Question

How do you find the margin of error for the 95% confidence interval used to estimate the following population portion: “In a client test with 2440 subjects, 70% showed improvement from the treatment”?

Answer 1

Hint: In the type of question that is stated above we can easily say that it is a type Hypotheses testing in which we will have to calculate the population portion, for this we will taking two cases into play with which we will be identifying which type of test we will be using, we then will find the test static value (z) from the table and then with the formula we will need to find the population portion.

Complete step by step solution:
For the above question it is already stated that the sample size (n) (which means the population size taken) which is 2440 i.e.
n = 2440
Now the mean of the sample proportion which is stated as 70% so we can also say that
p (sample proportion) = 0.7
now it is also stated the level of confidence which is 95% and we know that sum of level of confidence and level significance is 100, by using this relation we get,
level of confidence + level of significance (α) = 100
α = 5%
now from the above statement we can conclude that this is a case of two tailed test:
H0: there is no sign of improvement.
H1: there is a sign of improvement.
From the question we can also see that there is a huge sign of improvement so we are accepting the H1 statement which means that
The test static(z) from calculation > the test static from table (z( α))
To calculate the test static value we will be using the formula as stated below:
\[z=\dfrac{p-P}{\sqrt{\dfrac{PQ}{n}}}\]
In the above formula the value of Q=1-P.
Now the value of (z ( α)) for 5% level of significance from the table for two tailed test is1.96
Taking the above stated value into consideration we will get and substituting it in the formula we will get:
\[\begin{align}
  & \Rightarrow 1.96=\dfrac{0.7-P}{\sqrt{\dfrac{P\left( 1-P \right)}{2440}}} \\
 & \Rightarrow \sqrt{P\left( 1-P \right)}=\dfrac{\sqrt{2440}\left( 0.7-P \right)}{1.96}\left[ \text{now after solving we will get P as} \right] \\
 & \Rightarrow P=0.741\text{ or }0.655 \\
\end{align}\]
From the above value of P we can say that there will be two values of P which will satisfy the equation.

Note: Remember to take the cases that will describe the type of case which is used to solve the above hypotheses testing. The other thing is to also remember the relation between level of confidence and level of significance as the table for test static is with respect to level of significance.