SOLUTION: Hi, please help me understand how to tackle this problem: For a normal distribution with mean μ = 81.2 feet and standard deviation σ = 12.4 feet, what range of values

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Question 1080441: Hi, please help me understand how to tackle this problem:
For a normal distribution with mean μ = 81.2 feet and standard deviation σ = 12.4 feet, what range of values has approximately 95% of the data between them? Please explain your reasoning.
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
For a normal distribution with mean μ = 81.2 feet and standard deviation σ = 12.4 feet, what range of values has approximately 95% of the data between them? Please explain your reasoning.
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Center the 95% on the mean
Find the z-value with a left-tail of 2.5%
invNorm(0.025) = -1.96
Find the z-value with right-tail of 2.5%
By symmetry that answer is +1.96
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Find the corresponding raw score values
x = 81.2-1.96*12.4 = 56.90 (left limit)
x = 81.2+1.96*12.4 = 105.5 (right limit)
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Cheers,
Stan H.
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Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
By the empirical rule, roughly 95% of the distribution is between z = -2 and z = 2. In other words,

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Let's find the raw score (x) for the z score of -2

z+=+%28x-mu%29%2F%28sigma%29

z+=+%28x-81.2%29%2F%2812.4%29 Plug in the given mu (greek letter mu) and sigma (greek letter sigma) values.

-2+=+%28x-81.2%29%2F%2812.4%29

-2%2A12.4+=+x-81.2 Multiply both sides by 12.4

-24.8+=+x-81.2

-24.8%2B81.2+=+x-81.2%2B81.2 Add 81.2 to both sides

56.4+=+x

x+=+56.4

So if x+=+56.4, then the z-score is z+=+-2

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Let's repeat the same steps as above, but now for z+=+2

z+=+%28x-mu%29%2F%28sigma%29

z+=+%28x-81.2%29%2F%2812.4%29 Plug in the given mu and sigma values.

2+=+%28x-81.2%29%2F%2812.4%29 Plug in the given z score

2%2A12.4+=+x-81.2 Multiply both sides by 12.4

24.8+=+x-81.2

24.8%2B81.2+=+x-81.2%2B81.2 Add 81.2 to both sides

106+=+x

x+=+106

So if x+=+106, then the z-score is z+=+2

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In summary, the expression is the same as

Answer: The range of values is from 56.4 feet to 106 feet