A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.
Number of bees
0
1
2
3
4
5
6
7
\(\geqslant 8\)
Number of intervals
6
16
19
18
17
14
6
4
0
Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer.
Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a \(95 \%\) confidence interval for the overall mean time.