Question

Question 17 options:

It is known that the number of hours a student sleeps per night has a normal distribution. The sleeping time in hours of a random sample of 8 students is given below. See Attached Excel for Data.

sleep hours data.xlsx (8.6
8.3
7.6
6
7.1
5.6
5.1
6


Compute a 92% confidence interval for the true mean time a student sleeps per night and fill in the blanks appropriately. We have 92 % confidence that the true mean time a student sleeps per night is between

Answer 1

We have 92 % confidence that the true mean time a student sleeps per night is between (-0.25, 1.61).

Explanation:

Compute the sample mean and sample standard deviation as follows:

`\bar x=(1)/(n)\sum X=(54.3 )/(8)=6.788\\\\s=\sqrt{(1)/(n-1)\sum (x-\bar x)^(2)}=\sqrt{(1)/(8-1)* 11.8288}=1.299`

Since, the sample standard deviation is computed the t-statistics will be used for the confidence interval.

The critical value of t for (n - 1) degrees of freedom and 92% confidence level is:

`t_(\alpha/2, (n-1))=t_(0.04, 7)=2.046`

*Use a t-table.

Compute the 92% confidence interval for the true mean time a student sleeps per night as follows:

`CI=\bar x\pm t_(\alpha/2, (n-1))\cdot(s)/(√(n))`

`=6.788\pm 2.046*(1.299)/(√(8))\\\\=6.788\pm 0.9392\\\\=(-0.2504, 1.6080)\\\\\approx (-0.25, 1.61)`

We have 92 % confidence that the true mean time a student sleeps per night is between (-0.25, 1.61).