Question

Visit Jeju ~ In South Korea, "Jeju" is a famous holiday destination with breathtaking scenic views. A travel agent in Jeju wants to know the satisfaction levels of tourists who visit the island. The travel agent surveyed tourists at random and constructed a 95% confidence interval for the proportion of tourists who were extremely satisfied with their Jeju visit to be (0.6457, 0.8489). What is the margin of error? Give the exact answer as a decimal.

Answer 1

The 95% confidence interval for the true proportion is given on this case by (0.6457 , 0.8489)

For this case we can starting calculating an estimation for the proportion like this:

`\hat p =(0.6457 +0.8489)/(2) = 0.7473`

And we can calculate the margin of error like this:

`ME= Upper -\hat p = 0.8489- 0.7473= 0.10106`

`ME= \hat p -Lower = 0.7473- 0.6457= 0.10106`

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The population proportion have the following distribution

`p \sim N(p,\sqrt{(p(1-p))/(n)})`

Solution to the problem

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by and . And the critical value would be given by:

`z_(\alpha/2)=-1.96, z_(1-\alpha/2)=1.96`

The confidence interval for the mean is given by the following formula:

`\hat p \pm z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}`

The 95% confidence interval for the true proportion is given on this case by (0.6457 , 0.8489)

For this case we cab starting calculating an estimation for the proportion like this:

`\hat p =(0.6457 +0.8489)/(2) = 0.7473`

And we can calculate the margin of error like this:

`ME= Upper -\hat p = 0.8489- 0.7473= 0.10106`

`ME= \hat p -Lower = 0.7473- 0.6457= 0.10106`