Name ______
AP STATISTICS CHAPTER 9:
What is the difference between a parameter and a statistic?
What does the symbol represent?
Under what conditions is a statistic said to be
UNBIASED?
The SAMPLING DISTRIBUTION of a statistic is the taken by the statistic in all possible samples .
What determines the spread (variability) of the sampling distribution?
The methods we use for evaluating the variability of a sampling distribution in this course will apply if….
Recall that =
THE SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION
Choose an SRS of size n from a population with population proportion p having some characteristic ofinterest. Let be the proportion of the sample having that characteristic. Then:
1)The mean of the sampling distribution is
2)The standard deviation of the sampling distribution is given by
RULES OF THUMB (Conditions that need to be verified before we can proceed):
1)Use this formula for standard deviation when
2)Use the normal approximation when
Example: A polling organization asks an SRS of 1500 first year college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of the true value?
- STATE: What is our goal? Re-state the problem using mathematical symbols.
- PLAN: Summarize salient facts and conditions.
- DO: Perform necessary calculations.
- CONCLUDE: Answer the question, in context.
Example 2: Suppose one student tossed a coin 200 times and found only 42% heads. Do you believe that this is likely to happen?
Example 3: Assume that 30% of the students at HH wear contacts. In a sample of 100 students, what is the probability that more than 35% of them wear contacts?
Example 4: About 11% of American adults are black. Therefore, the proportion of blacks in an SRS of 1500 adults should be close to .11. If a national sample contains only 9.2% black, should we suspect that the sampling procedure is somehow under-representing blacks?
Sample Means:
How do you know if a statistic is unbiased?
General Properties of Sample Means:
Suppose that is the mean of an SRS of size n drawn from a large population with mean and standard deviation .
Rule 1:
Rule 2:
When are we allowed to use Rule #2?
Rule 3:
EX) The army reports that the
distribution of head circumference
among soldiers is approximately
normal with mean 22.8 inches and standard deviation of 1.1 inches.
a) What is the probability that a randomly selected soldier’s head will have a circumference that is greater than 23.5 inches?
b) What is the probability that a random sample of five soldiers will have an average head circumference that is greater than 23.5 inches?
Example 2:
Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let x represent the length of a single trout taken at random from the pond. This group of biologists has determined that the length has a normal distribution with mean of 10.2 inches and standard deviation of 1.4 inches.
What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?
What is the probability that the mean length of five trout taken at random is between 8 and 12 inches long?
What sample mean would be at the 95th percentile? (Assume n = 5)
Example 3
A soft-drink bottler claims that, on average, cans contain 12 oz of soda. Let x denote the actual volume of soda in a randomly selected can. Suppose that x is normally distributed with s = .16 oz. Sixteen cans are randomly selected and a mean of 12.1 oz is calculated. What is the probability that the mean of 16 cans will exceed 12.1 oz?
IF THE POPULATION HAS ANY DISTRIBUTION, WE MAY EMPLOY THE .
When n is , the sample mean has the distribution.
What are the conditions for using the CLT?
Ex: A hot dog manufacturer asserts that one of its brands has an average fat content of 18 grams per hot dog. The standard deviation is 1. In a sample of 36 hot dogs, what is the probability that the mean fat content will be at least 18.4 grams?
CHAPTER 9 – SUMMARY
SAMPLE PROPORTIONS
Standard deviation – use when the population is at least 10 times as large as the sample.
Use the normal approximation when and
Then
SAMPLE MEANS
Standard deviation - use when the population is at least 10 times as large as the sample.
If the population is normally distributed, then
Central limit theorem: Given any distribution, if n is large (), then
SUMMARY/QUESTIONS TO ASK IN CLASS