Suppose X is a uniform random variable with C = 20 and d = 90. Find the probability that a randomly selected observation is between 23 and 85.
I don't know what C and d indicate.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores.
To find the probability that a randomly selected observation is between 23 and 85, given a uniform random variable X with C = 20 and D = 90, you need to compute the cumulative distribution function (CDF) at these two values.
The CDF of a uniform random variable is given by:
CDF(x) = (x - C) / (D - C)
where x is the value of interest, and C and D are the lower and upper limits of the random variable.
Let's calculate the CDF at x = 23 and x = 85:
CDF(23) = (23 - 20) / (90 - 20) = 3 / 70
CDF(85) = (85 - 20) / (90 - 20) = 65 / 70
The probability that a randomly selected observation is between 23 and 85 is then given by the difference between these two CDF values:
P(23 ≤ X ≤ 85) = CDF(85) - CDF(23)
= 65/70 - 3/70
= 62/70
= 31/35
Therefore, the probability that a randomly selected observation is between 23 and 85 is 31/35.
To find the probability that a randomly selected observation is between 23 and 85, we need to calculate the cumulative probability for the upper bound (85) and subtract the cumulative probability for the lower bound (23).
To calculate the cumulative probability, we first need to standardize the values using the formula:
Z = (X - C) / d
Where X is the value you want to standardize, C is the mean, and d is the standard deviation.
For the upper bound (85):
Z_upper = (85 - 20) / 90 = 65 / 90 ≈ 0.72
Next, we need to calculate the cumulative probability associated with Z_upper. This can be done using a standard normal distribution table or a calculator. For example, using a calculator or software, the cumulative probability for Z_upper can be found to be approximately 0.7659.
For the lower bound (23):
Z_lower = (23 - 20) / 90 = 3 / 90 ≈ 0.03
Again, calculate the cumulative probability associated with Z_lower, which is approximately 0.5120.
Finally, subtract the cumulative probability of the lower bound from the cumulative probability of the upper bound to find the probability between 23 and 85:
P(23 ≤ X ≤ 85) = P(X ≤ 85) - P(X ≤ 23)
≈ 0.7659 - 0.5120
≈ 0.2539
Therefore, the probability that a randomly selected observation is between 23 and 85 is approximately 0.2539.