I need help with my STAT homework!

Question

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!

Answer 1

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.