Unknown distribution, CLT applied

2 The number of emergency calls received by a fire station may be modelled by a Poisson distribution. During a given period of 13 weeks, the station received a total of 108 emergency calls.

1. Construct an approximate \(98 \%\) confidence interval for the average weekly number of emergency calls received by the station.
2. Hence comment on the station officer's claim that the station receives an average of one emergency call per day.
   (2 marks)

7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8

1. A courier delivers parcels. The random variable \(X\) represents the number of parcels delivered successfully each day by the courier where \(X \sim \mathrm {~B} ( 400,0.64 )\)

A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { 100 }\) is taken.
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257

1. A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

1. There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2

1. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3

Estimate the probability that the mean of the sample is greater than 2.5