Search & Filtering
1) The maternity ward at a certain hospital in a certain country in Africa is one of the busiest in the world with an average of 60 births per day. Let X = the number of births in an hour.
a. Find the mean and standard deviation of X.
b. What is the probability that the maternity ward will deliver three babies in one hour?
c. What is the probability that the maternity ward will deliver at most three babies in one hour?
d. What is the probability that the maternity ward will deliver more than five babies in one hour?
1) Plastic is now surpassing aluminum as the packaging material of choice for soft drinks. According to a recent article, 80% of all new vending business is going to plastic. If a random sample of 10 new vending businesses is selected, what are P(X>2), P(X<5), and P(X<9)?
a) Suppose that 𝑋1,𝑋2,...,𝑋𝑛 is a random sample from a distribution with
probability density function:
𝑓(𝑥;𝜆)={
1
6𝜆4𝑥3𝑒−𝑥
𝜆, 𝑖𝑓 𝑥 ≥0 𝑎𝑛𝑑 𝜆>0
0, 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒
i) Show whether or not 𝑓(𝑥;𝜆) belongs to the 1-parameter exponential family.
(5)
ii) Derive the Maximum likelihood estimator of 𝜆. (8)
iii) Suppose that 𝑛=200, ∑ 𝑥𝑖200𝑖=1 =20,∑ 𝑥𝑖2200𝑖=1 =100, and ∑ 𝑥𝑖3200𝑖=1 =250.
Determine the Maximum likelihood estimate of 𝜆. (2)
1) 15 policyholders over 50 are selected at random; what is the probability that at least 10 will submit a claim during the year? What is the probability that 4 will submit a claim during the year? How many do you expect to submit a claim. What is the standard deviation?
Let 𝑋1,𝑋2,...,𝑋𝑛 be a random sample from a normal distribution with known
mean 𝜇. Using a distribution of 𝑈=∑ (𝑋𝑖−𝜇)2
𝜎2
𝑛𝑖=1 , Mr Mbalula showed that a
100(1−𝛼)% two sided confidence interval for 𝜎2 is give by
[∑ (𝑋𝑖 −𝜇)2𝑛𝑖=1
𝜒𝛼
2,𝑛
2 ;∑ (𝑋𝑖 −𝜇)2𝑛𝑖=1
𝜒1−𝛼
2,𝑛
2 ].
Prove whether or not Mr Mbalula is correct.
The number of fishing rods selling each day is given below. Perform analyses of the time series to determine which model should be used for forecasting.
a. 3 day moving average analysis
b. 4 day moving average analysis
c. 3 day weighted moving average analysis with weights w1=0.2, w2=0.3 and w3=0.5 with w1 on the oldest data
d. exponential smoothing analysis with a = 0.3.
e. Which model provides a better fit of the data?
f. Forecast day 13 sales of fishing rods using the model chosen in part (e).
Day
Rods sold
1 60
2 70
3 110
4 80
5 70
6 85
7 115
8 105
9 65
10 75
11 95
12 85
A textile producer has established that a spinning machine stops randomly due to thread breakages at an average rate of 5 stoppages per hour.
a) What is a probability that in a given hour, 3 stoppages will occur on this spinning machine
b) What is the probability that no more than 1 stoppage will occur in a given half-hour interval?
A population consists of three numbers (3, 6 ,9). Consider all possible samples of sizes 2 which can be drawn without replacement from the population. Find the following
Determine the area below the following
1. Z= 2
2. Z= 3.1
3. Z= -1.5
4. Z= 2.14
