.78
http://stattrek.com/online-calculator/normal.aspx
Suppose Y ∼ N(μ=0, σ). Find σ so that the normal population contains 90% of its probability between -1 and 1. Please round your answer to two decimals.
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[Hint: The normal distribution is symmetric around 0, so you can simplify the problem and only consider the part of the distribution less than 0. It may be helpful to draw a picture.]
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[Hint: After you find σ, compute Pr{-1 < Y < 1} to check your answer]
Thank you!
2 answers
So, we need 45% of the population to be between 0 and 1, which means we need 5% to be greater than 1 (or less than -1)
Using z = (1 - mu)/sigma, using 0 for mu and sigma being unknown. So z = 1/sigma.
From the standard normal table, we find that the z-score corresponding to a probability of 0.05 is 1.645.
z = 1.645 = 1/sigma, so sigma = 1/1.645 = 0.61, rounded to two significant digits.
As a rough check, the Empirical Rule tells us that 95% of the population should be within 2 SD of the mean, which would be between -1.22 and 1.22, so it's pretty easy to believe that 90% would be between -1 and 1.
Using z = (1 - mu)/sigma, using 0 for mu and sigma being unknown. So z = 1/sigma.
From the standard normal table, we find that the z-score corresponding to a probability of 0.05 is 1.645.
z = 1.645 = 1/sigma, so sigma = 1/1.645 = 0.61, rounded to two significant digits.
As a rough check, the Empirical Rule tells us that 95% of the population should be within 2 SD of the mean, which would be between -1.22 and 1.22, so it's pretty easy to believe that 90% would be between -1 and 1.