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110年 - 110 國立臺灣大學_碩士班招生考試_土木工程研究所交通組:統計學(C)#100575
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6. [10%] The cumulative distribution function for random variable X is given by:
(i) (5%) Determnine the median of X.
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(i) (10%) Suppose that the daily concentration follows a normal distribution. The critical level of pollutant concentration would be reached if the daily concentration exceeds 100 mgl. What is the probability that the daily concentration of pollutants in the river would not exceed the critical level?
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(ii) (10%) Suppose that the pollutant concentration between different days of a woek is statistically independent. What is the probability that the critical level of pollutant concentration would not be reached during a given week?
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(iii) (10%) Suppose that the daily co oncentration follows a log-normal distribution. What is the probability that the daily concentration of pollutants in the river would not exceed the critical level (100 mg/l)?
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(i)(10%) Determine the probability that a rod randomly chosen from this manufacturer will not meet the specification.
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3.[10%] Laboratory tests are performed to evaluate the compressive strengths for 2 types of concrete (A and (B). 10 specimens are used for each type of concrete. The sample standard deviati tions are 200 kPa and 250 kPa for concrete A and concrete B, respectively. Construct a 90% confidence interval for the ratio of the 2 population variances
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(i) (10%) Calculate the sample means and sample variances of the scores for classes A and B respectively.
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(ii) (10%) Assume that the populations of the test scores can be approximated with normal distributions having the same variance, test the claim that there is no difference between the scores for classes A and B. Use the P-value approach and 0.05 as the level of significance.
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(ii) (5%) Find the expected value of X.
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