114年 - 114 國立臺北大學_碩士班一般入學考試試題_企業管理學系甲組：統計學#146907

科目：研究所、轉學考(插大)◆統計學 | 年份：114年 | 選擇題數：0 | 申論題數：7

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所屬科目：研究所、轉學考(插大)◆統計學

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申論題 (7)

1. 假設線性迴歸式為：$Y = \beta0 + \beta1X + \varepsilon$，$\beta0$、$\beta1$ 是參數，$\varepsilon$ 是誤差，樣本大小為 n。

依據這個模式回答兩個問題：

(1) \(\beta_{1}\) 的估計滿足一致性(consistency)指的是什麼？(\(10\%\))

(2) 不滿足這個特性對 \(\beta_{1}\) 的估計有何影響？(\(10\%\))

2. 北大企業行銷部門進行新產品市場測試。行銷人員想知道包裝與口味對消費意願的影響。測試因子設計包括：三款包裝(P1, P2, P3)及兩種口味(T1, T2)。藉由隨機指派 180 位受測者到這六個組合，依據蒐集的資料進行二因子變異數分析(two-way ANOVA)。統計檢定發現：包裝與口味有交互效果(interaction effect)。請問：

(1) 這個發現對行銷人員有什麼意涵？(20%)

(2) 如果行銷人員想要得到更具體的結果，您認為接下來該進行什麼分析？(10%)

Part II

1. A group of 6 friends are exchanging Christmas gifts. In how many ways can they exchange gifts so that no one receives their own gift? (15%)

2. Given that the lifetime of a particular device follows an exponential distribution with a mean of \( x \) hours, find the expected lifetime of the device, given that it has lasted at least \( Y \) hours. (15%)

3. A rare medical condition affects only 1 in 50,000 people in a specific population. A sophisticated diagnostic test for this condition has the following characteristics:

If a person has the disease, the test is \(99.5\%\) accurate in detecting it (sensitivity).
If a person does not have the disease, the test has a \(1.5\%\) false positive rate.
Two independent tests are performed on the same patient.

What is the probability (%) that a person actually has the disease, given that they tested positive on the first test and also tested positive on the second test? (20%)

(Note: You are asked to answer in percentages, counted to two decimal places.)