Suppose that you have a normally distributed population with unknown μ and standard deviation σ = 20. Given that the probability that a random observation X will fall within the range μ±E is .95 or 95%, find E.
95% = mean ± 1.96 SD
1.96 (20) = ?
To find the value of E, we need to determine the critical value associated with the desired probability and then multiply it by the standard deviation.
Step 1: Find the critical value, denoted as z, using the Z-table or a statistical calculator. In this case, since we want to find the range within μ±E that covers 95% of the population, we need to find the critical value that leaves 2.5% in each tail of the distribution.
Looking up the standard normal distribution table, we find that the critical value associated with an area of 0.025 in the right tail is 1.96. This corresponds to an area of 0.975 in the left tail.
Step 2: Once we have the critical value (z = 1.96), we can set up the equation using the formula for the confidence interval:
E = z * σ
where σ represents the standard deviation of the population, which is given as 20.
Plugging in the values:
E = 1.96 * 20
E ≈ 39.2
Therefore, the value of E is approximately 39.2. This means that with a 95% confidence level, the range within μ±E will cover 95% of the population.