For a Normal distribution with mean, mu=2, and standard deviation, sigma = 4,
70% of all observations have values between x and y . (Round to 4 decimal places.)
Consult your standard Z table. 35% of all observations occur between 0 and 1.04σ. So, 70% occur between -1.04σ and 1.04σ.
So, that means that the actual values occur in
2-1.04(4) <= x <= 2+1.04(4)
-2.16 <= x <= 6.16
To find the values x and y for which 70% of the observations fall between them in a Normal distribution, we need to use the z-score. The z-score measures the number of standard deviations a particular value is from the mean.
First, we need to find the z-scores corresponding to the lower and upper percentiles. Since we want to find the values that include 70% of the observations, we subtract half of that percentage from 50%. Thus, we subtract 0.15 (i.e., half of 0.70) from 0.50.
Next, we need to find the corresponding z-scores using a standard normal distribution table (also called a z-table) or a statistical calculator. The z-score corresponding to the percentile is the value that has that percentile to its left in the standard normal distribution.
Using a z-table or a statistical calculator, we find that the z-score corresponding to the 0.35th percentile is approximately -0.3853 (rounded to 4 decimal places). To find the corresponding observation value (x), we use the formula:
x = mu + (z * sigma)
where mu is the mean and sigma is the standard deviation. Plugging in the given values, we get:
x = 2 + (-0.3853 * 4) = 0.8577 (rounded to 4 decimal places).
Similarly, for the upper value (y), we use the corresponding z-score to the 99.65th percentile, which is approximately 1.8364 (rounded to 4 decimal places). Plugging in the values:
y = 2 + (1.8364 * 4) = 9.3456 (rounded to 4 decimal places).
Therefore, 70% of all observations have values between approximately 0.8577 and 9.3456.