The table shows the results of a survey in which 3,545,286 pub...

Question

The table shows the results of a survey in which 3,545,286 public and 509,168 private school teachers were asked about their full-time teaching experience.

Table showing teaching experience of public and private school teachers with counts across four experience ranges and totals.

Are the events “being a public school teacher” and “having more than 20 years of full-time teaching experience” independent? Explain.

Accepted Answer

I read, hello, everyone. So this question says, a hospital has 3200 nurses and 1800 doctors. Of the nurses, 480 have completed advanced training, while 720 doctors have completed advanced training. Are the events being a nurse and having completed advanced training independent? Option A says yes and option B says no. So, to understand the answer to this question, whether or not these two events are independent from each other, we can use the definition of independent events from probability. So if we have two events, A and B. The probability of a. Intersection B should be equal to. The probability of a multiplied by the probability of B in order for them to be independent. So, let's define the events. Let's allow N, for example, to be the probability of the person being a nurse. Nurse with an N, not an M like I wrote earlier. And let's allow A to be the probability that that person completed advanced training. So first Let's compute the individual probabilities. So we have piece of N, which is the probability of the person being a nurse, and piece of A, which is the probability of the person having completed advanced training. So the probability of N would be equal to the 3200 nurses. In the hospital, divided by the sum of 3200 and 1800. Because the denominator represents the total, so you would include the total number. Of staff. Which means adding the 1800 doctors to the denominator. So The probability of N therefore, is equal to 3200 divided by 5000. And that gives you 0.64. So now, you would find the probability of A of the person having completed advanced training. So for the numerator you would add. The total number of advanced trained nurses and doctors, and then divide that by the total number of staff. So P of A, in this case is equal to the sum of 480 and 720. Divided by 5000. And so this Gives you 0.24. That's the probability of A. So now you want to find the probability of the intersection of N and A. So in this case, that's the probability of a person who is a nurse and has completed advanced training. In this case, the numerator would be 480, but the total would still be 5000. That's the total number of staff. And so the probability of N intersecting A is equal to 0.096. And now that you have the information necessary, you can check for independence using the formula that we talked about earlier. So, In this case, you would multiply the probability of N by the probability of A to see if that's equal to the probability of an intersection A. So 0.24 Multiplied by 0.096. Gives you 0.1536. Which in this case is not equal to the probability of an intersecting A. So Because these two values are not equal to each other, the events being a nurse and having completed advanced training are not independent. Which means that option B, no is the correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

The table shows the results of a survey in which 3,545,286 public and 509,168 private school teachers were asked about their full-time teaching experience.

Are the events “being a public school teacher” and “having more than 20 years of full-time teaching experience” independent? Explain.

Key Concepts

Independence of Events

Joint and Marginal Probabilities

Using Contingency Tables for Probability

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