Sampling Distribution of Sample Means Open Minitab Stat 212 – Day 21 Sampling Distribution of Sample Means Open Minitab

Last Time If n is large, sampling distribution of sample proportions will be approximately normal with mean p and standard deviation Test of Significance (H0: p=p0) Use normal distribution to approximate p-value Advantage: Also reports a test statistic Confidence interval for p Use normal distribution to approximate interval Margin of error = half-width of interval (z*SE) Some approximation methods are better than others

Interpretation of “confidence” p. 4-41: I have calculated a 95% confidence interval for p, so there is a .95 probability that p is in the interval

Methods Simulation (empirical p-value) If finite population, use hypergeometric distribution If process or large population, use binomial distribution Technical condition: population size > 20n If large sample size, use normal distribution Technical condition: np and n(1-p) > 10 Most flexible Most common in practice right now n > 10 and n(1- )>10 np0> 10 and n(1-p0)>10

Example: Bad News First p=proportion of all St. Olaf students that prefer bad news first H0: p=.5 (no preference in population) Ha: p>.5 (preference for bad news first) z = 3.0, p-value = .001, reject H0, significant pref Exact p-value = .002 Level of significance 95% z interval for p: (.643, .957) 95% wilson interval: (.603, .914) 95% binomial interval: (.593, .932)

Last Time – Uncertainty in Statistics Always the possibility are committing an error Type I Error = rejecting a true null hypothesis P(Type I Error) is controlled by level of significance Type II Error = failing to reject a false null hypothesis Power = probability of rejecting a false null hypothesis Determine rejection region, how often will sample be in that rejection region if alternative is true reject H0 reject H0

Power (Act 3-10)

Power (Act. 4-7, p. 4-32)

Power

Power

Power

Do the same thing for sample means! Open milititamen.mtw and examine the distribution of chest measurements Obs units? Variable? Type? Describe distribution Take random sample Did everyone get sample mean?

Sampling distribution of sample mean Run the macro militiamen.mtb Describe the distribution. How compare to population?

Sampling distribution of sample mean Open agemom.mtw Describe the population Re-initialize k1 and run macro again

Central Limit Theorem for Sample Mean Sampling distribution of sample mean will be Centered at m Standard deviation s/ Normally distribution if population is normal or approximately normally distribution is sample size is large Guideline: n>30

Confidence interval for m

Test of significant about m H0: m=m0 Ha: m m0 < > or ≠ Observe “xbar” Test statistic: (xbar – hypothesized mean) s/ Compare to t distribution with n-1 df

Activity 4-12

For Monday Practice problem 2,3, p. 4-52 Activity 4-12 p. 4-53 (a)-(i) www.rossmanchance.com/iscat/applets/ Use Internet Explorer Uniform distribution Activity 4-12 p. 4-53 (a)-(i) Review p. 4-56 Start reading Ch. 5