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Null Hypothesis Testing statistics problems
Null Hypothesis Test for MEAN example question
A literary critic wants to establish that the mean number of words per sentence, appearing in newly discovered short story is different from 9.1 words. A sample of 36 sentences provided a data with mean of 8.6 words and standard deviation of 1.2 words. Test at a level of significance α = 0.10.
Solution to this Hypothesis Testing Statistics practice problem is given in the video below!
Null Hypothesis Test for PROPORTION example problem
Out of a sample of n = 625 students interviewed, 139 missed at least one class last week. Let p = proportion of students that missed at least one class last week. Conduct a test with the intent of establishing that the proportion of students that missed at least one class last week is greater than 0.20. Test at a level of significance α = 0.05.
Solution to this Hypothesis Testing Statistics practice problem is given in the video below!
Variance problems
Married Couples Expected Value & Variance example question
If 10 married couples are randomly seated at a round table, compute
(a) the Expected number
(b) the Variance
of the number of wives that are seated next to their husbands.
Solution to this Expected Value & Variance practice problem is given in the video below!
Confidence Interval statistics problems
Confidence Interval for MEAN Normal Distribution example question
A manager at a power company monitored the employee time required to process high-efficiency lamp bulb rebates. A random sample of 40 applications gave a sample mean time of 3.8 minutes and a standard deviation of 1.2 minutes. Construct a 90% confidence interval for the mean time to process μ.
Solution to this Confidence Interval Statistics practice problem is given in the video below!
T-Distribution Confidence Interval for MEAN example
From a random sample of size 16, one has calculated the 95% confidence interval for μ (population mean) and obtained the result (39.51, 42.49). What are the x̄ (sample mean) and s (sample standard deviation) for the sample? Assume the population is normal.
Solution to this Confidence Interval Statistics practice problem is given in the video below!
Confidence Interval for PROPORTION Normal Distribution example problem
A telephone survey of 1,000 was taken after movie X came out in the national theater.
a) If 832 liked the movie, with what confidence can it be asserted that 83.2% ± 3% of the adult population liked this movie?
b) If the adult population in the nation is 150 million, what is the interval estimating the total number of people in the nation who liked the movie?
Solution to this Confidence Interval Statistics practice problem is given in the video below!
Expected Value problems
Expected Value example question
A box contains 8 green and 4 blue marbles. Two marbles are selected at once without replacement. What is the expected number of green marbles among the selected ones?
a) 1
b) 
c) 
d) 
e) 
f) none of the above
Solution to this Expected Value practice problem is given in the video below!
Expected Value example problem #2
You have 5 pairs of shoes. Four of them are worth $30 each, while the fifth is worth $2,000. You select a pair at random. What is the expected value of the pair you have selected?
a) $424
b) $30
c) $50.20
d) $432
e) $1,030
f) none of the above
Solution to this Expected Value practice problem is given in the video below!
Binomial Distribution Expectation example
A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.
Solution to this Binomial Random Variable Expected Value practice problem is given in the video below!
Hypergeometric Distribution Expectation example question
A ball is chosen at random from each of 5 urns. Each urn contains balls as follows:
urn 1: 1 white, 5 black
urn 2: 3 white, 3 black
urn 3: 6 white, 4 black
urn 4: 2 white, 6 black
urn 5: 3 white, 7 black
Compute the expected number of white balls selected.
Solution to this Hypergeometric Random Variable Expected Value practice problem is given in the video below!
Continuous Distribution Expected Value example problem
The density function of X is given by
f(x) = { 
0 otherwise
If E[X] = 
, find the values of constants 
and 
.
Solution to this Continuous Random Variable Expected Value practice problem is given in the video below!
Independent & Identically Distributed UNIFORM Random Variables Expected Value example
If X1, X2, …, Xn are independent and identically distributed random variables having uniform distributions over (0,1),
find
E [max(X1, …, Xn)]
and
E [min(X1, …,Xn)]
Solution to this Uniform Random Variable Expected Value practice problem is given in the video below!
SOA Exam P CAS Exam 1 Expected Value Variance Standard Deviation probability problems
Multiple Moment Generating Functions MGF example question
A company insures homes in three cities, J, K and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are
Let X represent the combined losses from the three cities. Calculate 
.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Expected Value & Probability Density Function PDF example problem
Let X be a continuous random variable with density function
{ 
Calculate the expected value of X.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Moment Generating Function MGF example
Let X1 , X2 , X3 be a random sample from a discrete distribution with probability function
{ 
Determine the moment generating function, M(t), of 
.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Expected Value & Probability Density Function PDF word problem
An insurance company’s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to
Determine the company’s expected monthly claims.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Percentiles & Probability Density Function PDF example question
An insurer’s annual weather related loss, X, is a random variable with density function
{ 
Calculate the difference between the 30th and 70th percentiles of X.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Standard Deviation & Moment Generating Function MGF example problem
An actuary determines that the claim size for a certain class of accidents is a random variable, X, with a moment generating function
Determine the standard deviation of the claim size for this class of accidents.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Standard Deviation Table example
A probability distribution of the claim sizes for an auto insurance policy is given in the table below:
| Claim Size | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
| Probability | .15 | .10 | .06 | .20 | .10 | .10 | .30 |
What percentage of the claims are within one standard deviation of the mean claim size?
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Variance & Cumulative Distribution Function CDF example question
A random variable has the cumulative distribution function
{ 
Calculate the variance of X.
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
Variance word problem
A recent study indicates that the annual cost of maintaining and repairing a car in a town to Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive), what will be the variance of the annual cost of maintaining and repairing a car?
Solution to this Society of Actuaries Exam P practice problem is given in the video below!
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