Question

Compute the range and sample standard deviation for strength of the concrete​ (in psi).

Answer 1

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Answer 2

Answer:

$Range = 1150\ psi$

$Standard\ Deviation = 442.3\ psi$

Explanation:

Given

3920​, 4090​, 3300​, 3100​, 2940​, 3830​, 4090​, 4030

Required

- Determine the Range

- Determine the Standard Deviation

Calculating the Range...

The Range is calculated using the following formula;

$Range = Highest\ Strength - Least\ Strength$

From the given data;

$Highest\ Strength = 4090\\ Least\ Strength = 2940$

Hence,

$Range = 4090 - 2940\\\\Range = 1150\ psi$

Calculating the Standard Deviation...

Start by calculating the mean

$Mean = \frac{\sum x}{n}$

Where x->3920​, 4090​, 3300​, 3100​, 2940​, 3830​, 4090​, 4030

n = 8

$Mean = \frac{3920 + 4090 + 3300 + 3100+ 2940+ 3830+ 4090+4030}{8}$

$Mean = \frac{29300}{8}$

$Mean = 3662.5$

Subtract the mean from each observation

$3920​ - 3662.5 = 257.5\\4090​ - 3662.5 = 427.5\\3300 - 3662.5 = -362.5\\3100 - 3662.5 = -562.5\\2940 - 3662.5 = -722.5\\3830 - 3662.5 = 167.5\\4090 - 3662.5 = 427.5\\4030 - 3662.5 = 367.5$

Square the result of the above

$257.5^2 =66,306.25\\427.5^2 =182,756\\-362.5^2 =131,406.25\\-562.5^2 =316,406.25\\-722.5^2 =522,006.25\\167.5^2 =28,056.25\\427.5^2 =182,756.25\\367.5^2 =135,056.25$

Add the above results together

$66,306.25+182,756+131,406.25+316,406.25+522,006.25+28,056.25+182,756.25+135,056.25 = 1564749.75$

Divide by n

$\frac{1564749.75}{8} = 195593.71875$

Take Square root of the above result to give standard deviation

$SD = \sqrt{195593.71875}$

$SD = 442.259786494$

$SD = 442.3\ psi\ (Approximated)$

Hence,

$Range = 1150\ psi$

$Standard\ Deviation = 442.3\ psi$