CH 4, 5 & 6: Normal/Sampling Distributions, Confidence Intervals

Random variable

a numerical value assigned to an event in the sample space

Expected value

Mean, average or expected value for discrete random variable, Y

Mx+y

Mx + My

Mx-y

Mx- My

If a & b are constants: M(a+bY)

a+ bµY

sigma(a+by)

b²sigma²y

IF x+y are independent: sigma²x+y

sigma²x + sigma²y

sigma²x-y

sigma²x + sigma²y

Binomial Random Variable Characteristics

1) There are only 2 possible outcomes for each trial: “success” and “failure”

2) There are N trials

3) Trials are independent

4) Prob. of success= P is constant

5) Interested in # of successes

Density Curves

Smooth histograms for continuous data

• Probability= area under the curve

• Total area under the curve= 1

• Most widely used continuous variable

• Center of distribution is the mean (mean= median= mode)

• Symmetric distribution

Normal Random Variable

Z score

represents the number of standard deviations above or below the mean

Why can’t we assume that P(Y=a) or P(Z=a) is an integer above zero?

We cannot find area at singular points underneath the curve, they always must be from a range

Ways to assess Normality

• Calculate the mean of Y ± 1sd, 2sd, 3sd, see where it approximates as a percentage (should be around 68%, 95%, 99.7%)

• Q-Q plot: theoretical data on x axis, actual data on y-axis→ closer the points are to straight line of theoretical data= more confident we are about normal assumption

Sample mean

• way to estimate the population mean, Y

• Random variable with some distribution

• Ybar= random variable denoting ALL possible values of sample mean, ybar= observed value based on sample

• Ybar= random, ybar= fixed→ sampling distribution of the sample mean

The spread of the means is always…

less than the spread of Y

Central Limit Theorem

• if sample of size n≥30, is taken, then Ybar= normal distribution regardless of distribution of Y.

• as long as sample size= large enough, the averages will distribute normally

SEybar vs sigma ybar

• sigma y bar= variability of the data from the mean

• as n→ ∞, ybar converges to Mu, s onverges to sigma, and sy/sqrtn converges to 0. This means there will be no difference between the mean and sd.

• IF we have the entire population, our error is 0 since ybar= mu, and s= sigma

t-distribution

the standardized distances of sample means to the population mean when the population sd= unknown, observations come from a normally distributed population.

Properties of t-distribution

• same general shape of curve; symmetric but spread wider

• spread relies on n; smaller n= more spread in distribution

• t⍺= t-value w ⍺ in the upper tail @ df= n-1

• since t is always wider than z, t⍺>z⍺