24-sa-inf-model-slr.Rmd

1. \(a) $H_0: \beta_1 = 0$, $H_A: \beta_1 \ne 0$. (b) The observed slope of 0.604 is not a plausible value, the p-value is extremely small, and the null hypothesis can be rejected. c. The p-value is also extremely small.
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1. \(a) Roughly 0.53 to 0.67. (b) For individuals with one cm larger shoulder girth, their average height is predicted to be between 0.53 and 0.67 cm taller, with 98% confidence.
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1. \(a) $H_0: \beta_1 = 0$, $H_A: \beta_1 \ne 0$. (b) The observed slope of 2.559 is not a plausible value, the p-value is extremely small, and the null hypothesis can be rejected. (c) The p-value is also extremely small.
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1. \(a) Rough 90% confidence interval is 1.9 to 3.1. (b) For a one unit (one percentage point) increase in poverty across given metropolitan areas, the predicted average annual murder rate will be between 1.9 and 3.1 persons per million larger, with 90% confidence.
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1. \(a) $H_0: \beta_1 = 0$, $H_A: \beta_1 \ne 0$. (b) The p-value is roughly 0.45 which is much bigger than 0.05.  The null hypothesis cannot be rejected.There is no evidence with these data that there is a linear relationship between a father's age and the baby's weight. (c) The p-value of 0.449 is quite similar. The hypothesis test conclusion is the same, the data do not support a linear model.
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1. \(a) Rough 95% confidence interval is (-.008, 0.016). (b) 95% confident that for individuals with fathers who are one year older, their average weight is predicted to be between -0.008 and 0.016 pounds heavier.
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1. \(a) $H_0$: The true slope coefficient of body weight is zero ($\beta_1 = 0$). $H_A$: The true slope coefficient of body weight is different than zero ($\beta_1 \neq 0$). (b) The p-value is extremely small (zero to 4 decimal places), which is lower than the significance level of 0.05. With such a low p-value, we reject $H_0$. The data provide strong evidence that the true slope coefficient of body weight is greater than zero and that body weight is positively associated with heart weight in cats. (c) (3.539, 4.529). We are 95% confident that for each additional kilogram in cats' weights, we expect their hearts to be heavier by 3.539 to 4.529 grams, on average. (d) Yes, we rejected the null hypothesis and the confidence interval lies above 0.
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1. \(a) $r = \sqrt{0.292} \approx -0.54$. We know the correlation is negative due to the negative association shown in the scatterplot. (b) The residuals appear to be fan shaped, indicating non-constant variance. Therefore a simple least squares fit is not appropriate for these data.
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