Answer to Question #231666 in Statistics and Probability for tam

Question

Suppose the students that attend a university get an average of 5.7 hours of sleep

each night with a standard deviation of 1.7 hours. A student is randomly selected.

(i) Find the probability that the student has less than 7 hours of sleep each

night. (4 marks)

(ii) Find the value of m if 5% of the students sleep more than m hours.

(4 marks)

- 2 -

A sample of 35 students are selected.

(iii) Find the mean and standard deviation of the sample mean sleep time.

(2 marks)

(iv) Find the probability that the randomly selected sample of 35 students

have more than 6 hours of sleep each night.

Answer 1

1.

Let $X=$ the number of hours of sleep each night: $X\sim N(\mu, \sigma).$

Given $\mu=5.7\ hr, \sigma=1.7\ hr.$

(i)

$$P(X<7)=P(Z<\dfrac{7-\mu}{\sigma})=P(Z<\dfrac{7-5.7}{1.7})$$

$$\approx P(Z<0.7647)\approx0.777777$$

(ii)

$$P(X>m)=1-P(Z\leq\dfrac{m-\mu}{\sigma})$$

$$=1-P(Z\leq\dfrac{m-5.7}{1.7})=0.05$$

$$\dfrac{m-5.7}{1.7}\approx1.6449$$

$$m\approx5.7+1.7(1.6449)$$

$$m\approx8.5\ hr$$

2.

(iii)

$$\mu_{\bar{X}}=\mu_{X}=5.7\ hr$$

$$\sigma _{\bar{X}}=\sigma_{X}/\sqrt{n}=1.7/\sqrt{35}\ hr\approx0.28735\ hr$$

(iv)

$$P(\bar{X}>6)=1-P(Z\leq\dfrac{6-\mu}{\sigma\sqrt{n}})$$

$$=1-P(Z\leq\dfrac{6-5.7}{1.7/\sqrt{35}})\approx1-P(Z\leq1.044014)$$

$$\approx0.14824$$