In Exercises 9–20, use the data in the following table, which ...

Question

In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Drinking and Driving If two of the high school drivers are randomly selected, find the probability that they both drove when drinking alcohol.
b. Assume that the selections are made without replacement. Are the events independent?

Survey table: High school drivers texting and drinking behavior. Texted while driving: Yes-731, No-3054. No texting: Yes-156, No-4564.

Drinking and Driving If two of the high school drivers are randomly selected, find the probability that they both drove when drinking alcohol.

b. Assume that the selections are made without replacement. Are the events independent?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says a study on screen habits and sleep patterns collected survey results from high school students aged 16 and older. The table below summarizes responses based on whether students used electronic devices right before bed and whether they reported poor sleep quality. And below the text here, we have a table. In the first row of the table, it is labeled used electronic devices before bed, and then it has two additional columns for reported poor sleep. First column is yes, and it has a value of 820, and the next column is no, and has a value of 2940. For the second row, it is labeled did not use electronic devices, and in two columns about reported poor sleep, under the yes column, we have a value of 210, and under the no column, we have 4030. If 2 students are randomly selected, find the probability that both report or sleep. Assume that the selection was made without replacement. Are the events independent? So for the first part of this problem, we need to find the probability that both students report poor sleep if the two students are randomly selected without replacement. So the first thing we need to do is determine the total number of students who were surveyed. And to do that, we'll just need to add up all the values that are in our table. So we'll call the total number of students surveyed in, and this is going to be equal to 820. Plus 2940. Plus 210 plus 40,030. And add all the values up, we get that this is going to be equal to. 8000. Now, next we need to determine the total number of students who reported for sleep. And so that means just adding up the values under the yes column under reported poor sleep. And so we'll call this some S. And this is going to be equal to 820 plus 210, which is going to be equal to 1030. So now we need to calculate the probability that both students reported poor sleep, and so we'll call that probability P. And we call that a probability, is the number of favorable outcomes divided by the total number of possible outcomes. Now for our favorable outcomes, that's gonna be all possible pairs of poor sleepers. And since we're looking at pairs here, we're gonna use combinatorics. So, the number of pairs of Of students that reported poor sleep is going to be 1030. Choose 2. And then this gets divided by the total number of all possible pairs. Again, we'll use a combinatoric, and so that's going to be 8000. Choose 2. So doing these common torques, this probability is going to be equal to. 529,935. Divided by 31 million 996,000. And during this division, this is going to be approximately equal to. 0.0166. So that is the probability that we were looking for for this problem. So now we need to answer the second half of this problem, which is, are the events independent? Now, since the first selection affects the 2nd, since there's no replacement, that means that the events are not independent. So the answer to the second half of this problem is going to be, no, they are not independent. So those are gonna be the answers for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Probability

Dependent and Independent Events

Conditional Probability

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