A cosmic ray observatory monitors the frequency of high-energy solar micro-bursts. Records show that the observatory detects an average of 48 micro-bursts per 10-hour cycle. The number of micro-bursts detected per hour is modeled using a Poisson distribution.
Determine the critical region for a two-tailed test of the mean number of detections per hour at a 10% level of significance. The probability of rejection in each tail must be less than 0.05.
Calculate the actual significance level of this test.
A new filtering lens is installed which the lead scientist claims has significantly reduced the sensitivity to these micro-bursts, thereby reducing the mean number of detections.
A random observation period of 50 hours is conducted, and a total of 215 micro-bursts are recorded.
Use a suitable approximation to test the scientist's claim at a 5% level of significance. You should state your hypotheses clearly.