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Zorluk: ZorDescriptive Statistics and Data Representations

A weather station recorded the daily high temperatures, in degrees Fahrenheit, for the first 10 days of a month. The mean high temperature for these 10 days was 72∘F72^\circ\text{F}. When the high temperatures for 2 additional days were included in the dataset, the mean high temperature for all 12 days increased to 74∘F74^\circ\text{F}. If the high temperature on the 12th day was 6∘F6^\circ\text{F} higher than the high temperature on the 11th day, what was the high temperature, in degrees Fahrenheit, on the 12th day?

  1. A
    81
  2. B
    84
  3. 87Cevap
  4. D
    90
  5. E
    93

Cevap

87
To find the temperature on the 12th day, first calculate the sum of the first 10 days (10×72=72010 \times 72 = 720) and the total sum of all 12 days (12×74=88812 \times 74 = 888). The difference (888−720=168888 - 720 = 168) is the sum of the high temperatures on the 11th and 12th days. Let T12T_{12} be the high temperature on the 12th day. Since the 12th day was 6∘F6^\circ\text{F} warmer than the 11th day, the 11th day's temperature was T12−6T_{12} - 6. Substituting gives (T12−6)+T12=168(T_{12} - 6) + T_{12} = 168, which simplifies to 2T12=1742T_{12} = 174, yielding T12=87∘FT_{12} = 87^\circ\text{F}.

Adım Adım Çözüm

1
Calculate the sum of high temperatures for the first 10 days.
10×72=720∘F10 \times 72 = 720^\circ\text{F}
The sum of values is equal to the number of data points multiplied by their mean.
2
Calculate the total sum of high temperatures for all 12 days.
12×74=888∘F12 \times 74 = 888^\circ\text{F}
The new mean for all 12 days is 74∘F74^\circ\text{F}.
3
Determine the sum of the temperatures on the 11th and 12th days.
888−720=168∘F888 - 720 = 168^\circ\text{F}
Subtracting the sum of the first 10 days from the total sum gives the combined total of the 2 additional days.
4
Set up an equation to find the temperature of the 12th day (T12T_{12}).
T11+T12=168T_{11} + T_{12} = 168 and T11=T12−6  ⟹  (T12−6)+T12=168  ⟹  2T12=174  ⟹  T12=87∘FT_{11} = T_{12} - 6 \implies (T_{12} - 6) + T_{12} = 168 \implies 2T_{12} = 174 \implies T_{12} = 87^\circ\text{F}
The 12th day temperature is 6∘F6^\circ\text{F} higher than the 11th day temperature, so substituting T11T_{11} allows us to solve directly for T12T_{12}.

Anahtar Kavram

Using the arithmetic mean to find missing data points in expanded datasets
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Descriptive Statistics and Data Representations

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