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.

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luisejr77 luisejr77 · Answer 1 · 2019-08-14T04:16:27+02:00

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]