Let’s say the random variable X is normally distributed with a mean (µ) = 35 and standard deviation (σ) = 5. Can you find P(25 < X < 45), i.e., the probability of the variable falling between 25 and 45? Note 1: Report the answer as a number rounded off to two digits after the decimal point.Note 2: You’re not expected to know the right answer at this. This question is given just to get you thinking in the right direction
Question
Let’s say the random variable X is normally distributed with a mean (µ) = 35 and standard deviation (σ) = 5. Can you find P(25 < X < 45), i.e., the probability of the variable falling between 25 and 45? Note 1: Report the answer as a number rounded off to two digits after the decimal point.Note 2: You’re not expected to know the right answer at this. This question is given just to get you thinking in the right direction
Solution
To find the probability of the variable falling between 25 and 45, we first need to convert these values to z-scores. The z-score is a measure of how many standard deviations an element is from the mean.
The formula to calculate the z-score is:
Z = (X - µ) / σ
Where: X is the value from the dataset, µ is the mean of the dataset, and σ is the standard deviation of the dataset.
First, let's calculate the z-score for 25:
Z_25 = (25 - 35) / 5 = -2
Next, let's calculate the z-score for 45:
Z_45 = (45 - 35) / 5 = 2
Now, we need to find the probability that the z-score is between -2 and 2. This is the same as finding the area under the standard normal curve between -2 and 2.
Looking up these values in the z-table, we find that:
P(Z < 2) = 0.9772 (this is the area to the left of Z = 2) P(Z < -2) = 0.0228 (this is the area to the left of Z = -2)
To find the probability that Z is between -2 and 2, we subtract the smaller probability from the larger one:
P(-2 < Z < 2) = P(Z < 2) - P(Z < -2) = 0.9772 - 0.0228 = 0.9544
So, the probability of the variable falling between 25 and 45 is approximately 0.95, or 95%, when rounded off to two digits after the decimal point.
Similar Questions
Let’s say the random variable X is normally distributed with µ = 35 and σ = 5. Find P(25 < X < 50), i.e., the probability of the variable taking a value between 25 and 50. (Report the answer as a number rounded off to two digits after the decimal point.)Note: You’re not expected to know the right answer at this point. This question is given just to get you thinking in the right direction.
State the approximate probability that a normal random variable will fall within the following intervals:Mean plus or minus one standard deviation
What is the probability that a normal random variable will take a value that is less than 1.05 standard deviations above its mean? In other words, what is P(Z < 1.05)?0.85310.14680.93320.0668What is the probability that a normal random variable will take a value that is between 1.5 standard deviations below the mean and 2.5 standard deviations above the mean? In other words, what is P(−1.5 < Z < 2.5)?0.99380.06680.92700.0730What is the probability that a normal random variable will take a value that is more than 2.55 standard deviations above its mean? In other words, what is P(Z > 2.55)?0.99450.99460.00550.0054
Probability of Normal Random VariablesLet’s say that you need to find the cumulative probability for a random variable X which is normally distributed. You do not know what the value of X is or, for that matter, what the value of µ and σ is. You only know that X = µ + σ. Can you find the cumulative probability, i.e. the probability of the variable being less than µ + σ?
If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to 5, what is the probability that a sample mean X¯ will fall in the interval from μX¯ −1.9σX¯ to μX¯ −0.4σX¯ ? Assume that the sample means can be measured to any degree of accuracy.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.