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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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[ACTION NAME]

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# Your action step 2

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...

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system
The number of eggs $y$ laid by a medicinal insect is related to the temperature $x$ within a certain range. Six sets of observation data of this medicinal insect have been collected as shown in the table below:

| Temperature $x/^{\circ}\mathrm{C}$ | 21 | 23 | 24 | 27 | 29 | 32 |
|------------------------------------|----|----|----|----|----|----|
| Number of eggs $y/$个              | 6  | 11 | 20 | 27 | 57 | 77 |

$(Ⅰ)$ If a linear regression model is used, find the regression equation of $y$ with respect to $x$, $\hat{y}=\hat{b}x+\hat{a}$ (rounded to 0.1).

$(Ⅱ)$ If a nonlinear regression model is used and the regression equation of $y$ with respect to $x$ is $\hat{y}=0.06e^{0.2303x}$ with a correlation coefficient $R^{2}=0.9522$.

(i) Compare the two models in $(Ⅰ)$ and $(Ⅱ)$ using $R^{2}$ to determine which model has a better fit.

(ii) Use the model with the better fit to predict the number of eggs laid by this medicinal insect at a temperature of $35^{\circ}\mathrm{C}$ (round the result to the nearest integer).

Given: For a set of data $(x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$, the least squares estimates of the slope and intercept of the regression line $\hat{y}=\hat{b}x+\hat{a}$ are $\hat{b}=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$ and $\hat{a}=\overline{y}-\hat{b}\overline{x}$, respectively. The correlation coefficient $R^{2}=1-\frac{\sum_{i=1}^{n}({y_i}-\hat{y_i})^2}{\sum_{i=1}^{n}({y_i}-\overline{y})^2}$. Given that $\sum_{i=1}^{6}(x_{i}-\overline{x})(y_{i}-\overline{y})=557$, $\sum_{i=1}^{6}({y}_{i}-\overline{y})^{2}=3930$, $\sum_{i=1}^{6}({y_i}-\hat{y_i})^2=236.64$, and $e^{8.0605}=3167$.

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