REVIEW Central Limit Theorem Central Limit Theoremand The t Distribution

Random Variables A random variable is a quantitative “experiment” whose outcome is not known in advance. All random variables have three things: –A distribution –A mean –A standard deviation

_ The Random Variables X and X _ The Random Variables X and X X = a random variable designating the outcome of a single event µσ Mean of X = µ; Standard deviation of X = σ _ X = a random variable designating the average outcome of n measurements of the event µσ/√n _ _ ‗ Mean of X = µ; Standard deviation of X = σ/√n THIS IS ALWAYS TRUE AS LONG AS σ IS KNOWN!

Example Attendance at a basketball game averages with a standard deviation of a X = Attendance at a game 20000= 4000 µ = 20000; σ = 4000 Averagen _ X = Average attendance at n games _ Mean of X = /√n ‗ ‗ Standard deviation of X = 4000/√n

The Central Limit Theorem σ, is knownAssume that the standard deviation of the random variable X, σ, is known. Two cases for the distribution of X Xis normal. THEN X (Attendance at a game) is normal. THEN _ X normal distribution for any sample size, n _ X (Average attendance at n games) has a normal distribution for any sample size, n Xis not normal. THEN X (Attendance at a game) is not normal. THEN _ Xunknown distribution _ X (Average attendance at n games) has an unknown distribution. But the larger the value of n, the closer it is approximated by a normal distribution. 1 2 Central Limit Theorem

What is a Large Enough Sample Size? n = 30 _ To determine whether or not X can be approximated by a normal distribution, typically n = 30 is used as a breakpoint. a –In most cases, smaller values of n will provide satisfactory results, particularly if the random variable X (attendance at a game) has a distribution that is somewhat close to a normal distribution.

Examples Attendance at a basketball game averages with a standard deviation of Assuming that attendance at a game follows a normal distribution, what is the probability that: 1.Attendance at a game exceeds 21000? 2.Average attendance at 16 games exceeds 21000? 3.Average attendance at 64 games exceeds 21000? Repeat the above when you cannot assume attendance follows a normal distribution.

Answers Assuming Attendance Has a Normal Distriubtion If X, attendance, has a normal distribution since σ is known to = 4000, THEN _ Average attendance, X, is normal with: ‗ Standard deviation of X = __ 4000/√16 = /√16 = 1000 in case 2 __ and __ 4000 /√64 = /√64 = 500 in case 3.

Calculations Case 1: P(X > 21000) Here, z = ( )/4000 =.25 So P(X > 21000) = =.4013 _ Case 2: P(X > 21000) Here, z = ( )/1000 = 1.00 _ So P(X > 21000) = =.1587

Calculations (Continued) _ Case 3: P(X > 21000) Here, z = ( )/500 = 2.00 _ So P(X > 21000) = =.0228

Answers Assuming Attendance Does Not Have a Normal Distriubtion Case 1 : Since X is not normal we cannot evaluate P(X > 21000) _ Case 2: Since X is not normal, n is small, X has an unknown distribution. Thus we cannot evaluate this probability either. Since n is large, case 3 can be evaluated in the same manner as when X was assumed to be normal: _ Case 3: P(X > 21000) Here, z = ( )/500 = 2.00 _ So P(X > 21000) = =.0228

What Happens When σ Is Unknown? -- t Distribution This is the usual case. _ If X has a normal distribution, X will have a t distribution with: n-1 degrees of freedom _ Standard deviation = s/√n But the t distribution is “robust” meaning we can use it even when X is only roughly normal – a common assumption. From the central limit theorem, it can also be used with large sample sizes when σ is unknown.

When to use z and When to use t z and t distributions are used in hypothesis testing and confidence intervals _ These are determined by the distribution of X. USE z Large n or sampling from a normal distribution Large n or sampling from a normal distribution σ is known σ is known USE t Large n or sampling from a normal distribution Large n or sampling from a normal distribution σ is unknown σ is unknown

REVIEW _ The random variables X and X –Mean and standard deviation Central Limit Theorem _ Probabilities For the Random Variable X t Distribution When to use z and When to Use t –Depends only on whether or not σ is known