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.

Texting While Driving If two of the high school drivers are randomly selected, find the probability that they both texted while driving.
a. Assume that the selections are made with replacement. Are the events independent?

Texting While Driving If two of the high school drivers are randomly selected, find the probability that they both texted while driving.

a. Assume that the selections are made with 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. Below the texts we're given a table where one row is labeled used electronic devices before bed, and then the next two columns are for reported poor sleep. Under the first column, that's labeled yes, we have a value of 820. And under the next column, which is labeled no, we have a value of 2940. The next row is labeled did not use electronic devices, and under the reported poor sleep columns, under yes, we have a value of 210, and under no, we have 40,030. If two students are randomly selected, find the probability that both report poor sleep. Assume that the selections are made with replacement. Are the events independent? So the first thing we need to do is find the probability that both reported poor sleep if two students are randomly selected. So in order to find this probability, we first need to determine the number of students that were surveyed. So we'll label that value as N. And to get the total number of students surveyed, we just need to add up all the responses that are listed in the table. So N is going to be equal to 820 plus 2940 plus 210 plus 4030. And when we add all these values up, this is going to be equal to 8000. Next, we need to determine how many students reported poor sleep. And so that's just gonna be adding up the values under the yes column under reported poor sleep. 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, we'll now use this information to calculate the probability that we're looking for. So we're looking for the probability that both of our randomly selected students reported poor sleep. So we'll call that probability P. And this is going to be equal to. So, the probability of selecting one student is just gonna be the number of favorable outcomes divided by the total number of outcomes. So, in this case, We will have 1030 as the number of favorable outcomes, and that gets divided by. 8000, which is the total number of outcomes. And so that's gonna be the probability of one student being selected and reporting poor sleep, and we'll just have to multiply this by the probability that a second student, I also selected. So we'll multiply this fraction by, again, the same probability, which is 1030, divided by 8000. And that's because with replacement, we still have the same number of favorable possible outcomes, and the same number of total possible outcomes. So now when we multiply these two quantities together, this probability is going to be approximately equal to 0.0166. So this here is the probability that we're looking for for this problem. Now we need to answer the second half of this question, which is, are the events independent? Now, since we made the selections with replacement, meaning that we replaced the student that we had selected first back into the pool in order to make the second selection, that means that these events are independent, since the selection of the first student doesn't affect the selection of the 2nd student. So that means that the Events are independent, so the answer to the second half of this question is yes, they are independent. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Probability

Independent Events

Sampling with Replacement

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