Question: Suppose that you obtained the following summary quantities to estimate the parameters in a regression study. Assume that x and y are related according to the simple linear regression model: \[\hat{y}\] = \[{{\hat{\beta }}_{0}}\] + \[{{\hat{\beta }}_{1}}\] x .

n = 14, \[\sum\limits_{i=1}^{n}{{{y}_{i}}}\] = 572, \[\sum\limits_{i=1}^{n}{y_{i}^{2}}\] = 23,530, \[\sum\limits_{i=1}^{n}{{{x}_{i}}}\] = 43, \[\sum\limits_{i=1}^{n}{x_{i}^{2}}\] = 157.42, and \[\sum\limits_{i=1}^{n}{{{x}_{i}}}{{y}_{i}}\] = 1697.80.

1. Calculate the least squares estimates of the slope ( \[{{\hat{\beta }}_{1}}\] ) and intercept ( \[{{\hat{\beta }}_{0}}\] ).
2. Estimate  ² .
3. Calculate r .
4. Calculate the t statistic to test the hypothesis H _o :  ₁ =0. Is the slope significantly different from zero, assuming =5%?
5. Calculate the t statistic to test the hypothesis H _o :  Is correlation significantly different from zero, assuming =5%?

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