Complete Question
Express the confidence interval 0.555 less than p less than 0.777 in the form Modifying above p with caret plus or minus Upper E.
Answer:
The modified representation is [tex]\r p \pm E = 0.666 \pm 0.111[/tex]
Step-by-step explanation:
From the question we are told that
The confidence interval interval is [tex]0.555 < p < 0.777[/tex]
Now looking at the values that make up the up confidence interval we see that this is a symmetric confidence interval(This because the interval covers 95% of the area under the normal curve which mean that the probability of a value falling outside the interval is 0.05 which is divided into two , the first half on the left -tail and the second half on the right tail as shown on the figure in the first uploaded image(reference - Yale University ) ) which means
Now since the confidence interval is symmetric , we can obtain the sample proportion as follows
[tex]\r p = \frac{0.555 + 0.777}{2}[/tex]
[tex]\r p =0.666[/tex]
Generally the margin of error is mathematically represented as
[tex]E = \frac{1}{2} * K[/tex]
Where K is the length of the confidence interval which iis mathematically represented as
[tex]K = 0.777 -0.555[/tex]
[tex]K = 0.222[/tex]
Hence
[tex]ME = \frac{1}{2} * 0.222[/tex]
[tex]ME = 0.111[/tex]
So the confidence interval can now be represented as
[tex]\r p \pm E = 0.666 \pm 0.111[/tex]
