The typical amount of sleep per night for undergraduate college students has a bell shaped distribution with a mean of 7 and a standard deviation of 1.90 hours. Based on this distributional information we can conclude that at least 95 % of this population sleep between___and_______hours a night
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.4750) and the related Z scores. Insert Z scores both into the equation above.
To find the range of hours in which at least 95% of the population sleeps, we need to calculate the z-scores corresponding to the lower and upper limits.
The z-score formula is given by:
z = (x - μ) / σ
Where:
- x is the value we are interested in (hours of sleep for this case)
- μ is the mean (7 hours)
- σ is the standard deviation (1.90 hours)
To find the lower limit, we need to find the z-score that corresponds to the 2.5th percentile (since we want at least 95% of the population to be higher than this value). Using a z-table or a calculator, we find that the z-score is approximately -1.96.
Now we can substitute the values into the formula to find the lower limit:
-1.96 = (x - 7) / 1.90
Solving for x, we get:
-1.96 * 1.90 = x - 7
-3.724 = x - 7
x ≈ 3.276
So, at least 95% of the college students sleep more than approximately 3.276 hours per night.
To find the upper limit, we need to find the z-score that corresponds to the 97.5th percentile (since we want at least 95% of the population to be lower than this value). Using a z-table or a calculator, we find that the z-score is approximately 1.96.
Now we can substitute the values into the formula to find the upper limit:
1.96 = (x - 7) / 1.90
Solving for x, we get:
1.96 * 1.90 = x - 7
3.724 = x - 7
x ≈ 10.724
So, at least 95% of the college students sleep less than approximately 10.724 hours per night.
Therefore, we can conclude that at least 95% of the population sleeps between 3.276 and 10.724 hours a night.