4. (10%) When you get accepted by NTPU, you might wish to commute to the campus and live in downtown Taipei City. You will likely need to take the metro and transfer by bus to the school. Suppose the travel time for metro and bus are both normally distributed and independent from each other and it takes you exactly 10 minutes in total for all the walking. | | Bus | Metro | | --- | --- | --- | | μ | 20 | 12 | | σ | 4 | 3 | Suppose you need to show up in class at 9:10AM on campus, what is the probability for you to be late when you walk out of your door at 8:25AM?

6. (20%) A manufacturing company produces electrical switches. To test the strength of the switches, robustness testing is carried out on 100 switches to determine how many times before the switch breaks down. The average of number of uses is 1.245 million times and the standard deviation is 0.3 million. The company would like to investigate whether or not the population mean of usage is more than 1.2 million times. A. At the 0.10 level of significance, is there evidence that the population mean of the number of usage is more than 1.2 million? Please form the null hypothesis and alternative hypothesis respectively before performing statistical tests. B. When the true population mean is 1.240 million and the level of significance is set as 0.05, what is the probability of making type I and type II errors respectively? Square root of numbers: | Number | 10 | 20 | 30 | 40 | 50 | 60 | | --- | --- | --- | --- | --- | --- | --- | | SQRT | 3.16 | 4.47 | 5.48 | 6.32 | 7.07 | 7.75 |

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