The Random Variable X is normally distributed with mean 560 and standard deviation 20.
Find P(X<530) which I calculated as 0.0668.
However... Next question asks
It is known that p(|X-560|<a)
Find the value of a
Pick some values for p, and play around at this website. As given, since you have no value for p, you cannot find a.
http://davidmlane.com/hyperstat/z_table.html
To find the value of "a" such that p(|X-560|<a) is known, we need to understand what this probability expression means.
The expression p(|X-560|<a) represents the probability that the absolute difference between the random variable X and its mean 560, denoted by |X-560|, is less than some value "a".
In other words, p(|X-560|<a) is the probability of X falling within a distance of "a" from the mean (560).
Since X follows a normal distribution with mean 560 and standard deviation 20, we can use properties of the normal distribution to find the value of "a".
The key idea is that approximately 95% of the data lies within two standard deviations of the mean in a normal distribution. Therefore, we can set the value of "a" to be 2 times the standard deviation (20) to capture 95% of the data:
a = 2 * standard deviation
a = 2 * 20
a = 40
So, the value of "a" is 40.
This means that p(|X-560|<40) is the probability of X falling within 40 units of the mean 560, which captures approximately 95% of the data.