Assume that the average full time college student in the U.S. requires a total of $10,500 per year to cover the costs of tutition, books and fees with a standard deviation of $1,300. If 50 full time students are randomly sampled in the U.S., what is the probability that their total annual costs for tutition, books and fees exceed $550,000? Round your answer to 4 decimal places. Remember to round your z-value to 2 decimal places.

Accepted Solution

Answer:[tex]P({\displaystyle {\overline {x}}}>11,000)=0.0033[/tex]Step-by-step explanation:We know that the mean [tex]\mu[/tex] is:[tex]\mu=10,500[/tex]The standard deviation [tex]\sigma[/tex] is:[tex]\sigma=1,300[/tex]The sample size is:[tex]n=50[/tex]Then the sample average [tex]{\displaystyle {\overline {x}}}[/tex] is:[tex]{\displaystyle {\overline {x}}}=\frac{550,000}{50}=11,000[/tex]In this case we look for[tex]P({\displaystyle {\overline {x}}}>11,000)[/tex]We calculate the Z score. In this case:[tex]Z=\frac{{\displaystyle {\overline {x}}}-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex][tex]Z=\frac{11,000-10,500}{\frac{1,300}{\sqrt{50}}}[/tex][tex]Z=2.72[/tex]Therefore:[tex]P({\displaystyle {\overline {x}}}>11,000)=P(Z>2.72)[/tex]Looking at the normal table we have to[tex]P(Z>2.72)=0.0033[/tex]