given that x~n(300,15), find the 70th percentile.
First, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion and its related Z score. Insert values in equation below to find the score.
Z = (score-mean)/SD
307.87
To find the 70th percentile for a normally distributed variable, you can use the cumulative distribution function (CDF) or z-score formula.
1. Using the CDF method:
The cumulative distribution function (CDF) gives you the probability that a random variable is less than or equal to a specific value. In this case, we want to find the value at which the cumulative probability is 0.70.
Step 1: Standardize the variable:
To use the CDF, we need to convert our x value to a standard normal distribution with a mean of 0 and a standard deviation of 1. We can do this by using the formula: z = (x - μ) / σ, where z is the standardized value, x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
Given that x ~ N(300, 15), we can plug in the values:
z = (x - 300) / 15
Step 2: Find the z-score:
To find the z-score, we look up the corresponding value in the standard normal distribution table or use a calculator. The z-score for a cumulative probability of 0.70 is approximately 0.5244.
Step 3: Solve for x:
To solve for x, we rearrange the z-score formula and substitute in the known values:
x = z * σ + μ
x = 0.5244 * 15 + 300
x ≈ 307.87
Therefore, the 70th percentile for x ~ N(300, 15) is approximately 307.87.
2. Using the z-score method:
Another way to find the percentile is by directly using the z-score.
Step 1: Find the z-score:
The z-score formula is still the same: z = (x - μ) / σ. Plugging in the values, we get:
z = (x - 300) / 15
Step 2: Solve for x:
Rearranging the formula to solve for x, we get:
x = z * σ + μ
Substituting the known values:
x = 0.5244 * 15 + 300
x ≈ 307.87
Again, we find that the 70th percentile for x ~ N(300, 15) is approximately 307.87.
Note: The percentile in this case represents the value below which 70% of the data falls.