copyright © 2010 pearson education, inc. slide 23 - 1


Solution: B

Copyright © 2010 Pearson Education, Inc.

Chapter 23Inference About Means


Example: Based on weighing thousands of animals, the American Angus Association reports that mature Angus cows have mean weight of 1309 pounds with a standard deviation of 157 pounds. This result was based on a very large sample of animals from many herds over a period of 15 years, so let’s assume these summaries are the population parameters and that the distribution of the weights was unimodal and reasonably symmetric. What does the CLT predict about the mean weight seen in random samples of 100 mature Angus cows?


Getting Started

The Central Limit Theorem told us that the sampling distribution model for means is Normal with mean μ and standard deviation

Since we don’t know the population standard deviation we will estimate the population parameter σ with the sample statistic s.

The shape of the sampling model changes—the model is no longer Normal.

SD yn

sSE y

n


Gosset’s t William S. Gosset, an employee of the Guinness

Brewery in Dublin, Ireland, worked long and hard to find out what the sampling model was.

The sampling model that Gosset found has been known as Student’s t.

The Student’s t-models form a whole family of related distributions that depend on a parameter known as degrees of freedom (df). We denote the model as tdf.


A Confidence Interval for Means

A practical sampling distribution model for means

When the conditions are met, the standardized sample mean

follows a Student’s t-model with n – 1 degrees of freedom.

We estimate the standard error with

y

tSE y

sSE y

n


When the conditions are met, we are ready to find the confidence interval for the population mean, μ.

The confidence interval is

where the standard error of the mean is

The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size.

1ny t SE y

1*nt

sSE y

n


One-sample t-interval for the mean



As the degrees of freedom increase, the t-models look more and more like the Normal.

In fact, the t-model with infinite degrees of freedom is exactly Normal.


TI-Tips Find t-model probability, critical values and confidence interval for a mean

The equivalent of normalcdf for a t-model is tcdf(left,right,df).

The equivalent of invNorm for a t-model is invT. The problem is most calculators don’t have it. It would be #4 under DISTR if yours has it. If it doesn’t, you must use a t-chart.

To find a confidence interval: STAT TESTS TInterval

You can have the data in a list or enter your statistics.


Assumptions and Conditions Independence Assumption:

Independence Assumption. The data values should be independent.

Randomization Condition: The data arise from a random sample or suitably randomized experiment. Randomly sampled data (particularly from an SRS) are ideal.

10% Condition: When a sample is drawn without replacement, the sample should be no more than 10% of the population.


Assumptions and Conditions

Normal Population Assumption: We can never be certain that the data are from

a population that follows a Normal model, but we can check the

Nearly Normal Condition: The data come from a distribution that is unimodal and symmetric.

Check this condition by making a histogram or Normal probability plot.


Assumptions and Conditions (cont.)

Nearly Normal Condition: The smaller the sample size (n < 15 or so),

the more closely the data should follow a Normal model.

For moderate sample sizes (n between 15 and 40 or so), the t works well as long as the data are unimodal and reasonably symmetric.

For larger sample sizes, the t methods are safe to use unless the data are extremely skewed.


Finding t-Values By Hand

The Student’s t-model is different for each value of degrees of freedom.

Because of this, Statistics books usually have one table of t-model critical values for selected confidence levels.


Example: In 2004 a team of researchers published a study of contaminants in farmed salmon. Fish from many sources were analyzed for 14 organic contaminants. The study expressed concerns about the level found. One of those was the insecticide mirex, which has been shown to be carcinogenic. One farm in particular produced salmon with very high levels of mirex. After those outliers are removed, summaries for the mirex concentrations in the rest of the farmed salmon are: n = 150, ybar = 0.0913ppm, s = 0.0495 ppm What does a 95% confidence interval say about mirex? Are the assumptions and conditions for inference satisfied?


A Test for the Mean

The conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval.

We test the hypothesis H0: = 0 using the statistic

The standard error of the sample mean is

When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t model with n – 1 df. We use that model to obtain a P-value.

0

1n

yt

SE y

sSE y

n

One-sample t-test for the mean


Sample Size

To find the sample size needed for a particular confidence level with a particular margin of error (ME), solve this equation for n:

The problem with using the equation above is that we don’t know most of the values. We can overcome this: We can use s from a small pilot study. We can use z* in place of the necessary t value.

1n

sME t

n


Degrees of Freedom If only we knew the true population mean, µ, we

would find the sample standard deviation as

But, we use instead of µ, though, and that causes a problem.

When we use to calculate s, our standard deviation estimate would be too small.

The amazing mathematical fact is that we can compensate for the smaller sum exactly by dividing by n – 1 which we call the degrees of freedom.

2

.( )y

sn

y

y y 2 instead of y 2


Example: Researches tested 150 farm-raised salmon for organic contaminants. They found the mean concentration of mirex to be 0.0913 ppm with a standard deviation 0.0495 ppm. As a safety recommendation to recreational fishers, the EPA recommended “screening value” for mirex is 0.08 ppm. Are farmed salmon contaminated beyond the level permitted by the EPA?


Example: A company claims its program will allow your computer to download movies quickly. We’ll test the free evaluation copy by downloading a movie several times, hoping to estimate the mean download time with a margin of error of only 8 minutes. We think the standard dev is about 10 minutes. How many trial downloads must we run if we want 95% confidence in our estimate with a margin of error of only 8 minutes?


Homework: Pg. 554 1-19 odd (Day 1)

Pg. 554 21 – 35 a,b

copyright © 2010 pearson education, inc. slide 23 - 1

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