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!
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.
To find the value of σ such that the normal population contains 90% of its probability between -1 and 1, we can use the properties of the standard normal distribution.
Step 1: Simplify the problem
Since the normal distribution is symmetric around 0, we only need to consider the part of the distribution less than 0. This means we can find the value of σ for which the standard normal distribution has 95% of its probability between 0 and 1.
Step 2: Locate the critical value
To find the critical value, we need to find the z-score associated with the cumulative probability of 0.95. We can look up this value in a standard normal distribution table or use a calculator. From the table, we find that the z-score corresponding to a cumulative probability of 0.95 is approximately 1.645.
Step 3: Calculate σ
Using the z-score formula, we have:
z = (x - μ) / σ
Since we are looking at the part of the distribution less than 0, we have:
0.95 = (0 - 0) / σ
Simplifying the equation, we get:
1.645 = 0 / σ
Since the standard deviation σ cannot be zero, we can conclude that the solution to the equation is σ = 0.
So, the value of σ that ensures 90% of the normal population's probability is between -1 and 1 is 0.
To double-check our answer, we can calculate the probability Pr{-1 < Y < 1} using the standard normal distribution formula. Substituting the values into the formula, we get:
Pr{-1 < Y < 1} = Φ(1) - Φ(-1)
Using the standard normal distribution table or a calculator, we can find the values of Φ(1) and Φ(-1), which are approximately 0.8413 and 0.1587 respectively.
Hence,
Pr{-1 < Y < 1} = 0.8413 - 0.1587 = 0.6826
The probability is approximately 0.6826, which confirms our earlier solution.
.78
http://stattrek.com/online-calculator/normal.aspx