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:
On a week night, what percentage of students sleep between 5 and 10 hours
Perfectly suited for David Lane's great stats calculator
Just enter the values from your problem
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I get 84.15%
If you must use your tables,
z-score for 10 = (10-7)/1.7 = 1.7647
z-score for 5 = (5-7)/1.7 = -1.17647
look up probability for 1.7647 in your tables
look up probability ofr -1.17647
subtract the two values to get .8415
To find the percentage of students who sleep between 5 and 10 hours on a week night, we need to calculate the z-scores for 5 and 10 using the mean and standard deviation given.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x is the value we are interested in (in this case, 5 and 10)
μ is the mean (7 hours)
σ is the standard deviation (1.7 hours)
Let's calculate the z-scores for both 5 and 10:
For 5 hours:
z = (5 - 7) / 1.7
= -2 / 1.7
≈ -1.18
For 10 hours:
z = (10 - 7) / 1.7
= 3 / 1.7
≈ 1.76
Now that we have the z-scores, we can use a z-table to find the corresponding percentages.
Looking up the z-values in the z-table:
For a z-score of -1.18, the table shows a percentage of approximately 0.119 a.k.a 11.9%.
For a z-score of 1.76, the table shows a percentage of approximately 0.960 a.k.a 96.0%.
To find the percentage of students who sleep between 5 and 10 hours, we subtract the lower percentage from the higher percentage:
Percentage = 96.0% - 11.9%
= 84.1%
Therefore, approximately 84.1% of college students sleep between 5 and 10 hours on a week night.
To find the percentage of college students who sleep between 5 and 10 hours on a week night, we need to calculate the z-scores for both values and then use a z-table to find the corresponding percentages.
Step 1: Calculate the z-score for 5 hours:
Z = (X - μ) / σ
Z = (5 - 7) / 1.7
Z = -2 / 1.7
Z ≈ -1.176
Step 2: Calculate the z-score for 10 hours:
Z = (X - μ) / σ
Z = (10 - 7) / 1.7
Z = 3 / 1.7
Z ≈ 1.765
Step 3: Use the z-table to find the corresponding percentages:
The percentage of students that sleep between 5 and 10 hours can be found by subtracting the area to the left of the z-score for 5 hours from the area to the left of the z-score for 10 hours.
Using a z-table, the area to the left of the z-score -1.176 is approximately 0.119 (or 11.9%).
The area to the left of the z-score 1.765 is approximately 0.960 (or 96.0%).
Step 4: Calculate the percentage of students that sleep between 5 and 10 hours:
Percentage = Area to the left of Z = 1.765 - Area to the left of Z = -1.176
Percentage = 0.960 - 0.119
Percentage ≈ 0.841
Therefore, approximately 84.1% of college students sleep between 5 and 10 hours on a week night.