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Math  /  Data & Statistics

QuestionIslam Spre. Copy of Pattems of - Unblocked Games -- Unblocked Garnes -- Vandalizing My Ow... Ne \#2b ur math teacher recorded the grades students received based the number of hours they studied for a test. The results are wn below and can be represented by a linear function. the table below to answer the questions. \begin{tabular}{|c|c|} \hline Hours Studied & Grade on Test \\ \hline 5 & 85 \\ \hline 7 & 92 \\ \hline 9 & 93 \\ \hline 9 & 95 \\ \hline 12 & 108 \\ \hline \end{tabular}
What is the correlation coefficie 0.15 −0.15-0.15 0.96 −0.96-0.96

Studdy Solution

STEP 1

1. The data points are (5,85),(7,92),(9,93),(9,95),(12,108)(5, 85), (7, 92), (9, 93), (9, 95), (12, 108).
2. The correlation coefficient measures the strength and direction of a linear relationship between two variables.

STEP 2

1. Calculate the means of the hours studied and the grades.
2. Compute the covariance of the variables.
3. Calculate the standard deviations of the hours studied and the grades.
4. Use the correlation coefficient formula to find the correlation.

STEP 3

Calculate the mean of the hours studied:
Mean of hours=5+7+9+9+125=425=8.4\text{Mean of hours} = \frac{5 + 7 + 9 + 9 + 12}{5} = \frac{42}{5} = 8.4
Calculate the mean of the grades:
Mean of grades=85+92+93+95+1085=4735=94.6\text{Mean of grades} = \frac{85 + 92 + 93 + 95 + 108}{5} = \frac{473}{5} = 94.6

STEP 4

Compute the covariance of the variables:
Covariance=(5−8.4)(85−94.6)+(7−8.4)(92−94.6)+(9−8.4)(93−94.6)+(9−8.4)(95−94.6)+(12−8.4)(108−94.6)5\text{Covariance} = \frac{(5-8.4)(85-94.6) + (7-8.4)(92-94.6) + (9-8.4)(93-94.6) + (9-8.4)(95-94.6) + (12-8.4)(108-94.6)}{5}
=(−3.4)(−9.6)+(−1.4)(−2.6)+(0.6)(−1.6)+(0.6)(0.4)+(3.6)(13.4)5= \frac{(-3.4)(-9.6) + (-1.4)(-2.6) + (0.6)(-1.6) + (0.6)(0.4) + (3.6)(13.4)}{5}
=32.64+3.64−0.96+0.24+48.245=83.85=16.76= \frac{32.64 + 3.64 - 0.96 + 0.24 + 48.24}{5} = \frac{83.8}{5} = 16.76

STEP 5

Calculate the standard deviation of the hours studied:
SD of hours=(5−8.4)2+(7−8.4)2+(9−8.4)2+(9−8.4)2+(12−8.4)25\text{SD of hours} = \sqrt{\frac{(5-8.4)^2 + (7-8.4)^2 + (9-8.4)^2 + (9-8.4)^2 + (12-8.4)^2}{5}}
=11.56+1.96+0.36+0.36+12.965=27.25=5.44≈2.33= \sqrt{\frac{11.56 + 1.96 + 0.36 + 0.36 + 12.96}{5}} = \sqrt{\frac{27.2}{5}} = \sqrt{5.44} \approx 2.33
Calculate the standard deviation of the grades:
SD of grades=(85−94.6)2+(92−94.6)2+(93−94.6)2+(95−94.6)2+(108−94.6)25\text{SD of grades} = \sqrt{\frac{(85-94.6)^2 + (92-94.6)^2 + (93-94.6)^2 + (95-94.6)^2 + (108-94.6)^2}{5}}
=92.16+6.76+2.56+0.16+179.565=281.25=56.24≈7.5= \sqrt{\frac{92.16 + 6.76 + 2.56 + 0.16 + 179.56}{5}} = \sqrt{\frac{281.2}{5}} = \sqrt{56.24} \approx 7.5

STEP 6

Use the correlation coefficient formula:
r=CovarianceSD of hours×SD of grades=16.762.33×7.5≈16.7617.475≈0.96r = \frac{\text{Covariance}}{\text{SD of hours} \times \text{SD of grades}} = \frac{16.76}{2.33 \times 7.5} \approx \frac{16.76}{17.475} \approx 0.96
The correlation coefficient is:
0.96 \boxed{0.96}

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