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Topics for Today

More on Confidence Intervals

Example

NAEP quantitative scores

The NEAP is a broad-scope survey conducted on a ______of students in grade 4, 8 and 12 in the United States.

http://en.wikipedia.org/wiki/National_Assessment_of_Educational_Progress

It’s used by law makers and educators to form policy and priorities.

Such as comparing
states: /

[ source http://www.schoolinfosystem.org/archives/naep_state.gif ]

and tracking the reading performance of students (scores are from the ‘reading’ component of the survey)

[ source: http://www.balancedreading.com/2005NAEP.gif ]

… but there are many components:

The NAEP mathematics section test basic arithmetic skills and gives a ‘quantitative score’.

Scores on the test range from 0 to ___.

A person who scores 233 can add the amounts of two cheques appearing on a bank deposit slip.

Someone scoring 325 can determine the price of a meal from a menu.

A person scoring ___ can transform a price in cents per ounce into dollars per pound.

In a recent year, ___ U.S. men 21 to 25 years of age were in the NAEP sample.

Their mean quantitative score was = ___. These ___ men are a ______from the population of all young men.

On the basis of this sample, what can we infer about the mean score μ in the population of all 9.5 million young men of these ages in the U.S.?

The ______tells us the sample mean from a random sample will be close to the population mean μ.

= 272 so μ is somewhere around 272

But how _____?

To determine how close it is, we need to remember the ______of , and use it to construct a confidence interval.

Sampling distribution of :

· ______: the mean of 840 scores has a distribution that is approximately ______.

· The mean of this normal sampling distribution is the same as the mean, μ of the entire population.

· The ______of from a sample of 840 is , where σ is the standard deviation of the distribution of individual scores.

Suppose σ = __. The standard deviation of is = ___

If we choose many samples of size 840 and find the means and display their distribution, it would look more like a ______distribution with mean μ and standard deviation = 2.1.

Statistical Confidence

The 68-95-99.7 rule says that in 95% of all samples, the mean will be within _ standard deviations of the ______score μ.

So in 95% of the ______the population parameter μ is between

- 4.2 and + 4.2

Our sample gave = ___

Statistically speaking we are 95% confident that the unknown mean (μ) score lies between

- 4.2 = 272 - 4.2 = ______

and

+ 4.2 = 272 + 4.2 = _____

95% confidence implies the method gives correct results 95% of the time.

______

is called a 95% confidence interval for μ.

Recognize there are 2 possibilities:

1. The interval between 267.8 and 276.2 ______the true population parameter μ.

2. Our sample was one of the few samples for which is ______4.2 points of μ. Only __ of the samples will give such results.

[ source http://www.southalabama.edu/coe/bset/johnson/lectures/lec16_files/image006.jpg ]

Confidence Interval

Estimate ± ______

Estimate: Our guess for the unknown parameter. The estimate in our case is the statistic .

Margin of Error: Measures the ______of our estimate, based on the variability of the estimate.

Confidence Level: ______that the interval will ______the true parameter. In repeated sampling we would expect 95% of the intervals to contain the true population parameter μ.

Margin of Error

A __% confidence interval for the population mean μ has a margin of error

____ * (standard deviation of )

= 1.96 *

The margin of error for a __% confidence interval for the mean μ is given by:

_____*

The margin of error for a __% confidence interval for the mean μ is given by:

____ *

Example: Interpretting a CI

A poll on voting preferences for candidates interviewed 1025 people randomly selected in the Vancouver area. The poll found that 46 % of the people surveyed said they preferred the Liberal party.

a) The poll announced a margin of error of ± 3 percentage points with a 95% confidence level. What is the 95% confidence interval for the percent of all people who will vote Liberal in the upcoming election?

b) What are your conclusions?

Example: Analyzing chemical data

A manufacturer of chemical products analyses a sample from each batch of a product to verify the concentration of a particular ingredient. The chemical analysis is not very accurate.

Repeated measurements on the same batch give different results and are approximately normally distributed.

The analysis procedure has no bias and the population mean μ is the true concentration of the sample. The standard deviation of this distribution is known to be σ = 0.0068 grams/litre.

The lab analysed each sample three times and reported the average reading.

Three analyses of one sample give the following concentrations:

0.8403 0.8363 0.8447

a) Construct a 95% confidence interval for the true concentration μ.

b) Management asks the lab to produce results that are accurate to within ± 0.001 with a 95% confidence. How many samples should be taken to comply with this request?

In reality, s is usually UNKNOWN

Up to now, all the examples have given you the standard deviation from the ______(s), but this is rarely (if ever) known. Instead, we will have to use the standard deviation from the ______(s)

Remember, the standard deviation s and s mean the same thing, just one is measured on the ______and the other is measured on the ______.

However, when the population standard deviation is unknown, we ____ to use the sample standard deviation.

This doesn’t come for free.

Since we’re estimating something else from the sample (instead of just ‘knowing’ it), our confidence intervals get _____.

The t-distribution

The t-distribution is a similar shape to the normal distribution.

(t in red, normal in blue)

[source http://www.mechanical-writings.com/img/gt/confidence-interval-t-distribution/T_distribution_1df.png ]

Unlike the normal distribution, (which depends only on µ and s), the t-distribution depends on something called the degrees of freedom.

[source http://02.edu-cdn.com/files/static/mcgrawhillprof/9780071621885/ESTIMATION_AND_CONFIDENCE_INTERVALS_07.GIF ]

or … you can check out this applet:

http://www.stat.tamu.edu/~jhardin/applets/signed/T.html

look at the panel on the left side and change the degrees of freedom (slider at the top of the page) to see how the t-distribution becomes more ‘normal’ as the degrees of freedom increase.

Our margin of error will use the t-distribution instead of the z.