Descriptive Statistics
Additional Sheet 5
1. Calls come into a call centre at the rate of 4/minute. Calculate the probability
that
i)
the time between calls is greater than one minute.
ii)
The time between calls is less than 10 seconds
iii)
The time between calls is between 15 and 45 seconds.
2. The length of a side of a square is chosen at random from the uniform
distribution on the interval [3, 6]. Calculate
i)
The probability that the length of the side is between 4 and 8
ii)
The probability that the area of the square is between 12.25 and
25.
iii)
The expected area of the square.
3. The density function of random variable X is given by f(x)=cx4 for xϵ[-1,1],
otherwise f(x)=0. Calculate
i)
the value of the constant c
ii)
P(0.5<X<1)
iii)
E(X) and σX.
4. The distribution of monthly salaries in Luxembourg is given by the Pareto
distribution with minimum wage 2000 Euros and concentration factor 2.
a) Calculate the probability that an individual earns
i) more than 6000 Euros/month
ii) between 3000 and 5000 Euros/month
b) Calculate the mean monthly salary in Luxembourg.
5. The mean maximum daily high in Wrocław on midsummer's day is 26
degrees with a standard deviation of 4 degrees. Assuming that this
temperature has a normal distribution.
a) Calculate the probability that the highest temperature on midsummer's day this year is
i) greater than 29
ii) less than 20
iii) between 21 and 28
b) The probability that the maximum daily high on midsummer's day exceeds t is 0.15, find t.
6. The variance of the height of adult Poles is 225cm2.
a) Find the probability that the mean of the heights of 50 randomly chosen adult Poles is
within 3cm of the mean height of all adult Poles.
b) How many observations are required to estimate the mean height of all
adult Poles to within 2cm with a probability of 0.9.