Final answer:
The population standard deviation of the daily amounts spent by a family during a 7-day summer vacation is approximately $17.728, calculated by finding the mean, summing the squared differences from the mean, dividing by the number of values, and then taking the square root.
Step-by-step explanation:
The student asked for the population standard deviation of daily amounts spent by a family during a 7-day summer vacation. To find the population standard deviation, we must first calculate the mean of the data set and then use this mean to find the variance before taking the square root to obtain the standard deviation.
- First, find the mean (average) of the dataset:
($96 + $125 + $80 + $110 + $75 + $100 + $121) ÷ 7 = $707 ÷ 7 = $101 - Next, calculate the squared difference between each data point and the mean, and sum them up:
($(96-101)^2 + (125-101)^2 + (80-101)^2 + (110-101)^2 + (75-101)^2 + (100-101)^2 + (121-101)^2 = 25 + 576 + 441 + 81 + 676 + 1 + 400 = 2200 - Since we are finding the population standard deviation, divide the sum by the number of values in the dataset, which is 7:
2200 ÷ 7 = 314.2857... - Finally, take the square root of 314.2857... to find the population standard deviation:
$√314.2857... ≈1 17.728
Therefore, the population standard deviation of the daily amount spent by the family during their vacation is approximately $17.728.