Interpreting Two-Way Frequency Tables

Interpreting Two-Way Frequency Tables

The Lesson Activities will help you meet these educational goals:

• Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively, and look for and make use of structure.
• STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.
• 21stCentury Skills—You willuse critical-thinking and problem-solving skills.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

______

Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1. Two-Way Frequency Tables and Conditional Probability

This two-way frequency table gives the results of a screening test for heart disease conducted by determining the cholesterol level in the blood samples of 750 patients.

1. Using the two-frequency table, compute the probability of having high cholesterol. Or P(high cholesterol).

Sample answer:

P(high cholesterol)

1. Using the two-way frequency table, compute the probability of having heart disease, or P(heart disease).

Sample answer:

1. Using the two-way frequency table, compute the probability of a patient having heart disease, given that the patient has high cholesterol, orP(heart disease|high cholesterol).

Sample answer:

Method 1:Use the standard definition of conditional probability,

P(heart disease|high cholesterol)

Method 2:Just by looking at the two-way table, I can see that the sample space of all people with high cholesterol is 372 and that those with high cholesterol who also have heart disease is 223, so:

P(heart disease|high cholesterol)

This is the same result as the first, but is much more easily seen and calculated.

1. Using the two-frequency table, compute the probability of a patient nothaving heart disease, given that the patient has normal cholesterol, or P(no heart disease|normal cholesterol).

Sample answer:

P(no heart disease|normal cholesterol)

1. Fill in the missing data in the following two-way frequency table for a different screening test.

Sample answer:

1. What is the probability of a false negative for this test,given that there was a negative test result?

Sample answer:

False negatives = 7 (the number of people who got a negative test result but have the disease).

To find the probability of a false negative given that there was a negative test result, we need to restrict the sample space to the number of people with a negative test result, that is, 574.

P(false negative|negative test result)

1. Two-Way Frequency Tables and Independence

1. In a recent survey for an upcoming city mayoral election, people were asked to name the political party they identified with and also the party of the candidate they were going to vote for.

• Of the 150 people who identified themselves as Democrats, 133 said they would vote for the Democratic candidate. The rest said they would vote for the Republican.
• Of the 160 people who identified themselves as Republican, 142 said they would vote for the Republican candidate. The rest said they would vote for the Democrat.

Complete the two-way frequency table for this situation.

Sample answer:

1. Use your two-way frequency tableto determine this conditional probability:

P(person will vote for the Republican candidate|person is a Democrat).

Sample answer:

P(person will vote for Republican candidate|person is a Democrat)

1. Use your two-way frequency tableto determine this conditional probability:

P(person will vote for the Republican candidate|person is a Republican).

Sample answer:

P(person will vote for the Republican candidate|person is a Republican)

1. Are being a Democrat and voting for the Democratic candidate independent?How do you know? Show your work.

Sample answer:

The probability of voting for the Democratic candidate can be calculated as:

P(person will vote for the Democratic candidate)

The probability of voting for the Democratic candidate given that the person is a Democrat can be calculated as:

P(person will vote for the Democratic candidate|person is a Democrat)

Since these two values are not equal, being a Democrat and voting for the Democratic candidate are not independent.

This two-way frequency table shows the results of a survey of users of the mass-transit system in a city.People were asked their age and whether they used the system regularly (three or more times per week).

1. Find P(uses mass transit|age is 30 or under)

Sample answer:

With the condition that age is 30 or under, the sample space is restricted to the first column. This simplifies the problem to:

P(uses mass transit |age is 30 or under)

1. Find P(age is over 30|does not use mass transit)

Sample answer:

With the condition does not use mass transit, the sample space is restricted to the second row. This simplifies the problem to:

P(age is over 30|does not use mass transit)

1. Are using mass transit and being over 30 years old independent? How do you know?

Sample answer:

The probability of using mass transit can be calculated as:

P(uses mass transit)

The probability of using mass transit given that a person is over 30 years old can be calculated as:

P(uses mass transit|age is over 30)

Since these two values are not equal, using mass transit and being over 30 years old are not independent.

1