Question

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Answer 1

Dear Student,

Please post the question one at a time.

4) X is normally distributed with population mean := $\mu = 7.2$

and Standard deviation = $\sigma = 1.8$

a) The probability that it takes at least 5 hours to write a response paper can be represented as = $P(X\geq 5)$

we will convert this to standard Z score to find the probability

$Z=\frac{X-\mu}{\sigma}$

$P(X\geq 5)=P(Z\geq \frac{5-7}{1.8})=P(Z\geq 1.11)=1-P(Z<1.11)$

$P(Z<1.11)=0.8665$

$P(X\geq 5)=1-P(Z<1.11)=1-0.8665=0.13335$

b) If 20 Students are selected at random then by definition of centra limit theorem,

the mean time of the students is normally distributed with

population mean := $\mu = 7.2$

and Standard deviation = $\sigma = 1.8$

now we will convert this to standard Z score.

$Z=\frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}$

Now we have to calculate the probability that the sample mean is not more than 8

$P(X\leq 8)=P(Z\leq \frac{8-7}{\frac{1.8}{\sqrt{20}}})=P(Z\leq 2.48)=0.9934$