When testing hypothesis there might be a situation where we reject but also won't accept the alternative hypothesis because the statistic calculated from the sample fell into a region that separates the hypothesis.

When is true, but we retain , then we are making a Type II error.

When the statistic, that is calculated from our sample, falls into the critical region, then we retain .

When testing hypothesis we retain , there is still a possibility that actually is true, but we are not able to show this with the given data;

The Law of Large Numbers and Central Limit Theorem both try to approximately tell us the behaviour of the sample mean .

Central Limit Theorem only holds if each is normally distributed (i.e. ).

The Central Limit Theorem gives us the approximate shape of the distribution of .

Central Limit Theorem is not restricted by the underlying distribution of -s.

If the sample size is small (e.g. ), then the Law of Large Numbers on does not hold, but the Central Limit Theorem on is still valid (i.e. if ).

, where is a constant.

Tell her to set it to 10 minutes

Tell her that with exponential distribution she cannot find such a number because exponential distribution is a discrete distribution.

Tell her that with exponential distribution she cannot find such a number for time slot length because exponential distribution has a uncountable set of possible values (infinitely many possible values).

Tell her to set it at 30 minutes

Determine the mean and standard deviation of . To get the time slot length needed, add standard deviation to the mean. Voilà!

Advise her to find the 95%-quantile using the exponential distribution.

Random variable that can take any value in some part of the real line (in an uncountable set)

is a normally distributed random variable

A continuous random variable can be transformed into a discrete random variable

If you know the probability density function (PDF) of a continuous random variable, then you can find the cumulative distibution function (CDF) by taking a derivative of the PDF.

Random variable with a finite number of possible values always has a finite expected value (i.e. )

Random variable with infinite countable set of possible values always has a finite expected value (i.e. )

Probability that among 5 randomly chosen people no-one is left-handed, is less than 0.5

Probability that among 3 randomly chosen people there is exactly 1 who is left-handed, is 0.243.

Probability that among 5 randomly chosen people at least one is left-handed, is less than 0.5

The number of left-handed people in a group of 20, is approximately normally distributed

Eye color of a randomly selected person

Body weight (kg)

Number of caries-affected teeth (of a dentist's patient)

Cups of coffee consumed yesterday

Net monthly salary (in EUR)