Answer
Verified
352.5k+ views
Hint: We solve this problem by calculating the percentage of area that represents the critical value of ${{Z}_{\alpha /2}}$ that corresponds to a 94% confidence. We use the formula to calculate the value of $\alpha $ that is,
$\left( 1-\alpha \right)\times 100=\text{Confidence level}$
Then the area to the left that represents the critical value of ${{Z}_{\alpha /2}}$ that corresponds to a 94% confidence is given as
$A=\left( 1-\dfrac{\alpha }{2} \right)$
Then we find the Z – score that corresponds to the area. To find the Z – scores we need to check the horizontal and vertical values and add them to get the Z – score corresponding to that area. We need to note that the Z – scores for areas greater than 0.5 are positive.
Complete step-by-step solution:
We are given that the confidence level as 94%
Now, let us find the value of $\alpha $
We know that the formula used to calculate the value of $\alpha $ is given as,
$\left( 1-\alpha \right)\times 100=\text{Confidence level}$
By substituting the required values in above formula we get,
$\begin{align}
& \Rightarrow \left( 1-\alpha \right)\times 100=94 \\
& \Rightarrow 1-\alpha =0.94 \\
& \Rightarrow \alpha =0.06 \\
\end{align}$
Now, let us find the area that corresponds to the critical value of ${{Z}_{\alpha /2}}$ that corresponds to a 94% confidence is given as
$A=\left( 1-\dfrac{\alpha }{2} \right)$
By using the above formula we get the required area as,
$\begin{align}
& \Rightarrow A=1-\dfrac{0.06}{2} \\
& \Rightarrow A=1-0.03=0.97 \\
\end{align}$
Now, let us find the Z – score that corresponds to the area of 0.97.
Let us assume that the required Z – score as $'z'$
Here, we can see that the table used for standard conversions of area under graph to Z – score does not include 0.97.
But we can see that there is an area equal to 0.9699 which is very close to the required area 0.97.
We know that we need to add horizontal and vertical values of Z – scores to get the Z – score corresponding to that respective area.
We know that for the area greater than 0.5 the Z – score will be positive.
So, we can say that the Z – score for 0.9699 is given as,
$\begin{align}
& \Rightarrow z=1.8+0.08 \\
& \Rightarrow z=1.88 \\
\end{align}$
Here, we can see that the value of Z – score corresponding to area 0.97 is 1.88.
Therefore, we can conclude that the critical value of ${{Z}_{\alpha /2}}$ that corresponds to 94% confidence is given as 1.88 that is,
$\Rightarrow {{Z}_{0.06/2}}={{Z}_{0.03}}=1.88$
Note: We need to note that if the area given is greater than 0.5 then the Z – score should be always positive. So, we need positive Z – scores from the table corresponding to the assumed areas.
Also we need to note that we take the Z – score of area equal to 0.97 as the Z – score of area equal to 0.9699 because they are almost equal. It is having very little difference.
But in the same case if the area is having some more difference then we need to use the interpolation theorem to find the required Z – score.
$\left( 1-\alpha \right)\times 100=\text{Confidence level}$
Then the area to the left that represents the critical value of ${{Z}_{\alpha /2}}$ that corresponds to a 94% confidence is given as
$A=\left( 1-\dfrac{\alpha }{2} \right)$
Then we find the Z – score that corresponds to the area. To find the Z – scores we need to check the horizontal and vertical values and add them to get the Z – score corresponding to that area. We need to note that the Z – scores for areas greater than 0.5 are positive.
Complete step-by-step solution:
We are given that the confidence level as 94%
Now, let us find the value of $\alpha $
We know that the formula used to calculate the value of $\alpha $ is given as,
$\left( 1-\alpha \right)\times 100=\text{Confidence level}$
By substituting the required values in above formula we get,
$\begin{align}
& \Rightarrow \left( 1-\alpha \right)\times 100=94 \\
& \Rightarrow 1-\alpha =0.94 \\
& \Rightarrow \alpha =0.06 \\
\end{align}$
Now, let us find the area that corresponds to the critical value of ${{Z}_{\alpha /2}}$ that corresponds to a 94% confidence is given as
$A=\left( 1-\dfrac{\alpha }{2} \right)$
By using the above formula we get the required area as,
$\begin{align}
& \Rightarrow A=1-\dfrac{0.06}{2} \\
& \Rightarrow A=1-0.03=0.97 \\
\end{align}$
Now, let us find the Z – score that corresponds to the area of 0.97.
Let us assume that the required Z – score as $'z'$
Here, we can see that the table used for standard conversions of area under graph to Z – score does not include 0.97.
But we can see that there is an area equal to 0.9699 which is very close to the required area 0.97.
We know that we need to add horizontal and vertical values of Z – scores to get the Z – score corresponding to that respective area.
We know that for the area greater than 0.5 the Z – score will be positive.
So, we can say that the Z – score for 0.9699 is given as,
$\begin{align}
& \Rightarrow z=1.8+0.08 \\
& \Rightarrow z=1.88 \\
\end{align}$
Here, we can see that the value of Z – score corresponding to area 0.97 is 1.88.
Therefore, we can conclude that the critical value of ${{Z}_{\alpha /2}}$ that corresponds to 94% confidence is given as 1.88 that is,
$\Rightarrow {{Z}_{0.06/2}}={{Z}_{0.03}}=1.88$
Note: We need to note that if the area given is greater than 0.5 then the Z – score should be always positive. So, we need positive Z – scores from the table corresponding to the assumed areas.
Also we need to note that we take the Z – score of area equal to 0.97 as the Z – score of area equal to 0.9699 because they are almost equal. It is having very little difference.
But in the same case if the area is having some more difference then we need to use the interpolation theorem to find the required Z – score.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Advantages and disadvantages of science
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Difference Between Plant Cell and Animal Cell
Which are the Top 10 Largest Countries of the World?
10 examples of evaporation in daily life with explanations
Give 10 examples for herbs , shrubs , climbers , creepers
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE