Nov 27, 2016 - 11/27/16. 2. Central Limit Theorem. â. The Central Limit Theorem (CLT) states that the sampling distribution model of the sample mean...
Sampling Distribution Model-Different random Samples give different values for a statistic. The sampling distribution model shows the behavior of the statistic over all the possible samples. Sampling Distribution Model for a Proportion-If assumptions of independence and random sampling are met, and we expect at least 10 successes and failures, then the sampling distribution of a proportion is modeled by a Normal model with a mean equal to the true proportion value, p, and a standard deviation equal to or
SD( p!) =
p(1 - p) n
SD( p!) =
pq n
Sampling Distributions l
Sampling Distribution Model for a mean- If assumptions of independence and random sampling are met and the sample size is large enough, the sampling distribution of the sample mean is modeled by a Normal model µ with a mean equal to the population mean, , and a standard deviation equal to
SD( y ) =
s n
1
11/27/16
Central Limit Theorem l
l
The Central Limit Theorem (CLT) states that the sampling distribution model of the sample mean (and proportion) is approximately Normal for large n, regardless of the distribution of the population, as long as the observations are independent.
Standard Error l
l
l
When we estimate the standard deviation of a sampling distribution using statistics found from the data, the estimate is called a standard error. For a proportion, we use and instead of pˆ qˆ p and q because these are our sample proportions. For a mean we use s instead of because s s is our sample standard deviation.
Conditions Sample proportion 10% condition
Success/Failure
Sample mean
Independence
Random Sampling
10% Condition
2
11/27/16
Examples Draw some Normal model for sample proportion based on the 68-95-99.7 rule.
l
• •
80% of all cars on the interstate exceed the speed limit. What portion of speeders might we see among the next 50 cars? We didn’t know the results yet but 31% of EHS voted for Donald Trump. We polled a random sample of 217 voters. What might the percentage of Trump supporters appear to be in our poll?
Examples Use the sample model to calculate some probabilities
l
• •
“Groovy” M& Ms are supposed to make up 30% of the candies sold. In a large bag of 250 M&Ms what is the probability that we get at least 25% groovy candies? What’s the probability that our survey predicts Trump ahead with 34% of the school vote?
Examples l
• • •
As you did with the sampling model for proportions, use the CLT together with the 68-95-99.7 Rule and Normal percentiles to examine what the mean of samples drawn from various populations might look like. SAT scores should have mean 500 and standard deviation 100. What about the mean of random samples of 20 students? Speeds of cars on a highway have a mean 52 mph and standard deviation 6mph, and are likely to skewed to the right. Describe what we might see in random sample of 50 cars? At birth babies average 7.8 pounds with standard deviation 2.1 pounds. A random sample of 34 babies born to mothers living near a large factory that may be polluting the air and water shows a mean birth weight of only 7.2 pounds. Is that usually low?
