Question

A journal published a study of the lifestyles of visually impaired students. Using diaries, the students kept track of several variables, including number of hours of sleep obtained in a typical day. These visually impaired students had a mean of 10.1 hours and a standard deviation of 1.82 hours. Assume that the distribution of the number of hours of sleep for this group of students is approximately normal.
Below what hours 90% of the visually impaired students have their sleep? [Answer to 2 decimal places]

A: 9.97   B: 12.43   C: 15.28   D: 16.05   E: 16.73   F: 18.30

What is the chance that a randomly selected visually impaired student has at least 8.5 hours of sleep? [Answer to 4 decimal places]

A: 0.0109   B: 0.2059   C: 0.3461   D: 0.6155   E: 0.8103   F: 0.9867

What is the inter-quartile range of sleep hours among visually impaired students. [Answer to 2 decimal places]

A: 2.35   B: 2.43   C: 2.46   D: 2.78   E: 2.85   F: 2.95

Answer 1

Solution :

Let X be a random variable which represents the number of hours of sleep obtained in a typical day.

Given that, X ~ N(10.1, 1.82²)

Mean (μ) = 10.1

SD (σ) = 1.82

a) Let 90% of the visually impaired students have their sleep below k hours.

Hence, P(X < k) = 0.90

We know that, if X ~ N(μ, σ²) then,

.........................(1)

Using "qnorm" function of R we get, P(Z < 1.282) = 0.90

Comparing, P(Z < 1.282) = 0.90 and (1) we get,

Hence, 90% of the visually impaired students have their sleep below 12.43 hours.

Option (B) is correct.

b) We have to find P(X ≥ 8.5).

We know that, if X ~ N(μ, σ²) then,

Using "pnorm" function of R we get, P(Z ≥ -0.8791) = 0.8103

Hence, the required probability is 0.8103.

Option (E) is correct.

c) The inter-quartile range is given as follows :

IQR = Q_{​​​​​​3} - Q_{​​​​​​1}

Where, Q_{​​​​​​3} is third quartile and Q_{​​​​​1} is first quartile.

Let first quatile is m.

Hence, P(X < m) = 0.25

We know that, if X ~ N(μ, σ²) then,

.........................(2)

Using "qnorm" function of R we get, P(Z < -0.6745) = 0.25

Comparing, P(Z < -0.6745) = 0.25 and (2) we get,

Hence, first quartile is, Q_{​​​​​​1} = 8.87.

Let third quatile is n.

Hence, P(X < n) = 0.75

We know that, if X ~ N(μ, σ²) then,

.........................(3)

Using "qnorm" function of R we get, P(Z < 0.67) = 0.75

Comparing, P(Z < 0.67) = 0.75 and (3) we get,

Hence, third quartile is, Q_{​​​​​}​​​​​_{​​​​​​3} = 11.31

Hence, IQR = (11.31 - 8.87) = 2.46

Hence, option (C) is correct.

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