4. A metaverse company analyzes the sales situation in 111 years, and determines the relationship between the payment method and the price category of consumers to purchase metaverse goods, and obtains the following joint probability table.

| Item | cash | credit card | debit card |
| --- | --- | --- | --- |
| Below 10,000 yuan | 0.09 | 0.03 | 0.04 |
| 10,000--100,000 yuan | 0.05 | 0.21 | 0.18 |
| More than 100,000 yuan | 0.03 | 0.23 | 0.14 |

Please answer: (15 %)

(1) What is the proportion of consumer payments made by using debit cards?

(2) What is the probability that consumers will spend more than 100,000 yuan when they use a credit card to pay for a transaction?

(3) What is the proportion of purchases paid by consumers using credit or debit cards for financial transactions?

5. Assuming that the operating income of the head office and branches of the international business is in line with the binary normal distribution, the operating income of the international business head office in 111 year is subject to the normal distribution of the mean value is 170 and the standard deviation is 10. The operating income of the branch is subject to the normal distribution of mean value is 170 and the variance is 10. The correlation coefficient between these two is 0.6. Today, considering a group of international businesses with a total operating income of 175. Please answer: (10 %) (1) What is the average operating income of its branches?

(2) What is the proportion of operating income exceeding 175?

6. An international business uses a simple linear regression model to estimate the relationship between the number of advertisements X and the sales amount Y. According to the monthly data of the business from January to October, 111, the average value of the 10 sales is 130, while the variation is 1748. As for the average number of 10 advertisements, it is 14. Additionally, the covariance between the sales amount Y and the number of advertisements X is 316. Also, the regression equation estimated using the classical least squares method (OLS) is: Y = 60 + 5X. Please answer: (25 %) (1) If the sales amount in March is 88, and the number of advertisements in this month is 8. Please estimate the fitted value for the sales amount on March, 111.

(2) Continuing from the previous question, please estimate the residual value of the sales amount in March.

(3) Estimate the coefficient of determination \((\mathbb{R}^2)\).

(4) Please estimate the correlation coefficient (CORR) between the sales amount Y and the number of advertisements X.

(5) In the case that the significant level alpha \((\alpha)\) is equal to 0.05, please use the T test approach to test the hypothesis that the number of advertisements X affects the sales amount Y. That is, please check whether the slope value of the regression linear relationship is zero or not.

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