Lucas recorded his lunch expenditure each day for one week in the table below.
Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Expenditure $4.85 $5.10 $5.50 $4.75 $4.50 $5.00 $6.00
Find the mean, standard deviation, and variance of Lucas’s lunch expenditures. Round to the nearest thousandth.
To find the mean, standard deviation, and variance of Lucas's lunch expenditures, we first need to calculate the total expenditure for the week:
Total expenditure = $4.85 + $5.10 + $5.50 + $4.75 + $4.50 + $5.00 + $6.00
Total expenditure = $35.70
Next, we calculate the mean:
Mean = Total expenditure / Number of days
Mean = $35.70 / 7
Mean = $5.100
Now, we calculate the variance. To do this, we first find the squared differences between each day's expenditure and the mean, then calculate the sum of these squared differences:
Squaring differences:
(Sunday - Mean)^2 = ($4.85 - $5.100)^2 = $0.0256
(Monday - Mean)^2 = ($5.10 - $5.100)^2 = $0.0000
(Tuesday - Mean)^2 = ($5.50 - $5.100)^2 = $0.2025
(Wednesday - Mean)^2 = ($4.75 - $5.100)^2 = $0.1102
(Thursday - Mean)^2 = ($4.50 - $5.100)^2 = $0.3600
(Friday - Mean)^2 = ($5.00 - $5.100)^2 = $0.0100
(Saturday - Mean)^2 = ($6.00 - $5.100)^2 = $0.8100
Sum of squared differences = $0.0256 + $0.0000 + $0.2025 + $0.1102 + $0.3600 + $0.0100 + $0.8100
Sum of squared differences = $1.5183
Variance = Sum of squared differences / (Number of days - 1)
Variance = $1.5183 / 6
Variance = $0.253
Finally, we calculate the standard deviation, which is the square root of the variance:
Standard deviation = √Variance
Standard deviation = √$0.253
Standard deviation = $0.503
Therefore, the mean of Lucas's lunch expenditures is $5.100, the variance is $0.253, and the standard deviation is $0.503.