Test #3 Review Sheet MATH 2600

Topics: The Uniform Distribution, The Normal Distribution, CLT, Confidence Intervals for the population mean.

Uniform Distribution X~ Uniform(c,d) ( the random variable X is uniformly distributed over the interval [c,d] )

Be able to provide the formula for the probability density function for a uniform random variable

Be able to find probabilities using a uniform distribution

P(X  a), P(X  b), P(a  X  b), etc.

Be able to expressthe mean and standard deviation of a uniform random variable in terms of c and d

Be able to find the expected value of a uniform random variable

Normal DistributionX ~ N(,) (the random variable X is normally distributed with mean  and standard deviation )

Be able to find probabilities using a normal distribution

P(X  a), P(X  b), P(a  X  b), etc.

TI-84+ support: normalcdf(low,high,, ) = P(low  X  high)

Be able to find boundaries for certain conditions

Find X-value(s) that cuts off upper x% (lower x %)

that serveas a boundary between the lower x% and upper (100-x)% (0<x<100)

that serve as boundaries for the middle x%

TI-84+ support: x0 = invNorm (Area, , )

where Area is the area under the normal curve to the left of x0.

Central Limit Theorem (CLT)

Understand the statement of the Central Limit Theorem and how to apply this theorem

Confidence Intervals for the Population Mean

  • Understand the relationship between the confidence level and the confidence interval width.
  • Be able to find the minimum sample size for a given confidence level and tolerance(desired proximity to P)
  • Be able to find a confidence interval for a population mean using a large sample or a small sample

Summary of confidence intervals for the population mean

Let P be a random variable for a population with mean P and std. dev

<TypicallyP is unknown and we want to find an interval estimator called a confidence interval

Let S be a random (representative) sample of size with sample mean and sample standard deviations

Assume S is selected (with replacement) from the values of P

TI-84+ Calculator support: Zinterval and TInterval as appropriate (found on the TESTS submenu of STAT menu)

For any population P,

If n  30 and is known, then a 100(1-)% confidence interval for P is

(– E, + E) where

If n  30 and is unknown, then use s to approximate, and a 100(1-)% confidence interval for P is

(– E, + E) where

If n  30 and P is approximately (or exactly) normally distributed

and is known,then a 100(1-)% confidence interval for P is

(– E, + E) where

Ifn  30 and P is approximately (or exactly) normally distributed

and is unknown, then use s to approximate, and the100(1-)% confidence interval for P is

(– E, + E) where

If n  30 and P is not approximately normally distributed, other methods must be applied, but we're STUCK!