In Exercises 1–5, assume that 4.2% of workers test positive wh...

Question

In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.

Workplace Drug Testing If four of the ten workers test positive for illegal drugs, is that a significantly high result?

Accepted Answer

All right, hello, everyone. So this question says, a local health department finds that 3% of the community is allergic to a common vaccine. Assume that a group of 12 individuals is randomly selected for vaccination. If 2 out of the 12 individuals experience an an allergic reaction, is this result considered significantly high? All right, so first, let's take a moment to outline the information we've been given. First, our sample size that's N is equal to 12. And then of those 12, X, which is the number of people that experience the allergic reaction in this case, is equal to 2. Now the expected. Value, right, that's P is equal to 0.03. That's 3% as a decimal. So first, what we want to do is calculate mu, which is the mean, and that's equal to N multiplied by P. So 12 multiplied by 0.03. Equals 0.36. And now we can use this to calculate the standard deviation. Sigma, that's the standard deviation. is equal to N multiplied by P multiplied by 1 subtracted by P. So this gives you the square root of 12, multiplied by 0.03. Multiplied by 1 subtracted by 0.03. And so after evaluating this expression, sigma is approximately equal to 0.591. Now, before we go any further, let's take a moment to actually define what a significantly high result even is. In this context, a significantly high result. Would be if the probability of X being greater than or equal to 2. Is less than or equal to 0.05. So now we have to actually calculate the probability of X being greater than or equal to 2. Now to do this. There are 2 probabilities that we need to find. We have to find the probability of X being equal to 0. And the probability of X being equal to 1. And we'll see why shortly. But for now, recall that the probability of X being equal to 0 equals the following. That's 12 divided by. 0 Or rather 12 and 0. Multiplied by 0.03 to the power of 0. And multiplied by 0.97, multiplied to the power of 12 rather. Recall that 0.97 is equal to 1 subtracted by P, which was 0.03. So now, this expression is equal to approximately 0.694. And now we do the same thing but with the probability of X being equal to 1. So that's 12 divided by 1. Multiplied by 0.03 to the first power. And multiplied by 0.97 to the 11th power. And this gives you approximately 0.257. So now, the two values that we just found are going to be used to find the probability of X being greater than or equal to 2. So that Is going to be equal to. 1, subtracted by the probability of X being equal to 0. Subtracted by the probability of x being equal to 1. So this expression simplifies to 1 subtracted by 0.694. Subtracted by 0.257. Which gives you 0.049. And now we compare Right, here we can see that our calculated value, let me actually move this. More to the left. We can see now that our calculated value of 0.049 is less than or equal to 0.05. Which means that getting 2 or more allergic reactions is in fact significantly high. Which brings us to our final answer. With just states, yes. It is significantly high. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Binomial Distribution

Significance Level

Hypothesis Testing

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