Name ______
AP STATISTICS CHAPTER 10: INTRODUCTION TO INFERENCE
Confidence Intervals:
A Confidence interval for a population characteristic is an ______of ______for the characteristic. It is constructed so that, with a chosen ______of ______, the value of the characteristic will be ______inside the ______.
Margin of error:
The confidence level associated with a confidence interval estimate is the ______of the ______used to construct the ______.
Critical Value:
SOME COMMON CONFIDENCE LEVELS AND THEIR z* VALUES
1. Confidence level 90%
2. Confidence level 95%
3. Confidence level 99%
Other z* values can be found
Formula for Confidence Interval:
TO CONSTRUCT A CONFIDENCE INTERVAL.
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Ex1:. A random sample of the SAT math scores of 500 California high school students yields a mean of 461. We know that the standard deviation for all seniors is 100. Construct a 95% confidence interval for the mean of all California students. Comment on the meaning of this interval.
Interpretations (2 different ways )
What are 3 things you should NOT say?
Ex2: A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with s = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level
3 ways to make Margin of error smaller:
Ex3: A random sample of 50 HH students was taken and their mean SAT score was 1250. (Assume s = 105) What is a 95% confidence interval for the mean SAT scores of HH students?
Ex4: Suppose that we have this random sample of SAT scores:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for the true mean SAT score? (Assume s = 105)
Ex5: The heights of HH male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?
What are Hypothesis Tests?
What are the 4 steps in a Hypothesis Test?
The statement being tested in a test of significance is called the
NOTATION:
The statement that we suspect is true is called the .
NOTATION:
Depending on the context of the problem, the null and alternate hypotheses will be written in one of three ways:
1.2.3.
Example 1: A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces).
Example 2: A car dealer advertises that is new subcompact models get 47 mpg. You suspect the mileage might be overrated.
Example 3: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses :
What is a p-value?
What does it mean to be statistically significant?
What are the 2 steps to writing conclusions?
Memorize this phrase!!
Formula for the test-statistic:
Example 4: The Fritzi Cheese Company buys milk from several suppliers as the essential raw material for its cheese. Fritzi suspects that some producers are adding water to their milk to increase their profits. Excess water can be detected by determining the freezing point of milk. The freezing temperature of natural milk varies normally, with a mean of -0.545 degrees and a standard deviation of 0.008. Added water raises the freezing temperature toward 0 degrees, the freezing point of water (in Celsius). The laboratory manager measures the freezing temperature of five randomly selected lots of milk from one producer with a mean of -0.538 degrees. Is there sufficient evidence to suggest that this producer is adding water to his milk?
Example 5: K-Mart brand light bulbs state on the package “Average Life 1000 Hr.”. A class decided to test this claim, and randomly selected a sample of 60 bulbs to test. For these 60 bulbs, the average bulb life was 970 hours. Is this evidence enough to dispute the claim K-Mart places on its packages? In other words, can we say that our results here are statistically significant? Assume that we know that the population standard deviation for K-Mart light bulbs is 120 hours.
Ex: Because of variation in the manufacturing process, tennis balls produced by a particular machine do not have identical diameters. Suppose that the machine was initially calibrated to achieve the design specification in with in. However, we now have reason to suspect the machine parts are now producing tennis balls that are too large. A random sample of 90 balls yields in. Does this provide evidence that the machine is producing tennis balls with too large a diameter? What if a new sample yielded in?
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For the problem above, let (alpha) be 0.05
If the is as small or smaller than , then we say that the data are
When we perform a hypothesis test, we make a decision to either
Reject the Null Hypothesis or
Fail to Reject the Null Hypothesis.
There is always the possibility that we made an incorrect decision.
We can make an incorrect decision in two ways:
If we when in fact , this is a .
If we when in fact , this is a .
VISUAL ORGANIZER OF ERRORS
Ex: Let , and . Describe both types of error and their consequences.
Ex:Let , and . Describe both types of error and their consequences.
THE PROBABILITY OF TYPE I AND TYPE II ERRORS
A new car had been released for which the manufacturer reports that the car gets 23mpg for city driving. A consumer group feels that the true mileage is lower and performs the following hypothesis test, using α = .05.
Assumptions: The gas mileage in the sample of 30 cars comes from an SRS drawn from the population of all mileage and the standard deviation is known, δ = 1.2.
Null and alternate hypothesis:
Decision matrix:
What is a Type I error in the context of this problem?
What is a Type II error in the context of this problem?
The Probability of Type I error is , !
The Probability of Type II error is , and is dependent upon the following
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2.
3.
Ex: the mean salt content of a certain brand of potato chip is supposed to be 2 mg. The salt contents vary normally with standard deviation .1 mg. For each batch produced, an inspector takes a sample of 5 chips and measures the salt content. The entire batch is rejected if the mean salt content of the sample is significantly higher than 2mg, at the 5% significance level. The company has decided that having a mean salt content that is .05mg higher than believed is entirely unacceptable.
Name the null and alternate hypothesis. Explain the meaning of a type I error in this problem.
Find the probability of a type I error.
Rule:
The is the probability of
Ways to increase .
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3.
SUMMARY/QUESTIONS TO ASK IN CLASS