Assign your numbers for mean μ and standard deviation σ.
Then select one number "a" bellow μ and one number "b" above μ.
Choose these numbers not far from μ so we can use z.
Find probability that x is between a and b: P(a < x < b)
To find the probability that x is between a and b, denoted as P(a < x < b), we can use the standard normal distribution with mean μ and standard deviation σ.
1. First, assign numerical values to μ and σ. Let's say μ = 0 and σ = 1 for simplicity (these are the parameters for the standard normal distribution).
2. Next, select a number "a" below μ and a number "b" above μ. Make sure these numbers are not far from μ so that we can use z-scores.
Let's assume a = -1 and b = 1 for this example.
3. Calculate the z-scores for a and b using the formula z = (x - μ) / σ.
For a: z1 = (a - μ) / σ = (-1 - 0) / 1 = -1
For b: z2 = (b - μ) / σ = (1 - 0) / 1 = 1
These z-scores represent the number of standard deviations away from the mean.
4. Look up the z-scores in the standard normal distribution table or use a calculator to find the area under the curve between the z-scores.
From the standard normal distribution table or a calculator, the area between -1 and 1 is approximately 0.6827.
Therefore, the probability that x is between -1 and 1 (P(-1 < x < 1)) is 0.6827 or 68.27%.
Note: If you chose different values for μ and σ, make sure to adjust the z-scores accordingly. The process remains the same.