the table shows the time a patient spends at the dentist and the amount of the bill. bill amount for time…

the table shows the time a patient spends at the dentist and the amount of the bill. bill amount for time spent at the dentist\n| time spent at the dentist (in hours) | bill amount |\n| ---- | ---- |\n| 1.4 | $235 |\n| 2.7 | $867 |\n| 0.75 | $156 |\n| 1.6 | $215 |\nwhat is the correlation coefficient for the data in the table?\n- -0.93\n- -0.27\n- 0.27\n- 0.93

the table shows the time a patient spends at the dentist and the amount of the bill. bill amount for time spent at the dentist\n| time spent at the dentist (in hours) | bill amount |\n| ---- | ---- |\n| 1.4 | $235 |\n| 2.7 | $867 |\n| 0.75 | $156 |\n| 1.6 | $215 |\nwhat is the correlation coefficient for the data in the table?\n- -0.93\n- -0.27\n- 0.27\n- 0.93

Answer

Answer:

We will use a statistical software or a calculator with statistical functions to calculate the correlation coefficient. Let (x) be the time spent at the dentist (in hours) and (y) be the bill amount.

  1. First, calculate the means of (x) and (y):
    • (n = 4)
    • (\bar{x}=\frac{1.4 + 2.7+0.75 + 1.6}{4}=\frac{6.45}{4}=1.6125)
    • (\bar{y}=\frac{235 + 867+156 + 215}{4}=\frac{1473}{4}=368.25)
  2. Then, calculate the following sums:
    • (\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})):
      • ((1.4 - 1.6125)(235 - 368.25)=(- 0.2125)(-133.25)=28.315625)
      • ((2.7-1.6125)(867 - 368.25)=(1.0875)(498.75)=542.0859375)
      • ((0.75 - 1.6125)(156 - 368.25)=(-0.8625)(-212.25)=183.065625)
      • ((1.6 - 1.6125)(215 - 368.25)=(-0.0125)(-153.25)=1.915625)
      • (\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=28.315625 + 542.0859375+183.065625 + 1.915625=755.3828125)
    • (\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}):
      • ((1.4 - 1.6125)^{2}=(-0.2125)^{2}=0.04515625)
      • ((2.7-1.6125)^{2}=(1.0875)^{2}=1.18265625)
      • ((0.75 - 1.6125)^{2}=(-0.8625)^{2}=0.74405625)
      • ((1.6 - 1.6125)^{2}=(-0.0125)^{2}=0.00015625)
      • (\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=0.04515625+1.18265625 + 0.74405625+0.00015625=1.972025)
    • (\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}):
      • ((235 - 368.25)^{2}=(-133.25)^{2}=17755.5625)
      • ((867 - 368.25)^{2}=(498.75)^{2}=248751.5625)
      • ((156 - 368.25)^{2}=(-212.25)^{2}=45050.5625)
      • ((215 - 368.25)^{2}=(-153.25)^{2}=23485.5625)
      • (\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=17755.5625+248751.5625+45050.5625+23485.5625=335043.25)
  3. Now, calculate the correlation coefficient (r):
    • (r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}})
    • (\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{1.972025\times335043.25}\approx\sqrt{660407.77}\approx812.65)
    • (r=\frac{755.3828125}{812.65}\approx0.93)

So the correlation - coefficient is (0.93), and the answer is (0.93).

Explanation:

Step1: Calculate means

(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}), (\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n})

Step2: Calculate product - sum

(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y}))

Step3: Calculate (x) - variance sum

(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2})

Step4: Calculate (y) - variance sum

(\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2})

Step5: Calculate correlation coefficient

(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}})