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“Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.”

+ REFLECTION… In Chapter 7 they GAVE us either the population

mean or the mean of the sampling distribution

If we had the mean of the sampling distribution we could predict the population

In real life, you seldom have this information

So….how do you estimate your population mean?!

Point Estimates vs. Interval Estimates

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We Don’t KNOW the sampling mean (which would be equal to our population mean) So we don’t KNOW where our statistic is RELATIVE to the true population parameter.

+ Making connections… In Chapter 2 we learned about the Empirical rule:

68% of values lie within (+1) σ of the mean, 95% of values lie within (+2) σ of the mean, 99% of values lie within (+3) σ of the mean.

In Chapter 4, we learned if we randomly select the sample, we should be able to generalize our results to the population of interest.

In Chapter 7 we learned that if we take multiple samples each may vary by a certain amount, but if we take all the possible combinations of samples, then, we can construct a sampling distribution.

+ Confidence Intervals: The Confidence Intervals: The BasicsBasics

Definition:

A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our “best guess” at the value of an unknown parameter.

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WHAT IS THE POINT ESTIMATOR?

WHAT IS THE POINT ESTIMATE?

CAN YOU IDENTIFY EACH IN AN

APPLICATION?

LET’S TRY THE FOLLOWING …

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What proportion p of U.S. high school

students smoke? A 2007 Survey

questioned a random sample of 14,041

students in grades 9-12. Of these 2,808

said they had smoked cigarettes a least

one day in the past month

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The math department wants to know what

proportion of students own a graphing

calculator, so they take a random sample

of 100 students and find that 28 own a

graphing calculator.

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The makes of a new golf ball want to estimate

the median distance the new balls will travel

when hit by a mechanical driver. They select a

SRS of 10 balls and measure the distance

travelled after being hit by a mechanical

driver.

The following are the distances in yards:285 286 284 285 282 284 287 290 288 285

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Big Idea We are estimating the mean of a sampling

distribution, and transitively the population parameter, by using a sample’s statistic [mean/ proportion]!

+INTERVAL ESTIMATEINTERVAL ESTIMATE

An interval estimate is defined by two numbers, between which a population parameter is said to lie.

For example, (a, b) is an interval estimate that states the population mean is between a and b.

+ CONFIDENCE INTERVALS: THE CONFIDENCE INTERVALS: THE BASICSBASICS

The confidence level describes the uncertainty associated with a sampling method.

Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample.

Some interval estimates would include the true population parameter and some would not.

A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter.

A 95% confidence level means that 95% of the intervals would include the parameter.

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Example: You give the SAT Test to a SRS of 500 high school seniors

in California. The mean x for the math portion is 461. The standard deviation of the population σ= 100.

What is the sample Standard deviation?

Would the sample mean vary if we took many samples of 500 seniors from the same population?

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*68-95-99.7 Rule says that in 95% of all samples, x of the samples will be within 2σ of the population mean μ.

So the mean x of 500 SAT Math scores will be within 4.5*2 = 9 points of the unknown μ in 95% of all samples.

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−

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• So in 95% of all samples, the unknown μ lies between x-9 and x+9….

• Or x + 9

• Or (x-9, x+9)

−− −

− −

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CONFIDENCE LEVEL: To say that we are 95% confident is shorthand for

“95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.”

CONFIDENCE INTERVAL: To interpret a C% confidence interval for an unknown parameter, say:

“We are C% confident that the interval from _____ to _____ captures the actual value of the [population parameter in context].”

Interpreting Confidence Level and Confidence Intervals

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Confidence Intervals

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Interpreting Confidence Levels and Confidence Intervals

The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times.

Instead, the confidence interval gives us a set of plausible values for the parameter.

We interpret confidence levels and confidence intervals in much the same way whether we are estimating a population mean, proportion, or some other parameter.

The confidence level does not tell us the chance that a particular confidence

interval captures the population parameter.

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CONFIDENCE INTERVAL DATA REQUIREMENTS

To express a confidence interval, you need three pieces of information.

•Confidence level

•A confidence level C gives the probability that the interval will

capture the true parameter value in repeated samples C% of the

time.

•C is usually 90%, 95% or 99% but can be any %.

•Statistic

•x or p

•Margin of error

•Critical value (z score) * Standard deviation of the statistic

− ^

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We usually choose a confidence level of 90% or higher because we want to be quite sure of our conclusions. The most common confidence level is 95%.

+ Calculating a Confidence IntervalC

on

fiden

ce Inte

rvals: Th

e B

asics

The confidence interval for estimating a population parameter has the form

statistic ± (critical value) • (standard deviation of statistic)

where the statistic we use is the point estimator for the parameter.

Calculating a Confidence Interval

Properties of Confidence Intervals: The “margin of error” is the (critical value) • (standard deviation of statistic) The user chooses the confidence level, and the margin of error follows from this choice. The critical value depends on the confidence level and the sampling distribution of the statistic.

Greater confidence requires a larger critical value The standard deviation of the statistic depends on the sample size n

The margin of error gets smaller when: The confidence level decreases The sample size n increases

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ConditionsBefore calculating a confidence interval for µ or p there are

three important conditions

1) Random: The data should come from a well-designed random sample

or randomized experiment.

2) Normal:

The sampling distribution of the statistic is approximately Normal.

For means: If the population distribution is Normal. If the population distribution is not Normal, then the CLT tells us the sampling distribution will be approximately Normal if n ≥ 30.

For proportions: We can use Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ 10.

3) Independent: Individual observations are independent. When sampling

without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use our formula for the standard deviation of the statistic.