Question

select the correct answer.
a researcher for a travel company is looking into the prices people are willing to pay for airplane tickets. the company has communicated that the overall population mean is $265 with a standard deviation of $40.53. the researcher has a sample of 130 ticket - purchases from one location. by the central limit theorem, which interval can the researcher be 95% certain that the sample mean will fall within?

o a. $254.23 and $275.77

o b. $257.82 and $272.18

o c. $261.41 and $268.59

o d. $264.37 and $265.63

Explanation:

Step1: Identify relevant values

Population mean $\mu = 265$, standard - deviation $\sigma=40.53$, sample size $n = 130$, and for 95% confidence, the z - score $z = 1.96$.

Step2: Calculate the standard error

The standard error of the mean $SE=\frac{\sigma}{\sqrt{n}}=\frac{40.53}{\sqrt{130}}\approx3.56$.

Step3: Calculate the margin of error

The margin of error $E = z\times SE=1.96\times3.56\approx6.98$.

Step4: Find the confidence interval

The lower limit is $\mu - E=265 - 6.98 = 258.02$ and the upper limit is $\mu+E=265 + 6.98 = 271.98$. The closest interval among the options is B.

Answer:

B. $257.82$ and $272.18$