Random variable X is normally distributed with mean 10 and standard deviation 2.
Compute the following probabilities.
a. Pr(X<10)
b. Pr(X<11.04)
I don't know where to start.
a) P(X<10) gives a z-score of (10-10)/2 = 0
b) P(X<11.04) gives a z-score of (11.04-10)/2 = .52
You must be given a table of "normal distribution" or have access to a program that duplicates the table or chart.
I use this one
http://davidmlane.com/hyperstat/z_table.html
You can use it directly without finding the z-scores
e.g. for b)
1.
enter: mean: 10 , Sd: 2, click on below and enter 11.04 to get .698468
2.
enter: mean: 0 , Sd: 1 , click on "below" and enter the z-score of .52 to get .698468
To compute the probabilities, you can use the standard normal distribution table or a calculator with built-in normal distribution functions. However, we need to standardize the random variable X first. Here's how you can do it:
Step 1: Standardize the random variable X.
To standardize a random variable, you subtract the mean and divide by the standard deviation. In this case, X has mean 10 and standard deviation 2, so we standardize X as follows:
Z = (X - mean) / standard deviation
= (X - 10) / 2
Step 2: Use the standard normal distribution table or a calculator to find the probabilities.
a. Pr(X < 10)
To compute Pr(X < 10), we need to find the probability of the standardized value being less than 0.
So, we calculate Pr(Z < 0) using the standard normal distribution table or a calculator. The result is the probability of Z being less than 0, which is approximately 0.5.
Therefore, Pr(X < 10) is also approximately 0.5.
b. Pr(X < 11.04)
First, we need to standardize the value 11.04:
Z = (11.04 - 10) / 2
= 1.04 / 2
= 0.52
Then, we calculate Pr(Z < 0.52) using the standard normal distribution table or a calculator. The result is the probability of Z being less than 0.52.
For example, if we use a standard normal distribution table, we find that Pr(Z < 0.52) is approximately 0.699. Therefore, Pr(X < 11.04) is approximately 0.699.
Note: The calculation may vary slightly depending on the level of precision desired and the method or table used to compute the probabilities.
To compute probabilities for a normal distribution, we usually use the standard normal distribution table, also known as the Z-table or the cumulative distribution function (CDF) table.
However, before we can use the standard normal distribution table, we need to standardize the random variable X.
Standardizing a random variable involves subtracting the mean and dividing by the standard deviation. This process transforms X into a standard normal distribution with mean 0 and standard deviation 1.
For any random variable X normally distributed with mean μ and standard deviation σ, the standardization formula is:
Z = (X - μ) / σ
Now let's apply this to the given problem:
a. Pr(X < 10)
To calculate Pr(X < 10), we first need to standardize the value 10.
Z = (10 - 10) / 2 = 0 / 2 = 0
Since we standardized the value to 0, we need to find the probability associated with Z = 0 in the standard normal distribution table.
Using the standard normal distribution table, the probability of Z being less than or equal to 0 is 0.5000.
Therefore, Pr(X < 10) = 0.5000.
b. Pr(X < 11.04)
To calculate Pr(X < 11.04), we need to standardize the value 11.04 using the same formula:
Z = (11.04 - 10) / 2 = 1.04 / 2 = 0.52
Now we can look up the probability associated with Z = 0.52 in the standard normal distribution table. In the table, we find that the probability of Z being less than 0.52 is 0.6985.
Therefore, Pr(X < 11.04) = 0.6985.