L. Data Interpretation, Statistics and Probability

Because major areas of study at postsecondary institutions have different prerequisites, certain mathematics benchmarks are marked with an asterisk (*). These asterisked benchmarks represent content that is recommended for all students, but is required for those students who plan to take calculus in college, a requisite for mathematics and many mathematics-intensive majors. The high school graduate can:

L1. Explain and apply quantitative information:

L1.1. Organize and display data using appropriate methods (including spreadsheets) to detect patterns and departures from patterns.

(Associated Workplace Task: #4)

L1.2. Read and interpret tables, charts and graphs.

(Associated Workplace Tasks: #3 and 4)

L1.3. Compute and explain summary statistics for distributions of data including measures of center (mean, median) and spread (range, percentiles, variance, standard deviation).

L1.4. Compare data sets using graphs and summary statistics.

Example:

Create a box plot of a school's test scores for 1995 and for 2000 and ask what changes occurred in the five years.

L1.5. Create scatter plots, analyze patterns and describe relationships in paired data.

L1.6. Know the characteristics of the Gaussian normal distribution (bell-shaped curve).

Example:

If a set of data is approximately normally distributed, know that approximately 95% of the data in the set are within two standard deviations of the mean and that approximately 99% of the data in the set are within three standard deviations of the mean.

L2. Explain and critique alternative ways of presenting and using information:

L2.1. Evaluate reports based on data published in the media by considering the source of the data, the design of the study, and the way the data are analyzed and displayed.

L2.2. Identify and explain misleading uses of data.

Example:

Explain why the following graphic misrepresents the data it is intended to illustrate.

L2.3. Recognize when arguments based on data confuse correlation with causation.

Example:

Researchers have noticed that the number of golf courses and the number of divorces in the United States are strongly correlated and both have been increasing over the last several decades. Can you conclude that the increasing number of golf courses is causing the number of divorces to increase? Explain your answer.

L3. Explain the use of data and statistical thinking to draw inferences, make predictions and justify conclusions:

L3.1. Explain the impact of sampling methods, bias and the phrasing of questions asked during data collection and the conclusions that can rightfully be made.

L3.2. Design simple experiments or investigations to collect data to answer questions of interest.

(Associated Workplace Task: #4)

L3.3. Explain the differences between randomized experiments and observational studies.

(Associated Workplace Task: #4)

L3.4. Construct a scatter plot of a set of paired data and if it demonstrates a linear trend, use a graphing calculator to find the regression line that best fits this data; recognize that the correlation coefficient measures goodness of fit and explain when it is appropriate to use the regression line to make predictions.

Example:

The following table gives the winning speeds (in miles per hour) at the Indianapolis 500 race for 20 years. Explain why it is not appropriate to use the linear regression equation for these data to estimate what the winning time was in 1920 or to predict the winning speed in 1990. Source: The World Almanac

L4. Explain and apply probability concepts and calculate simple probabilities:

L4.1. Explain how probability quantifies the likelihood that an event occurs in terms of numbers.

(Associated Workplace Task: #3)

L4.2. Explain how the relative frequency of a specified outcome of an event can be used to estimate the probability of the outcome.

Example:

Typically, 35 out of every 100 teenagers in a certain community have received a traffic ticket. Of those teenagers who have received a ticket, 55% were charged with speeding. What is the probability that a teenager chosen at random will have received a speeding ticket?

L4.3. Explain how the law of large numbers can be applied in simple examples.

Example:

Toss a fair coin 10 times, record the number of heads and apply the data to estimate the probability of getting heads on a single toss of the coin. Toss the coin 20 more times, add the results to the previous data and apply the 30 tosses to estimate the probability of getting a heads. Toss the coin 30 more times and make another estimate of the probability of getting a heads. What can you observe about the probability as the number of tosses increases?

L4.4. Apply probability concepts such as conditional probability and independent events to calculate simple probabilities.

Example:

A fair coin is tossed three times, and three heads are obtained. Understand that the probability of obtaining a head on the fourth toss is 1/2 because this event is independent of outcomes of the three previous tosses.

Example:

If two marbles are drawn randomly one after the other without replacement from a bag containing 4 red and 6 blue marbles, the probability that both marbles drawn are red is 4/10 x 3/9 = 2/15 because the probability of drawing a red marble on the second draw depends, or is conditional upon, the color of the first marble drawn.

L4.5. Apply probability concepts to practical situations to make informed decisions.

(Associated Workplace Tasks: #3 and 4)

Example:

A company has 6 telephone lines coming into its business. Efficiency experts performed a study for a week and determined that the following table could give the number of lines in use at any one time. Determine the probability that at most four lines were in use at one time during the week.