If 95 % confidence interval for the population mean was reported to be 160 to 170 and σ=25, then size of the sample used in this study is: (Given…

Let's find $|\vec{b}|$∣b∣ using the given conditions.

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We start by expanding the dot product $(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b})$(a−b)⋅(a+b):

$$(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b}) = \vec{a} \cdot \vec{a} + \vec{a} \cdot \vec{b} - \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b}$$(a−b)⋅(a+b)=a⋅a+a⋅b−b⋅a−b⋅b

Since dot product is commutative ($\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$a⋅b=b⋅a):

$$(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b}) = \vec{a} \cdot \vec{a} - \vec{b} \cdot \vec{b} = |\vec{a}|^2 - |\vec{b}|^2 = 27$$(a−b)⋅(a+b)=a⋅a−b⋅b=∣a∣2−∣b∣2=27

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Using the relationship $|\vec{a}| = 2|\vec{b}|$∣a∣=2∣b∣:

$$|\vec{a}|^2 = (2|\vec{b}|)^2 = 4|\vec{b}|^2$$∣a∣2=(2∣b∣)2=4∣b∣2

Substituting into our equation:

$$4|\vec{b}|^2 - |\vec{b}|^2 = 27$$4∣b∣2−∣b∣2=27

$$3|\vec{b}|^2 = 27$$3∣b∣2=27

$$|\vec{b}|^2 = 9$$∣b∣2=9

$$|\vec{b}| = 3$$∣b∣=3

Therefore, $|\vec{b}| = 3$∣b∣=3.