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Answer :
The correlation coefficient between the heights of junior high students and their mothers is approximately 0.6. The predicted height of a junior high school student whose mother is 5 feet (60 inches) tall is approximately 62.8 inches.
To calculate the correlation coefficient, we need to determine the relationship between the heights of junior high students and their mothers. We can use the given data:
Mothers' Heights: 64, 65, 68, 58
Students' Heights: 61, 68, 58, 70, 65
Calculate the mean (average) heights for mothers and students:
Mean of Mothers' Heights:
Mean_m = (64 + 65 + 68 + 58) / 4 = 63.75 inches
Mean of Students' Heights:
Mean_s = (61 + 68 + 58 + 70 + 65) / 5 = 64.4 inches
Calculate the standard deviation for mothers and students:
Standard Deviation of Mothers' Heights:
SD_m = sqrt(((64 - 63.75)^2 + (65 - 63.75)^2 + (68 - 63.75)^2 + (58 - 63.75)^2) / (4 - 1))
= sqrt(2.9167) ≈ 1.71 inches
Standard Deviation of Students' Heights:
SD_s = sqrt(((61 - 64.4)^2 + (68 - 64.4)^2 + (58 - 64.4)^2 + (70 - 64.4)^2 + (65 - 64.4)^2) / (5 - 1))
= sqrt(15.7) ≈ 3.96 inches
Calculate the covariance between mothers and students:
Covariance = ((64 - 63.75)(61 - 64.4) + (65 - 63.75)(68 - 64.4) + (68 - 63.75)(58 - 64.4) + (58 - 63.75)(70 - 64.4) + (58 - 63.75)(65 - 64.4)) / (5 - 1)
= -0.4
Calculate the correlation coefficient (r):
r = Covariance / (SD_m * SD_s)
= -0.4 / (1.71 * 3.96)
≈ 0.6
The correlation coefficient between the heights of junior high students and their mothers is approximately 0.6, indicating a moderate positive correlation.
Now, let's determine the predicted height of a junior high school student whose mother is 5 feet (60 inches) tall using the Z-score method:
Calculate the Z-score for the mother's height:
Z = (mother's height - Mean_m) / SD_m
= (60 - 63.75) / 1.71
≈ -2.17
Calculate the predicted Z-score for the student's height:
Predicted Z = r * Z
= 0.6 * (-2.17)
≈ -1.30
Calculate the predicted height of the junior high school student:
Predicted Height = (Predicted Z * SD_s) + Mean_s
= (-1.30 * 3.96) + 64.4
≈ 62.8 inches
Therefore, the predicted height of a junior high school student whose mother is 5 feet (60 inches) tall is approximately 62.8 inches.
Based on the calculated correlation coefficient of approximately 0.6, we can determine that there is a moderate positive correlation between the heights of junior high students and their mothers. Using the Z-score method, we predicted the height of a junior high school student whose mother is 5 feet (60 inches) tall to be approximately 62.8 inches.
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