A normal distribution has mean 102 and sigma = 17. What percentile is x = 127
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We are given a normal distribution with mean 102 and standard deviation 17. Our goal is to find what percentile the value x equals 127 represents. This means we need to determine what percentage of values in this distribution fall below 127.
A percentile tells us what percentage of data falls below a certain value. In our case, finding the percentile for x equals 127 means calculating the area under the normal curve to the left of this point. This shaded area represents that percentage.
To find this percentile, we standardize our value using the z-score formula: z equals (x minus mean) divided by standard deviation. Plugging in our values, z equals (127 minus 102) divided by 17, which is 25 over 17, approximately 1.47. This transformation allows us to use the standard normal distribution.
In the standard normal distribution, which has a mean of 0 and standard deviation of 1, we now look for the area to the left of z equals 1.47. This area directly gives us the percentile we're seeking.
Looking up z equals 1.47 in a standard normal table or using a calculator, we find that the probability of Z being less than or equal to 1.47 is approximately 0.9292. This means x equals 127 is at approximately the 93rd percentile of our original distribution.
To summarize: We started with a normal distribution with mean 102 and standard deviation 17. We wanted to find the percentile for x equals 127. We calculated the z-score to be approximately 1.47. Using the standard normal distribution, we found the area to the left of this z-score is about 0.9292, or 92.92 percent. Therefore, x equals 127 is at approximately the 93rd percentile of this distribution.