110年 - 110 國立臺北大學_碩士班一般入學考試試題_國際企業研究所：統計學#147633

(1) If the sales in the third month are y3 = 88 and the volume of advertisements performed is x3 = 8, please try to calculate the fitted value and residual value of this month's sales. (5 points)

(2) Please calculate the coefficient of determination R², and sales data y1,...,y10, and the correlation coefficient of the volume of advertisements x1,..., x10. (5 points)

(3) At the significance level α=0.05, use the t test method to test whether the volume of advertisements will affect sales, that is, test whether the slope of the linear relationship of the regression is 0. (5 points)

2. Assuming that there are 3 possible variants during a disease epidemic related to COVID-19: A, British variant, B, South African variant, and C, American variant of COVID-19 disease.

Among them, the probability of infected patients is that the probability of the A British variant new coronary pneumonia disease is 0.6, the B South African variant new coronary pneumonia disease probability is 0.3, and the C American variant new coronary lung disease probability is 0.1.

These three types of new coronary pneumonia diseases may have fever and cough symptoms. Among them, 20% of patients with A: British variant have fever, 60% of patients with B: South African variant have fever, and 80% of patients with C: American variant have fever; And 50% of patients with variant A in the UK have cough symptoms, 35% of patients with variant B in South Africa have cough symptoms, and 25% of patients with variant C in the United States have cough symptoms.

An existing patient who is infected has fever symptoms. Please calculate the probability of this patient to have these three types of new coronary pneumonia, and determine which type of new coronary pneumonia the patient is most likely to have. (10 points)

3. A Taiwan mask manufacturer received an order worth 50 million yuan from a politically unstable country (India). Given a probability of 0.7 this manufacturer can receive payment for the 50 million yuan order. There is a probability of 0.15 that you can only receive a payment of 30 million yuan, a probability of 0.1 that you can only receive a payment of 10 million yuan, and a probability of 0.05 that you will not receive any payment. For the sake of insurance, this Taiwanese manufacturer insures an accident insurance and pays an insurance premium of 10 million yuan first. The Japanese insurance company underwriting will pay any shortfalls in the company's receivables. If the random variable X represents the amount of money spent or earned by this Taiwanese mask manufacturer finally insuring here. Please calculate the expected value E(X) of X. (10 points)

4. An international business wants to choose one of the two recommended by its A branch and B branch to be promoted to a supervisory position. The performance information of the employees of these two branches is as follows: (1) Branch A: The z-score for the branch with a performance score of 84 points is 0.5, and the z-score for the branch with a performance score of 72 points is -2.5. The recommended employee's performance score in his branch is 90.0 points. (2) Branch B: The z-score for the branch with a performance score of 84 points is 3.0, and the z-score for the branch with a performance score of 72 points is -1.0. The recommended employee's performance score in his branch is 82.5 points.

The HR supervisor decided to use the z score to determine the candidate for promotion. May I ask which branch company recommended the employee to be promoted, and please explain the reason (10 points).

5. To compare whether the average lifespan of cell phone batteries of three different brands is the same, randomly test 5 batteries of each brand of cell phone batteries and record their life.

The average life spans of the three groups of samples are 3.0, 4.5, and 3.6, and the standard deviations of life spans are 1.5, 1.0, and 15.0, respectively. At the significance level α=0.05, please use one-way ANOVA to check whether the average battery life of these three brands of mobile phones is consistent, that is, check:

The null hypothesis H0: μ1 = μ2 = μ3, the opposite hypothesis H1: μ1, μ2, and μ3 are not completely equal.

Among them, μ1, μ2 and μ3 are the average lifespans of these three brands of mobile phone batteries. (5 points)

a. If the manager wants to limit the probability of being out of stock of the Tea Party paper bookmark to no more than \( 5\% \) in a week, how many units of the bookmarks should he keep at the start of the week? (5 points)

b. If the Shop has 2500 units on hand at the start of week, what is the probability that weekly demand will be greater than the inventory? (5 points)

c. What is the probability that the Shop sells between 2000 and 2500 Tea Party paper bookmarks in a typical week? (6 points)

d. If the standard deviation of weekly demand for the toy increases from 200 units to 350 units, how many more Tea Party paper bookmarks would have to be stocked to ensure that the probability of weekly demand exceeding inventory is no more than \( 10\% \)? (5 points)

a. First of all, from the age figures of the infected cases above, can you tell us if the infected cases' age is normally or skewed distributed (please show the calculations)? (5 points)

c. In terms of the infected cases, we would like to check them more closely. We would you to create a box and whisker plot for each of the case: the survival cases, and the confirmed death cases. (8 points)

b. What does the above result mean in plain language? (3 points)

d. Check the two box plots, can you tell us if you can determine the claim "survivors are mostly younger"? (5 points)

e. We run an independent samples t-Test (alpha=0.05) to check whether age matters in the survival/death cases, the results are shown below. Please tell us your conclusion. (3 points)

f. Which one (box and whisker plot or independent samples t-Test) should be employed to test Claim B? Why? (5 points)