The number of hours that college students sleep on a week night is approximated by a normal curve with a mean of 7 hours and a standard deviation of 1.7 hours. Answer the question below using a z table:
About what proportion of students sleep less than 5 hours on a week night? (round to whole number)
About 16%
About 24%
About 12%
About 32%
About 32%
Z = (score-mean)/SD
Use the Z table.
To find the proportion of college students who sleep less than 5 hours on a weeknight, we need to use a z-table.
To begin, we need to standardize the value of 5 hours using the formula:
z = (x - μ) / σ
Where:
x is the value we want to standardize (5 hours)
μ is the mean of the distribution (7 hours)
σ is the standard deviation of the distribution (1.7 hours)
Plugging in the values, we have:
z = (5 - 7) / 1.7
z ≈ -1.18
Now, we can use the negative z-value (-1.18 in this case) to look up the corresponding proportion in the z-table.
When we look up -1.18 in the z-table, we find that the area to the left of -1.18 is approximately 0.119. This represents the proportion of students who sleep less than 5 hours on a weeknight.
However, the z-table gives us the area to the left of a given z-value, so to find the proportion of students who sleep less than 5 hours, we need to subtract this value from 0.5 (since 0.5 represents the middle of the curve).
0.5 - 0.119 ≈ 0.381
So, about 38% of college students sleep less than 5 hours on a weeknight.
Since the options provided are all rounded to whole numbers, the closest option is "About 32%".