The brain volumes (cm cubed) of 20 brains have a mean of 1169.2 cm cubed and a standard deviation of 122.4 cm cubed. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1384.0 cm cubed be significantly high?

Not significantly high

Step-by-step explanation:

Solution:-

- A sample of n = 20 brains was investigated for the total occupied volume in ( cm^3 ).

- The random variable ( X ) would be assigned to the occupied volume of a brain.

- The random variable ( X ) is said to be normally distributed with the following parameters:

                                X ~ Norm ( μ , σ^2 )

- The normal distribution parameters mean ( μ ) and standard deviation ( σ ) are given:

                               X ~ Norm ( 1169.2 , 122.4^2 )

- The rule of thumb for outliers ( significantly high or low ) value with respect to the normal distribution is defined as:

          Significantly High: X > μ + 3*σ

          Significantly Low : X < μ - 3*σ

- These limits corresponds to the property of normal distribution that 99.7% of the data points lie within 3 standard deviations about the mean.

- Any value that lies outside this bound have statistical probability of 0.003 or ( 1.3 % ) significance to the data. Very low significance or value that is considered to be an outlier ( odd value ).

- The bound for the given data can be determined:

                     [ μ - 3*σ < X < μ + 3*σ ]

                     [ 1169.2 - 3*122.4 < X < 1169.2 + 3*122.4 ]

                     [ 802 < X < 1536.4 ] cm^3

- The value of X = 1384.0 cm^3 lies well within the outlier bound, and in fact it lies within 2 standard deviations. This can be determined by computing the standardized Z-score value:

                    Z = ( X -  μ ) / σ

                    Z = ( 1384.0 - 1169.2 ) / ( 122.4 )

                    Z = 1.755 .. (1.75 standard deviation)

- Therefore, for such data a brain volume of 1384.0 cm^3 would NOT be considered significantly high.

5y

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