Question

The mean of two sample large samples of 1000 and 2000 members are 67.5 inches and 68 inches respectively. Can the samples be regarded as drawn from the same population of standard deviation of 2.5 inches? Test at 5% level of significance.
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Answer 1

Step-by-step explanation:

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Answer 2

Given :

Mean of two samples $1000$ and $2000$ are $67.5$ and $68$ $inches$ respectively.

To find :

The samples that can be regarded as drawn when the standard deviation is 2.5 inches.

Step-by-step explanation:

• Let the samples be $n_{1}$ and $n_{2}$.
• Let the mean of the samples be $\bar{x_{1} }$ and $\bar{x_{2} }$.
• Then, the values will be

        $n_{1} =1000,n_{2} =2000$ and $\bar{x_{1} }=67.5,\bar{x_{2} }=68$

• The means must be zero in this case.
• But, we must prove it hypothetically in the case of nullity.
• In this case, the $\bar{x_{1} }$ must not be equal to $\bar{x_{2} }$
• Therefore, the equation will be        

        $\bar{x_{1} }-\bar{x_{2} }$ ≠ $0$

• Substitute the values

        $67.5-68.0$

• The answer would be

        $-0.5$

• Since, the standard deviation is already given
• The formula used is

        $S.E = \sqrt{\frac{a_{1}^{2} }{n_{1} } +\frac{a_{2}^{2} }{n_{2} }}$

• Take the common terms out.

         $S.E = a\sqrt{\frac{1 }{n_{1} } +\frac{1 }{n_{2} }}$

• Substitute the values of $n_{1}, n_{2}$

• Then, the equation would be

        $S.E = a\sqrt{\frac{1 }{1000 } +\frac{1 }{2000 }}$

• Value of $a$ is 1
• Then, the value of $S.E$ would be

        $S.E = \sqrt{\frac{1 }{1000 } +\frac{1 }{2000 }}$

• Take LCM for the numbers

        $S.E = \sqrt{\frac{2+1}{2000} }$

• Sum it all up

        $S.E = \sqrt{\frac{3}{2000} }$

• The final answer would be

        $S.E=0.097$

• Then, the value of $z$ would be

        $z=\frac{\bar{x_{1} }-\bar{x_{2} }}{S.E}$

• Substitute the values

        $z=\frac{-0.5}{0.097}$

• Divide the values

        $z= -5.15$

• $z$ cannot be negative. Therefore, the value of $z$ is $5.15$
• The value of $z$ is more than the critical value.
• Therefore, the samples can not be regarded as drawn from same population.

Final answer :

The samples be regarded as drawn from the same population of standard deviation of 2.5 inches.