Christopher B. answered • 05/07/16
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I'm sorry, Shirley, I would be glad to help, but I'm not exactly sure what you are asking. Are you stating that the probability of the sample mean being 105 minutes is approximately nothing or are you asking if it is? Maybe the communication breakdown is because you are trying to rephrase the question asked on your assignment, but you aren't quite certain what exactly is being asked. If you would try to word the question as closely as you can to how it is stated on your assignment then I would be happy to help interpret the question and provide you with the information that you will need to solve the problem on your own.
Christopher B.
No problem. Makes way more sense now, thanks. I'll answer for you in a little bit when I get home.
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05/07/16
Christopher B.
Okay, so we are given that the time between eruptions is normally distributed so we can use z-scores to calculate the area under a standardized normal curve. A z-score gives us the area to the left of the value in question and that area represents the probability that our normally distributed random variable will assume a value less than that value. In this question, we want the probability that the mean time between eruptions is greater than 105 minutes, so it will be easier to calculate the the complementary probability of the sample mean being below 105 minutes. That is:
P (X > 105) = 1 - P ( X ≤ 105) = 1 - P ( z ≤ (105 - 96)/ 20) = 1 - Φ(.45)
The term Φ(.45) represents the area to the left of .45 on the standard normal curve. It is usually evaluated via a chart that lists outputs of this function for corresponding z-scores. It can also be calculated by evaluating the corresponding integral that defines the area under the standard normal curve. I suspect that your instructor will only require you to use the table to find the output. Some things to remember that may be helpful are: the area under the standard normal curve is not actually 1, but is taken to be 1 in order to represent total probability, z-scores denote how many standard deviations to the right or left of the population mean a value is and the function Φ(z) gives the area of the curve to the left of a z-score for a particular value and is equivalent to the probability of the normal random variable assuming values less than or equal to that value. So, if you want to compute the probability that a normal random variable will take on values greater than a certain value, you will need to take the complementary probability. That is, you will have to take the z score of that value and use a table to find the corresponding area under the curve. Then subtract that value from one and you will have the desired probability, like we have done here.
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05/08/16
Shirley V.
05/07/16