Chapter 11 Section 2.pptx

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Comparing Inference

Intervals

Process Z - Interval

P Parameter –population measurement of interest Population – Who are you trying to reach/talk about

A Conditions: SRS & Large Sample Size (normal curve) Verify: by CLT

How was the sample taken and/or given…

N 1 Sample Z-Interval For Population Means

I

C From the data we are _______ % confident the the true mean, “context of real mean,” lies between

___________ and ____________ (provide units)

n³30

Population³10n

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Process T- Interval

A Conditions: SRS & Large Sample Size (t curve)

Verify: 15<n use t-procedures if data is close to normal but no outliers

15 < n<40 use t-procedures except in presence of outliers and strong skewness

40 >n normal t-procedures, even if clearly a skewed distribution

N 1 Sample t-Interval For Population Means

I

C From the data we are _______ % confident the the true mean, “context of real mean,” lies between ___________ and

____________ (provide units)

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Process T- Interval

P Parameter –population measurement as a difference

Population –Difference within the understanding of who you are trying to reach/talk about

15 < n<40 use t-procedures except in presence of outliers and strong skewness

40 >n normal t-procedures, even if clearly a skewed distribution

Independent ~ the way trials were taken, one did not effect the other

N Matched Pairs t-Interval For Population Means

I

C From the data we are _______ % confident the the true mean, “context of real mean,” lies between ___________ and

____________ (provide units)

x±t* s

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Comparing Inference Test

Process Z - Test

H H₀ : No Change H_a: Change

A Conditions: SRS & Large Sample Size (normal curve) Verify: by CLT

N 1 Sample Z-Test For Population Means

T

O Probability of z score (Area To Left)

M _{Fail to Reject H}_o_{Or Reject H}_o

S _{From the data we have/or do not have significant evidence at} the_______ significance level, talk about decision and evidence in terms of H_a

n³30

Population³10n

z= x-m

s n

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Process T - Test

15 < n<40 use t-procedures except in presence of outliers and strong skewness 40 >n normal t-procedures, even if clearly a skewed distribution

N 1 Sample T-Test For Population Means

T

O Probability of t score (Area To RIGHT), Know degrees of freedom and significance level

M Fail to Reject H_o Or Reject H_o

S From the data we have/or do not have significant evidence at the_______ significance level, talk about decision and evidence in terms of H_a

t= x-m

s n

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P Parameter –population measurement as a difference

Population –Difference within the understanding of who you are trying to reach/talk about

Independent ~ the way trials were taken, one did not effect the other How was the sample taken and/or given…

N Matched Pairs T-Test For Population Means

T

t= x-m

s n

m = 0

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Example #1: Workers at a factory asked their supervisor to provide music during their shift. The supervisor wanted to know whether the music really helped improve the worker’s performance. Because all the workers were assigned to different rooms at random, the supervisor randomly selected one room. For one week, he provided music in this room, and for one

week he provided none. He flipped a coin to determine which week to provide music. Afterward, he recorded the productivity of the workers, using the average number of items assembled per day:

Estimate the difference between the workers’ performance in the

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Example #1: Workers at a factory asked their supervisor to provide music during their shift. The supervisor wanted to know whether the music really helped improve the worker’s performance. Because all the workers were assigned to different rooms at random, the supervisor randomly selected one room. For one week, he provided music in this room, and for one

week he provided none. He flipped a coin to determine which week to provide music. Afterward, he recorded the productivity of the workers, using the average number of items assembled per day:

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Comparing Two Means

Comparative studies are more convincing than single single-sample investigations, so one-sample inference is not as common as

comparative (two-sample) inference.

In a comparative study, we may want to compare two treatments, or we may want to compare two populations. In either case, the samples must be chosen _________________ and _________________ in order to perform statistical inference.

Since matched pairs are NOT chosen independently, we will NOT

use two-sample inference for a matched pairs design. For a

matched pairs design, apply the one-sample t procedures to the observed differences.

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Otherwise, we may use two-sample inference to compare two

treatments or two populations.

Before you begin, check your assumptions! For comparing two means,

both samples must come from an _______ and must be chosen

_______________. Also, both populations must be

__________________________. (Always check for ____________ or

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If each population is __________, it turns out that the difference of

distributions will be __________; and if the populations are

approximately __________, then the difference will be approximately

__________. Of course, this all hinges on knowing the values of σ_X and

σ_Y, which (in all likelihood) we will not know. So what do we do?

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When we did that earlier in this chapter, we stopped looking for at the

distribution of , and instead started looking at the

distribution of

Naturally, that's what we're going to do here! The statistic is now

OR The more common notation

t

=

x

-

m

x

s

_x

n

x

t= x1 - x2

s₁2

n₁ + s₂2

n₂

1-C

2 t=

x₁ - x₂ s₁2

n₁ + s₂2

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There's just one problem—this statistic doesn't have a

__________________________.

Fortunately, it's close. We can still use the t distribution to answer

probability questions (and conduct inference) for this new variable.

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Degrees of Freedom

Back before computers and calculators were common, there was a

conservative rule of thumb: use the smaller sample size (minus one) as the df. This is called a conservative approach, because it tends to fail to reject the null hypothesis more often than it should.

Now that we have calculators and computers, we can use a different approach.

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Confidence Intervals

A Level C confidence interval for the difference of means is

__________________________________, where t*_{is the upper}

critical value from the t distribution with k degrees of freedom.

1 -

C

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Hypothesis Testing

The null hypothesis is that there is no difference between the two parameters.

The alternative hypothesis could be as follows:

One-Sided Two-Sided

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P Parameter –population measurement for comparison between two groups

Population – Who we are trying to reach/ talk about, through comparing groups

Independent ~ the way trials were taken, one did not effect the other How was the sample taken and/or given…

N 2 Sample T-Test For Population Means

T

t= x1-x2

s₁2

n₁ + s₂2

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Example #1: A study in Stockholm concerned the work conditions of the city's bus drivers—in particular, a program was designed to help relieve stress on the job by changing the bus routes. Drivers were randomly assigned to either an old route (control), or a new route (intervention). The heart rates of the drivers were measured after driving the routes.

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Is there any evidence that the improved routes have changed heart rates?

Here are the variables:

μ_C = mean heart rate for drivers of the ______________

μ_I = mean heart rate for drivers of the ______________

H₀: H_a:

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There are some requirements to conduct this test:

(a) SRS?

Each population variable must have a

________________________________. This is not given, so…

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Since ____________ are

unknown,______________________________________________.

 Find the test statistic:

 Calculate the P-value:

 Decision:

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Example #2: Some research has been conducted comparing the leg strengths of males and females.

Here are the data (Force, in Newtons) for a random sample of males:

Here are the data for a random sample of females: