For a particular group of people, the number of hours of sleep per night is normally distributed with a mean of 7.7 and a standard deviation of 1.6. To the nearest tenth 67% of people in this group will sleep over blank hours?

that's 1σ below the mean.

To find the number of hours that approximately 67% of people in this group will sleep over, we can use the properties of a normal distribution.

Step 1: Calculate the z-score corresponding to the desired percentile.
To find the z-score, we need to determine how many standard deviations away from the mean our desired percentile falls. We can use a standard normal distribution table or a calculator with a normal distribution function.

We want to find the z-score that corresponds to the 67th percentile. The percentage between the mean and the desired percentile is (100% - 67%) / 2 = 16.5%. Using a standard normal distribution table or calculator, the z-score corresponding to 16.5% is approximately -0.43.

Step 2: Calculate the raw score (number of hours) using the z-score formula.
The formula to calculate the raw score based on a given z-score is:
raw score = (z-score * standard deviation) + mean.

In this case, the mean is 7.7 hours and the standard deviation is 1.6 hours. Plugging in the values:
raw score = (-0.43 * 1.6) + 7.7 = 7.014.

Step 3: Round the answer to the nearest tenth.
Rounding 7.014 to the nearest tenth gives us approximately 7.0 hours.

Therefore, approximately 67% of people in this group will sleep over 7.0 hours.