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The data in the table below are the gestation periods, in months, of randomly selected animals and their corresponding life-spans in years.
Gestation
8, 2.1, 1.3, 1 11.5, 5.3, 3.8, 24.3
Life span
30 12 6 3 25 12 10 40
For these questions, round values to the nearest tenth.
What is the correlation between the values, rounded to the nearest hundredth?
What is the slope of the regression line
What is the y-intercept?
What is the predicted lifespan for an animal with a gestation period of 6 months?
Bot GPT 3.5
answered
To find the correlation between the values, we first need to calculate the mean for both sets of data.
Mean of gestation periods:
(8 + 2.1 + 1.3 + 1 + 11.5 + 5.3 + 3.8 + 24.3) / 8 = 6.95
Mean of life spans:
(30 + 12 + 6 + 3 + 25 + 12 + 10 + 40) / 8 = 15.5
Next, calculate the covariance and standard deviation for both sets of data. Then use these to calculate the correlation coefficient.
Covariance:
Cov(X,Y) = Σ[(X - mean of X)(Y - mean of Y)] / n-1
Cov(X,Y) = [(8-6.95)(30-15.5) + (2.1-6.95)(12-15.5) + ... + (24.3-6.95)(40-15.5)] / 7
Cov(X,Y) = 78.5357
Standard deviation of gestation periods:
(Σ[(X - mean of X)^2] / n-1)^0.5
St. dev. of gestation = 6.88
Standard deviation of life spans:
(Σ[(Y - mean of Y)^2] / n-1)^0.5
St. dev. of life spans = 12.3
Correlation coefficient:
Correlation = Cov(X,Y) / (St. dev. of gestation * St. dev. of life spans)
Correlation = 78.5357 / (6.88 * 12.3) = 0.9549
For the slope of the regression line:
Slope = Cov(X,Y) / St. dev. of gestation^2
Slope = 78.5357 / 6.88^2 = 1.97
For the y-intercept, we can use the formula:
y-intercept = mean of Y - slope * mean of X
y-intercept = 15.5 - 1.97 * 6.95 = 1.58
To find the predicted lifespan for an animal with a gestation period of 6 months, we use the regression line equation:
predicted lifespan = slope * 6 + y-intercept
predicted lifespan = 1.97 * 6 + 1.58 = 13.5 years.