(b) \((5\%)\) Find a \(95\%\) confidence interval for the different between the two population means.

6. (10%) The following table contains observed frequencies of a sample of 500. Test for independence of the row and column variables using α=0.01 level of significance.

7. (10%) In an experiment to investigate the performance of four different brands of tires for 125-cc motorcycles, five tires of each brand were tested and number of miles until failure was observed. A partially completed ANOVA table is given: | Source of variation | Sum of Squares | Degrees of Freedoms | Mean Square | F-value | | --- | --- | --- | --- | --- | | Treatments | (?) | (?) | (?) | (?) | | Error | 235419 | (?) | (?) | | | Total | 310501 | (?) | | | (a) \((7\%)\) Complete the ANOVA table.

(b) \((3\%)\) At the \(\alpha = 0.05\) level of significance, test whether the average number of miles until failure for the four different brands of tires are equal.

8. (20%) Given are following observations collected in a regression study on two variables. Consider the linear regression model $$Y = \beta_0 + \beta_1X + \varepsilon$$ | $$x_t$$ | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | | $$y_t$$ | 3 | 7 | 5 | 11 | 14 | (a) \((5\%)\) Find the estimated regression equation for these data.

(b) \((5\%)\) Use the estimated regression equation to predict the value of \(y\) when \(x = 7\).

(c) \((5\%)\) Find the coefficient of determination \(R^2\)

(d) (5%) Use F test to test the hypotheses \( H_0 \): \( \beta_1 = 0 \) vs. \( H_a \): \( \beta_1 \neq 0 \) at 0.05 level of significance.

The following table is the cumulative probabilities for the standard distribution

作文題目一：Effects of Coronavirus Disease 2019 (COVID-19) in Taiwan (250-300 words; 70%)

作文題目二：Brand-Name Products: Are They Worth the Money? (150-200 words; 30%)

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