Download 1 AP STATISTICS NOTES ON CHAPTER 10 DEFINITION: Statistical

AP STATISTICS NOTES ON CHAPTER 10
DEFINITION: Statistical Inference provides methods for drawing conclusions about a
population from sample data.
Notes:
1. Statistical Inference uses probability to express the strength of our conclusions.
2. Probability allows us to take chance variation into account and so to correct our judgment
by calculation. This protects us from jumping to conclusions.
TWO MOST FORMAL TYPES OF STATISTICAL INFERENCES:
1. Confidence Intervals – used for estimating the value of a population parameter.
2. Tests of Significance – used to assess the evidence for a claim about a population.
Notes:
1. Both types of inferences are based on the sampling distributions of statistics; that is, both
report probabilities that state what would happen if we used the inference method many
times. (long-run regular behavior that probability describes)
2. Inference is most reliable when the data are produced by a properly randomized design.
(Otherwise, conclusions may be challenged.)
Read and discuss page 537-539 SAT Math Scores in California
CONFIDENCE INTERVALS:
A level C confidence interval has two parts:
1. An interval calculated from the data, of the form estimate (plus or minus) margin of
error.
2. A confidence level C, which gives the probability that the interval will capture the true
parameter value in repeated samples.
Notes:
1. A 90% or higher confidence level is usually used.
2. C will stand for the confidence level in decimal form.
Discuss Figure 10.4 on page 541
CONDITIONS FOR CONSTRUCTING A CONFIDENCE INTERVAL FOR µ:
The construction of a confidence interval for a population mean µ is appropriate when:
1. the data comes from an SRS from the population of interest.
2. the sampling distribution of x is approximately normal.
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Notes:
1. To construct a level C confidence level interval, we want to catch the central probability
under a normal curve. To do this, go out z* standard deviations on either side of the
mean. You can then get the value z * from the standard normal table.
2. Values of z * that mark off a specified area under the standard normal curve are often
called critical values of the distribution.
DEFINITION: The number z * with probability p lying to its right under the standard normal
curve is called the upper p critical value of the standard normal distribution.
Note: The estimate of the unknown µ is x, and the margin of error is z * times σ /
n.
CONFIDENCE INTERVAL FOR A POPULATION MEAN:
Choose an SRS of size n from a population having unknown mean µ and known standard
deviation σ. A level C confidence interval for µ is x ± z ∗
σ
n
.
Notes:
1. z * is the value with area C between - z * and z * under the standard normal curve.
2. This interval is exact when the population distribution is normal and is approximately
correct for large n in other cases.
STEPS TO FOLLOW:
1. Identify the population of interest and the parameter you want to draw conclusions about.
2. Choose the appropriate inference procedure. Verify the conditions for using the selected
procedure.
3. If the conditions are met, carry out the inference procedure. (CI = estimate ± margin of
error.
4. Interpret your results in the context of the problem.
Read and work out problem 10.7 on. Page 549.
BEHAVIOR OF THE CONFIDENCE INTERVAL:
Goal: High confidence that says our method almost always gives correct data and small margin
of error that says we have pinned down the parameter quite precisely.
Margin of Error Gets Smaller When:
• z* gets smaller
• σ gets smaller
• n gets larger
Read and work out problem 10.8 on Page 550.
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CHOOSING THE SAMPLE SIZE: To obtain a desired margin of error m:
m = z∗
σ
n
1. Substitute the value of z* for your desired confidence level.
2. Set the expression for m less than or equal to the specified margin of error.
3. Solve the inequality for n.
Notes:
1. It is the size of the sample that determines the margin of error.
2. When n ≥ 15, the confidence level is not greatly disturbed by nonnormal populations
unless extreme outliers or quite strong skewness are present.
3. The margin of error in a confidence interval covers only random sampling errors.
4. The margin of error indicates how much error can be expected because of chance
variation in randomized data production.
TESTS OF SIGNIFICANCE:
Note:
1. An outcome that would rarely happen if a claim were true is good evidence that the claim
is not true.
2. Works by asking how unlikely the observed outcome would be if the null hypothesis
were true.
Read and discuss Example 10.9 on page 560.
BEGINNING STEPS FOR A TEST OF SIGNIFICANCE:
1. Always draw conclusions about some parameter of the population.
2. State the null hypothesis. The null hypothesis says that there is no effect or no change in
the population. If the null hypothesis is true, the sample result is just chance at work.
Notation: H 0: µ =0
3. The alternative hypothesis to “no effect” or “no change” Notation: H 0: µ > 0
Go over reasoning of a significance test on page 561.
TO MEASURE THE STRENGTH OF EVIDENCE AGAINST THE NULL
HYPOTHESIS, LOOK AT THE PROBABILITY UNDER THE NORMAL CURVE TO
THE RIGHT OF THE OBSERVED SAMPLE MEAN. THIS PROBABILITY IS CALLED
THE P-value.
Notes:
1. Small P-values are evidence against the null hypothesis because they say that the
observed result is unlikely to occur just by chance.
2. Large P-values fail to give evidence against the null hypothesis.
3. The level 0.05 is a common rule of thumb.
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4. A result with a P-value less than 0.05 is called statistically significant.
Discuss the reasoning of a significance test (page 563)
Notes:
1. The first step of significance is to state a claim that we will try to find evidence against.
This claim is the null hypothesis.
2. The alternative hypothesis H0 is the claim about the population that we are trying to find
evidence for.
3. Alternative hypothesis are one-sided when we are interested only in deviations from the
null hypothesis in one direction.
4. When the direction is not specified then the alternative hypothesis is two-sided.
5. Hypotheses refer to populations, not to particular outcomes. Therefore, state them in
terms of population parameters.
P-VALUES AND STATISTICAL SIGNIFICANCE:
1. P-value – the probability, computed assuming that H0 is true, that the observed outcome
would take a value as extreme or more extreme than that actually observed is called the
P-value of the test. The smaller the P-value is, the stronger is the evidence against H0
provided by the data.
2. Significance Level (α) – a value, which we regard as decisive, with which we compare
the P-value. (Essentially what we are doing is announcing in advance how much evidence
against H0 we will insist on.)
3. If the P-value is as small or smaller than alpha (α), then the data is statistically significant
at level α.
Discuss Inference Toolbox for Significance Tests (page 571)
DIFFERENT TYPES OF TESTS
1. z-test for a population mean
2. Tests with fixed significance level.
3. Tests from confidence intervals.
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