Impact of Changes in Mean and Standard Deviation
1. Suppose that in certain chemical plant the reaction time between activation of a warning light and the operator's sounding a buzzer alarm is normally distributed with a mean μ and a standard deviation σ.
a. Assume that the mean reaction time μ is set at 30 seconds. Suppose the value of the standard deviation σ increases from 3 to 4 seconds. What does the increase mean to the reaction time? Does the risk of a chemical accident increase or decrease by the change? Explain briefly.
b. Now assume that the mean μ increases from 30 to 35 with a constant standard deviation σ. How does the change in the value of μ affect the reaction time? What does that mean to the risk of a chemical accident? Explain your answer.
c. If an alarm is not sounded within 50 seconds after a hazardous event, the cooling system may have a problem to handle the situation with unpredictable consequences. Therefore, it is critical that the appropriate switches are turned on within the time limit. If μ is 30 and σ is 4. What fraction of time the response time will exceed 50 seconds? What would be the fraction when the value of μ were 35 with the same value of σ?
d. Now suppose that the mean reaction time is 35 seconds with the value of σ equal to 5. What fraction of time the response time will exceed 50 seconds?
e. What is the probability that the reaction time will be within 1 standard deviation of the mean of 30 seconds? What is the probability that the reaction time will be within 2 standard deviations of the mean? Assume here that σ = 4 seconds.
f. Assume μ=30 seconds and σ=4 seconds. What is the reaction time exceeded 95% of the time? What would be the reaction time exceeded 99 % of the time?
g. Suppose the value of the mean μ is unknown and σ = 5 seconds. What is the largest value of the mean μ be so that the probability of not responding within 50 seconds is smaller than 0.001? (Note: answer for this question should have 2 digits after the point).
2. The investigators in this chemical plant have found that occurrences of emergency situations closely fit a Poisson process. Let λ be the average number of emergency alerts per week.
a. Find the probability distribution for the number of emergency situations using values of λ=1, 2, 3, 4, 5 and 6. Create a bar plot with the probability distribution for each case (for number of emergencies from 0 to 30). Describe briefly the change in the appearance of the probability distribution function as λ increases from 1 to 6.
b. Calculate the probability that no emergency situations will occur in a given week. Assume that there are approximately 180 emergency alerts occurring over a year.
3. As above we assume that there are approximately 180 emergency alerts occurring over a year. How likely is that in one of them the operator will not be able to respond on time? Assume that the alerts occur independently of one another and that the reaction time follows a normal distribution with the mean of μ=30 and σ=4 seconds.
a. What is the probability that the reaction time exceeds 50 seconds in none of 180 emergency alerts? Specify the binomial distribution parameters you have entered in the template to answer the above question.
b. What is the probability that the reaction time exceeds 50 seconds in exactly one of 180 emergency alerts? What are the binomial distribution parameters used to answer the question?
4. The following table contains the reaction times for 5 different panel configurations for the emergency system and 20 operators responding to a particular emergency situation. The quality of a configuration panel can be measured by the mean reaction time and its standard deviation. Calculate the mean and standard deviation of the reaction time for each of the 10 panels. Which panel configuration would you recommend to minimize the risk of a nuclear accident caused by human error? Provide brief explanation of your answer referring to the numerical summaries.