answer:To estimate the number of people who would be expected to be under slept at 6 hours, we can use the z-score formula. The z-score is calculated as (X - μ) / σ, where X is the specified value (6 hours), μ is the mean (7.4 hours), and σ is the standard deviation (1.57 hours). Z = (6 - 7.4) / 1.57 ≈ -0.89. Using a standard normal distribution table (z-table), we find that the area to the left of z = -0.89 is approximately 0.1867. This represents the proportion of the population expected to sleep less than 6 hours. Since the sample size is 30, we can multiply this proportion by the sample size to estimate the number of people who would be expected to be under slept at 6 hours: 0.1867 × 30 ≈ 5.6. However, given the options 1, 2, or 3, and considering the calculated value of 5.6 is closer to the lower end options but the exact option isn't available, we consider that at least a few people would be expected to sleep less than 6 hours based on our sample's mean and standard deviation, with 2 people being the closest low-end estimate from the available options, taking into account some rounding error from our estimate.

answer:I shouldn't have directly concluded 2 people from the calculation of 5.6 people. The correct interpretation is that approximately 5.6 people would be expected to sleep less than 6 hours based on the sample's mean and standard deviation. Given the options 1, 2, or 3, and considering the calculated value of 5.6, it would be more accurate to say that the closest option provided is not a good representation of the expected number of people. However, if I had to choose from the available options, I should have stated that the calculated value is actually closer to the higher end of the options, which would be more than the highest option provided (3).

answer:To determine if the mean sleep hours falls within the recommended range, we can directly observe that the mean of 7.4 hours is indeed within the 7-9 hours recommended range. However, considering the standard deviation, we can expect a significant portion of the population to fall outside of this range. A commonly cited rule of thumb is the 68-95-99.7 rule, which states that about 68% of the data falls within 1 standard deviation, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations. Using this rule, we can estimate the spread of the data: - 1 standard deviation from the mean would be 7.4 ± 1.57 hours, resulting in a range of approximately 5.83 to 8.97 hours. This range does extend below the recommended lower limit. - 2 standard deviations would be 7.4 ± 3.14 hours, resulting in a range of approximately 4.26 to 10.54 hours. This range extends significantly beyond the recommended limits on both ends. Given this spread, while the mean of 7.4 hours is within the recommended range, a considerable portion of the population can be expected to fall outside of the 7-9 hours per night range due to the standard deviation of 1.57 hours.

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