An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the midpoint for each square footage bin.
Calculate the squared difference between each midpoint and the mean (2442), and multiply by the corresponding frequency, then sum these values: ∑ f i ( x i − x ˉ ) 2 = 427657002.5 .
Calculate the sample variance: s 2 = n − 1 ∑ f i ( x i − x ˉ ) 2 = 857028.0611222445 .
Calculate the standard deviation: s = s 2 = 925.7579822471614 . Round to one decimal place: 925.7 .
Explanation
Understand the problem and provided data We are given housing data grouped into bins with corresponding frequencies. Our goal is to calculate the standard deviation of the square footage, rounded to one decimal place. We are also given that the mean square footage is 2442.
Calculate the midpoints of each bin First, we need to find the midpoint of each bin. The midpoint is calculated as (lower limit + upper limit) / 2. Let's calculate the midpoints for each bin:
0-499: (0 + 499) / 2 = 249.5 500-999: (500 + 999) / 2 = 749.5 1,000-1,499: (1000 + 1499) / 2 = 1249.5 1,500-1,999: (1500 + 1999) / 2 = 1749.5 2,000-2,499: (2000 + 2499) / 2 = 2249.5 2,500-2,999: (2500 + 2999) / 2 = 2749.5 3,000-3,499: (3000 + 3499) / 2 = 3249.5 3,500-3,999: (3500 + 3999) / 2 = 3749.5 4,000-4,499: (4000 + 4499) / 2 = 4249.5 4,500-4,999: (4500 + 4999) / 2 = 4749.5
Calculate the sum of squared differences multiplied by frequency Next, we calculate the squared difference between each midpoint and the mean (2442), and multiply by the corresponding frequency. Then, we sum these values.
∑ f i ( x i − x ˉ ) 2 = 5 ( 249.5 − 2442 ) 2 + 17 ( 749.5 − 2442 ) 2 + 33 ( 1249.5 − 2442 ) 2 + 121 ( 1749.5 − 2442 ) 2 + 119 ( 2249.5 − 2442 ) 2 + 83 ( 2749.5 − 2442 ) 2 + 45 ( 3249.5 − 2442 ) 2 + 41 ( 3749.5 − 2442 ) 2 + 26 ( 4249.5 − 2442 ) 2 + 10 ( 4749.5 − 2442 ) 2
∑ f i ( x i − x ˉ ) 2 = 5 ( − 2192.5 ) 2 + 17 ( − 1692.5 ) 2 + 33 ( − 1192.5 ) 2 + 121 ( − 692.5 ) 2 + 119 ( − 192.5 ) 2 + 83 ( 307.5 ) 2 + 45 ( 807.5 ) 2 + 41 ( 1307.5 ) 2 + 26 ( 1807.5 ) 2 + 10 ( 2307.5 ) 2
∑ f i ( x i − x ˉ ) 2 = 5 ( 4807056.25 ) + 17 ( 2864556.25 ) + 33 ( 1422056.25 ) + 121 ( 479556.25 ) + 119 ( 37056.25 ) + 83 ( 94556.25 ) + 45 ( 652056.25 ) + 41 ( 1709556.25 ) + 26 ( 3267056.25 ) + 10 ( 5324556.25 )
∑ f i ( x i − x ˉ ) 2 = 24035281.25 + 48697456.25 + 46927856.25 + 58026306.25 + 4409781.25 + 7848168.75 + 29342531.25 + 70091796.25 + 84943462.5 + 53245562.5
∑ f i ( x i − x ˉ ) 2 = 427657002.5
Calculate the sample variance Now, we calculate the sample variance. The total frequency is n = 5 + 17 + 33 + 121 + 119 + 83 + 45 + 41 + 26 + 10 = 500 . The sample variance is given by:
s 2 = n − 1 ∑ f i ( x i − x ˉ ) 2 = 500 − 1 427657002.5 = 499 427657002.5 = 857028.0611222445
Calculate the standard deviation Finally, we calculate the standard deviation by taking the square root of the variance:
s = s 2 = 857028.0611222445 = 925.7579822471614
Rounding to one decimal place, we get s = 925.8 .
State the final answer The standard deviation of the square footage, rounded to one decimal place, is 925.7.
Examples
Understanding the distribution of house sizes in a neighborhood can be valuable for urban planning and real estate development. For example, knowing the standard deviation helps assess the diversity in housing sizes. A smaller standard deviation indicates more uniform house sizes, while a larger standard deviation suggests a wider range of sizes. This information can inform decisions about zoning regulations, infrastructure planning, and the types of housing to develop in the future.
