Finding Probability In Exercises 47–56, find the indicated pro...

Question

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)

Accepted Answer

Welcome back, everyone. Determine the probability that the standard normal variable Z is between negative 2.05 and 2.05. A says it's 0.9596. B 0.9802, C is 0.9798, and D 0.9400. Now to solve this problem, it would be best for us to look at the standard normal distribution to make sense of the problem. I know we know that the standard normal distribution, let me put it to the right here, is really a bell-shaped curve. That's symmetric about the mean mu equals 0 and has a standard deviation sigma that's equal to 1, OK? Now, we would like to figure out the probability that the standard normal variable is between -2.05 and 2.05. So both of these values represent the number of standard deviations from the mean. So on our curve, -2.05 would be to the left of the mean, and the 2.05 would be to the right of the mean. And essentially we're trying to figure out the probability that our standard normal variable is going to fall within this region that's shaded red. So the question is, how can we do that? Let me finish shading it here. Cause now the question is how can we find that probability out? Well, we can use a Z table to help us because the Z table gives us values for probabilities or, or the probability that Z is going to be less than a certain value. OK. So, for example, here. In our Z table, it will give us the probability that Z, let me make some space actually. It's gonna give us the probability that Z. OK. I probably say that Z. Is less than 2.05, OK? And which again, if I put this in blue, it would be all of this shaded region. And it would also give us the probability, OK. It would also give us the probability. That Z The probability that Z is less than -2.05, OK? In other words, all of this shaded region here in red. So now by looking at it you might notice that here the probability. That Z is gonna be between -2.05 and 2.05 would be equivalent to the probability that Z is less than 2.05. Minus the probability that Z is less than -2.05, OK? Because if we subtract this shaded region from this entire one, then we'll get the area that we would like, the area that's, again, let me make some space here, but the area that's essentially shaded in red. So all we need to do is to find these probabilities from our table. Now, from the Z table, the probability that Z is less than 2.05 is 0.9798. And the probability it's less than negative 2.05 is 0.0202. So that means this probability will be equal to 0.9798 minus 0.0202, which equals 0.9596. This is the probability that Z is between -2.05 and 2.05. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)

Key Concepts

Standard Normal Distribution

Z-scores

Probability Calculation

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Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.P(- 1.54 < z < 1.54)