Given that x is a normally distributed random variable with a mean of 28 and a standard deviation of 7, find the following probability:
P(x<28)
I understand that the means is 28 and the standard deviation is 7. However what I don't understand is how to get the value of x.
Do I subtract 28 from 7 to get 21 and square it for the answer?
or do I use the formula of z= x-mean/standard deviation? and if so how do I work the formula with out the value of x?
Use Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion below that Z score.
To find the probability P(x < 28) for a normally distributed random variable with a mean of 28 and a standard deviation of 7, you don't need to calculate the value of x directly. Instead, you can use the properties of the normal distribution and the concept of z-scores.
The z-score formula you mentioned, z = (x - mean) / standard deviation, is helpful here. It allows you to standardize any given value of x to its corresponding z-score.
In this case, since you want to find P(x < 28), you are looking for the probability of getting a value less than 28. Using the formula, you can calculate the z-score as follows:
z = (28 - 28) / 7 = 0 / 7 = 0
Since the mean is 28 and x = 28, the z-score is 0.
Next, you can refer to the standard normal distribution table (also called the z-table) or use statistical software to find the probability associated with the z-score 0. For a standard normal distribution, the probability associated with a z-score of 0 is 0.5000.
Therefore, P(x < 28) = 0.5000 or 50%.
To summarize:
1. Calculate the z-score using the formula: z = (x - mean) / standard deviation.
In this case, z = (28 - 28) / 7 = 0 / 7 = 0.
2. Refer to the z-table or use statistical software to find the probability associated with the z-score of 0.
The probability associated with z = 0 is 0.5000.
3. Therefore, P(x < 28) = 0.5000 or 50%.