1. 悉尼希尔顿酒店的总经理正在评估一项就业筛选测试

70% 通过测试；其余的失败。稍后，在新的文员工作之后

(a) 根据给定的信息，确定以下事件的概率： (i) 通过-

(b) 使用您对 (a) 部分的回答，确定文员通过

(c) 使用您在 (a) 和 (b) 部分的答案，确定文职雇员的概率：
(i) 考试不及格且表现令人满意， (ii) 考试不及格且成绩不合格

(d) 使用您在 (c) 部分的答案，确定文职员工的概率：(i) 失败

(v) 如果他们通过了测试，表现令人满意， (vi) 表现不令人满意

(e) 使用您在 (d) 部分的答案，确定以下百分比： (i) 文员

(f) 政府指南要求筛选测试达到 (e) 中的 (i) 部分至少达到 20% 和
(e) 中的 (ii) 部分至少为 60%。该测试是否符合政府的要求？解释。
[2 分]

2. 在设计卫星电源系统时，工程师需要确定为系统供电的电池发生早期故障的概率。需要随机抽取电池样本进行早期失效测试。那些被采样的电池都是相同类型的，适合在卫星上使用。工程师希望找到能够估计不会过早失效的电池的真实比例 p 的样本量，该比例将在总体比例的 0:05 范围内，概率不低于 95%。假设电池的数量很大。

(a) 如果最初假定 p 为 0.5，请找出工程师所需的样本量，解释您的

(b) 在测试 (a) 中发现的电池数量后，只有 250 个证明是令人满意的。是什么
p 的点估计？如果这是 p 的真实值，则不需要测试多少电池-

1. The general manager of the Hilton Hotel in Sydney is evaluating an employment screening test for
the front ofﬁce clerical staff. During this evaluation all new clerical employees are given the test.
70% pass the test; the rest fail. At a later time, after the new clerical employees have been working
for a while, their performance is evaluated as being satisfactory or unsatisfactory. Historically, 80%
of all clerical employees have been found to be satisfactory, and 75% of the satisfactory clerical
employees in the evaluation of the employment screening test have passed the screening test.

(a) From the given information, determine the probabilities of the following events: (i) pass-
ing the test, (ii) having satisfactory performance and (iii) passing the test given satisfactory
performance. [3 marks]

(b) Using your answers to part (a), determine the probability of a clerical employee passing the
test and having satisfactory performance. [1 mark]

(c) Using your answer in parts (a) and (b), determine the probabilities of the a clerical employee:
(i) failing the test and having satisfactory performance, (ii) failing the test and having unsat-
isfactory performance, (iii) passing the test and having unsatisfactory performance, and (iv)
having unsatisfactory performance. [4 marks]

(d) Using your answers in part (c), determine the probabilities of a clerical employee: (i) failing
the test given they are found to have unsatisfactory performance, (ii) failing the test given
they are found to have satisfactory performance, (iii) passing the test given they are found to
have unsatisfactory performance, (iv) unsatisfactory performance given they failed the test,
(v) satisfactory performance given they passed the test, (vi) unsatisfactory performance given
they passed the test, and (vii) satisfactory performance given they failed the test. [7 marks]
(e) Using your answers in part (d), determine the following percentages: (i) clerical employees
who failed the test and prove to be unsatisfactory, and (ii) clerical employees who passed the
test and who prove to be satisfactory. [2 marks]

(f) Government guidelines require screening tests to achieve at least 20% for part (i) in (e) and
at least 60% for part (ii) in (e). Does this test meet those government requirements? Explain.
[2 marks]

2. In designing the power system for a satellite, an engineer needs to determine the probability of early failure for the batteries powering the system. A random sample of batteries needs to be selected for testing for early failure. Those sampled batteries will all be of the same type, suitable for use in the satellite. The engineer wishes to ﬁnd the sample size that will provide an estimate of the true proportion of batteries that do not fail too early, p, that will be within 0:05 of the population proportion with probability no less than 95%. Assume that the population of batteries is large.

(a) If p is initially assumed to be 0.5, ﬁnd the engineer’s required sample size, explaining your
reasoning. [5 marks]

(b) After testing the number of batteries found in (a), only 250 prove satisfactory. What is the
point estimate of p? If this were the true value of p how many batteries were tested unneces-
sarily? [5 marks]