Answer the following questions. (a) The random variable x is distributed normally, with x~N(80,100) Find the probability that x is greater than 90. x >90. (b) Find P(x<85). (c) Find P(76<x<84).
Do not understand "x~N(80,100)".
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
To find the probabilities in the given normal distribution, we can use the standard normal distribution table or z-scores.
(a) To find the probability that x is greater than 90 (x > 90), we need to convert the given x-value into a z-score and find the corresponding probability.
Step 1: Calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we want to convert, μ is the mean, and σ is the standard deviation.
In this case, x = 90, μ = 80, and σ = 10.
z = (90 - 80) / 10 = 1
Step 2: Use the standard normal distribution table or a calculator to find the probability corresponding to the z-score. The standard normal distribution table provides probabilities for z-values up to a certain decimal place. We need to find the probability for z > 1.
Looking up the z-value of 1 in the standard normal distribution table, we find that the probability is approximately 0.8413.
So, P(x > 90) ≈ 1 - 0.8413 = 0.1587
Therefore, the probability is approximately 0.1587 or 15.87%.
(b) To find P(x < 85), we follow a similar approach.
Step 1: Calculate the z-score:
z = (x - μ) / σ
In this case, x = 85, μ = 80, and σ = 10.
z = (85 - 80) / 10 = 0.5
Step 2: Use the standard normal distribution table or a calculator to find the probability corresponding to the z-score. The standard normal distribution table provides probabilities for z-values up to a certain decimal place. We need to find the probability for z < 0.5.
Looking up the z-value of 0.5 in the standard normal distribution table, we find that the probability is approximately 0.6915.
So, P(x < 85) ≈ 0.6915
Therefore, the probability is approximately 0.6915 or 69.15%.
(c) To find P(76 < x < 84), we need to find the probability between two x-values.
Step 1: Calculate the z-scores for both values:
z1 = (76 - μ) / σ
z2 = (84 - μ) / σ
In this case, x1 = 76, x2 = 84, μ = 80, and σ = 10.
z1 = (76 - 80) / 10 = -0.4
z2 = (84 - 80) / 10 = 0.4
Step 2: Use the standard normal distribution table or a calculator to find the probability between the two z-scores. The standard normal distribution table provides probabilities for z-values up to a certain decimal place. We need to find the probability for -0.4 < z < 0.4.
Looking up the z-values of -0.4 and 0.4 in the standard normal distribution table, we find that the respective probabilities are approximately 0.3446 and 0.6554.
So, P(76 < x < 84) ≈ 0.6554 - 0.3446 = 0.3108
Therefore, the probability is approximately 0.3108 or 31.08%.