olympics take place every 4 years, we would expect the finishing time to decrease by 4(0.0106)= 0.0424 seconds from one Olympic games to the next.

Front 3

This scatterplot shows the finishing times for the Olympic gold medalist in the men's 100-meter dash for many previous Olympic games. The data shows a linear trend, and the least squares regression line (best fit line) is also shown. C. What is the vertical intercept of the function's graph? What does it mean in the context of the 100-meter dash?

Back 3

The vertical intercept of the regression equation's graph is 10.878. In context, this would be the predicted finishing time in seconds for the 100-meter dash gold medalist in the 1900 Olympic games.

Front 4

This scatterplot shows the finishing times for the Olympic gold medalist in the men's 100-meter dash for many previous Olympic games. The data shows a linear trend, and the least squares regression line (best fit line) is also shown. D. Note that the gold medalist finishing time for the 1940 Olympic games is not included in the scatterplot. Use the model to estimate the gold medalist's finishing time for that year.

Back 4

f=10.878-0.0106Y f=10.878-0.0106(40) f=10.454 The predicted finishing time for the 1940 gold medalist is 10.454 seconds.

Front 5

This scatterplot shows the finishing times for the Olympic gold medalist in the men's 100-meter dash for many previous Olympic games. The data shows a linear trend, and the least squares regression line (best fit line) is also shown. E. What is a realistic domain for the linear model? Comment on how your answer pertains to using this function to make predictions about future Olympic 100-meter dash race times.

Back 5

One thing we know is that the model will fail to be realistic once we obtain predicting racing times of zero or less. substituting 0 into the equation for f, we can solve for years after 1900 approx 1026.2. This equates roughly to the year 2926. If we take into account the current 4-year rotation for the summer Olympic games, however, we see that the model will only provide a positive prediction up through the Olympic games in the year 2924. To be even more realistic, we should expect any 100-meter dash to be completed in some positive amount of time; however, it is difficult to put a specific value on a reasonable result.

Front 6

A teacher suspects that there is a relationship between the number of text mesages high school students send per day and their academic achievement. To explore this, the teacher asked a random sample of 52 students at the school how many text mesages they sent yesterday and what their GPA was during the most recent marking period. This scatterplot summarizes the data. The best fit line is also shown. If g= grade point average and t= texts sent, the equation for the best fit line is g=3.8-0.005t. Interpret the quantities -0.005 and 3.8 in the context of these data.

Back 6

-0.005 is the slope of the best fit line. For students at this school, the predicted GPA decreases by 0.005 for each additional text message sent. (This can also be interpreted as the GPA decreases, on average, by 0.005, for each additional text message sent. 3.8 is the y-intercept of the best fit line. This linear model predicts that students at this school who send no text messages have, on average, a GPA of 3.8.

Front 7

Researchers have noticed that the number of golf courses and the number of divorcees in the U.S. are strongly correlated and both have been icreasing over the last several decades. Can you conclude that the increasing number of golf courses is causing the number of divorcees to increase? Either justify why a causation can be inferred, or explain what might account for the correlation other than a casual relation.

Back 7

No, it cannot be concluded that the increasing number of golf courses is causing the number of divorcees to increase. In general, correlation does not imply causation. There are a number of factors that may be increasing the number of golf courses and a number of factors causing the rise in number of divorcees. These factors may be (and often are) different. For example, there might be a rise in the popularity of golf that is causing the increase in the number of golf courses. However, the number of divorcees might be increasing due to the relative ease with which one can obtain a divorce now as opposed to, say, 10 years ago. Though there are factors that may cause both numbers to increase, for example, rise in population, this does not necessarily create a link between the sport of golf and divorcees.

Front 8

At a recent public County Council meeting, one resident expressed concern that 3 new coffee shops from a popular coffee shop chain were planning to open in the county, and the resident believed that this would create an increase in property crimes in the county. To support this claim, the resident presented the following data and scatterplot with best-fit line for 8 counties in the state. A. Would you consider the relationship betwee coffee shops and crimes to be strong, moderate, or weak?

Back 8

The relationship would be considered a strong and positive one given how closely the points adhere to a line with positive slope.

Front 9

At a recent public County Council meeting, one resident expressed concern that 3 new coffee shops from a popular coffee shop chain were planning to open in the county, and the resident believed that this would create an increase in property crimes in the county. To support this claim, the resident presented the following data and scatterplot with best-fit line for 8 counties in the state. B. Compute the correlation coefficient. Does the value of the correlation coefficient support your claim that the relationship is strong, moderate, or weak?

Back 9

r=0.968. Since the pattern shown is one of very strong, positive, linear association, a correlation coefficient value near +1 makes sense. This supports the claim of a strong relationship.

Front 10

At a recent public County Council meeting, one resident expressed concern that 3 new coffee shops from a popular coffee shop chain were planning to open in the county, and the resident believed that this would create an increase in property crimes in the county. To support this claim, the resident presented the following data and scatterplot with best-fit line for 8 counties in the state. C. The equation for the best-fit line as calculated by the resident (let p= predicted crimes, s= shops) p=1434+415.7s. Based on this line, what is the estimated number of additional annual property crimes for a given county that has 3 more coffee shops than any other county?

Back 10

Association does not necessarily imply causation. It is unlikely that building a new coffee shop would cause crime rates to increase, for such logic would imply that coffee drinkers engage in more criminal behavior than non-coffee drinkers, the coffee shop attracts criminals to the county, etc. From a perspective of context, there are other variables that may be responsible for the association, such as the fact that counties with higher population or population density may have both more coffee shops and more property crimes.

Front 11

At a recent public County Council meeting, one resident expressed concern that 3 new coffee shops from a popular coffee shop chain were planning to open in the county, and the resident believed that this would create an increase in property crimes in the county. To support this claim, the resident presented the following data and scatterplot with best-fit line for 8 counties in the state. D. If two counties were added to the data set as shown here, would you still consider using a line to model the relationship? If not, what other types of model would you consider?

Back 11

With the addition of the two observations, the scatterplot now displays a curved relationship with one outlier at (50, 20800). The scatterplot still shows a positive relationship between coffee shops and crimes, but the relationship no longer appears to be linear. The relationship may now be more accurately modeled using a quadratic or exponential curve.