Question

Researchers measured oxygen consumption in 10 seals. They measured oxygen consumption for each seal under two conditions: (1) when they did a dive for food (referred to as a feeding dive) and (2) when they did a dive that was not food-related (referred to as nonfeeding dives).   They wanted to learn about the relationship between oxygen consumption for feeding dives and oxygen consumption for nonfeeding dives so they fit a linear regression model with Y = Oxygen consumption in feeding dive and X = Oxygen consumption in nonfeeding dive. The model was fit in R and the results are given below:

Coefficients:

   Estimate Std. Error t value    Pr(>|t|)

Intercept    16.7693     8.6483    1.939 0.0885 .

OxygenNonfeeding 1.2010 0.1124 10.681 5.18e-06 ***

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Residual standard error: 6.542 on 8 degrees of freedom

Multiple R-squared: 0.9345,Adjusted R-squared: 0.9263

F-statistic: 114.1 on 1 and 8 DF, p-value: 5.18e-06

1. Find a 95% confidence interval for the slope of the regression equation. Notice that you are given the standard error in the R output (you do not need to calculate it by hand). Based on this confidence interval, does there appear to be an association between oxygen consumption in feeding dives and oxygen consumption in nonfeeding dives? Justify your response.

2. The coefficient of determination is 0.9345. Find a numerical value for the correlation coefficient. How do you know if it is positive or negative?

Answer 1

a) From output,

95% confidence interval for slope is

Where SE is standard error = 0.1124

Where tc is t critical value for alpha = 0.05 and degrees of freedom = n - 2 = 10 - 2 = 8

tc = 2.306         (From statistical table of t values)

95% confidence interval is

Since zero does not include in above interval then we can say that there is linear relationship between Oxygen feeding and oxygen non feeding.

b)

Correlation coefficient = r= = 0.9667

r = 0.9667

Correlation coefficient is positive because here slope = 1.2010 is positive.

Sign of correlation coefficient depends on sign of slope