Suppose that hours of sleep per night for single adults between 30 and 40 years of age are normally distributed with a mean of 6.7 hours and a standard deviation of 1.1 hours. What proportion of adults sleep longer than 4.5 hours per night? (Report to 3 decimal places.)

http://davidmlane.com/hyperstat/z_table.html

7.54

0.977

To find the proportion of adults who sleep longer than 4.5 hours per night, we need to calculate the area under the normal distribution curve to the right of 4.5.

First, we need to standardize the value of 4.5 using the formula:

z = (x - μ) / σ

Where:
x = 4.5 (value of interest)
μ = 6.7 (mean)
σ = 1.1 (standard deviation)

Substituting the values:

z = (4.5 - 6.7) / 1.1
z = -2 / 1.1
z ≈ -1.82

Now, we need to find the proportion of the area to the right of z = -1.82. We can use a normal distribution table or a statistical calculator to find this value.

Using a normal distribution table, we can find the proportion corresponding to z = -1.82. The table gives us the proportion for the area to the left of z, so we need to subtract it from 1 to get the area to the right.

Looking up the value of z = -1.82 in the table, we find that the corresponding proportion is approximately 0.0344.

However, this value only represents the area to the left of z = -1.82. To find the area to the right, we subtract it from 1:

Proportion to the right = 1 - 0.0344
Proportion to the right ≈ 0.9656

So, approximately 0.966 (rounded to 3 decimal places) or 96.6% of adults sleep longer than 4.5 hours per night.