Chapter 16: Inference in Practice STAT 1450. Connecting Chapter 16 to our Current Knowledge of...

Chapter 16: Inference in Practice

STAT 1450

Connecting Chapter 16 to our Current Knowledge of Statistics▸ Chapter 14 equipped you with the basic tools for confidence interval construction.

▸ Chapter 15 equipped you with the basic tools for tests of significance.

▸ Chapter 16 addresses some of the nuances associated with inference

(our owner’s manual of sorts).

16.0 Inference in Practice

Conditions for Inference

▸ Random sample:

Do we have a random sample?

If not, is the sample representative of the population?

If not, was it a randomized experiment?

▸ Large enough population: sample ratio

Is the population of interest at least 20 times larger than the sample?

▸ Large enough sample:

Are the observations from a population that has a Normal distribution, or one where

we can apply principles from a Normal distribution? Look at the shape of the

distribution and whether there are any outliers present.

16.1 Conditions for Inference

Cautions about Confidence Intervals

The margin of error covers only sampling errors.

▸ Undercoverage, nonresponse, or other biases are not reflected in

margins of error.

▸ The source of the data is of utmost importance.

▸ Consider the details of a study before completely trusting a confidence

interval.

16.2 Cautions about Confidence Intervals

Example: Parental Monitoring Software

▸ Many parents elicit the use of various software and passwords to

monitor the ways children use their computers. In a survey of a

random sample of high school students, 16.7% with 3.45% margin of

error expressed an ability to circumvent their parent’s security efforts.

Would you trust a confidence interval based upon this data? Explain.

The Confidence Interval would be (.1325, .2015).

Yes, it is from a random sample.

But, there is likely some under-reporting by the teens.

As mentioned in Chapter 8, people tend to provide conservative answers to provocative

questions.

16.2 Cautions about Confidence Intervals

Cautions about Significance Tests

▸ When H0 represents an assumption that is widely believed,

small p-values are needed.

▸ Be careful about conducting multiple analyses for a fixed a.

It is preferred to just run a single test and reach a decision.

16.3 Cautions about Significance Tests


▸ When there are strong consequences of rejecting H0 in favor of HA, we

need strong evidence.

▸ Either way, strong evidence of rejecting H0 requires small p-values.

▸ Depending on the situation, p-values that are below 10% can lead to

rejecting H0.

▸ Unless stated otherwise, researchers assume the de-facto

significance level of 5%.



▸ The P-value for a one-sided tests is half of the P-value for the

two-sided test of the same null hypothesis and of the same data.

▸ The two-sided case combines two equal areas. The one-sided case

has one of those areas PLUS an inherent supposition by the

researcher of the direction of the possible deviation from H0.


A Connection between Confidence Intervals and Significance Tests

▸ Analogous to how we use high levels of confidence for confidence

intervals, we need strong evidence (and very small p-values) to reject

null hypotheses.

▸ Standard levels of confidence are 90%, 95%, and 99%.

▸ Standard levels of significance are 10%, 5%, and 1%. Recall from last chapter: more than 10% was a “likely” event.

5% to 10% was an “unlikely” event.

< 5% was an “extremely unlikely” event.


Sample Size affects Statistical Significance

▸ Very large samples can yield small p-values that lead to rejection of H0.

▸ Phenomena that are “statistically significant” are not always “practically

significant.”


Example: Carry-on luggage

▸ Airlines are now monitoring the amount of carry-on luggage passengers bring with them.

It is believed that the mean weight of carry-on luggage for passengers on multiple hour

flights is 30 lbs. with a standard deviation of 7.5 lbs. A random sample of 500,000

passengers who had recently flown on multiple hour flights had an average carry-on

luggage weight of 29.9 lbs.

▸ The test statistic is -9.43 with a P-value of 0.

▸ There is a statistically significant reason to reject the H0 and believe that the mean weight

of carry-on luggage is not 30 lbs. But, practically, the sample mean (29.9) and the

population mean (30.0) are quite comparable.




𝑧=𝑥−𝜇0

𝜎 /√𝑛=¿



𝑧=𝑥−𝜇0

𝜎 /√𝑛= 29.9−30.0

7.50 /√500000



𝑧=𝑥−𝜇0

𝜎 /√𝑛= 29.9−30.0

7.50 /√500000=−9.43


▸ Be advised that it is better to design a single study and conduct one

test of significance - (yielding one conclusion) than to design 1 study,

and perform multiple analyses until a desired result is achieved.


Sample Size for Confidence Intervals

▸ In Chapter 14, we noticed that the sample size impacts the margin of

error (and thus the width of the confidence interval).

▸ The margin of error is expressed as where

▸ Some algebra leads us to a formula for the minimum sample size for a

desired margin of error:

The z confidence interval for the mean of a Normal population will have a specified

margin of error m when the sample size is

16.4 Planning Studies: Sample Size for Confidence Intervals


▸ In the carry-on luggage example from earlier, a random sample of

500,000 passengers yielded a standard deviation for the sample mean

that was extremely small; resulting in |z| ≈ 9.43.

Poll: Would you expect that ________ a) more or b) fewer passengers would need to be sampled to estimate the mean weight of carry-on luggage within a margin of error of 0.2 lbs. with 95% confidence?



Poll: Would you expect that ________ a) more or b) fewer passengers would need to be sampled to estimate the mean weight of carry-on luggage within a margin of error of 0.2 lbs. with 95% confidence?29.9 – 30 = | -.1| = .1= m




Larger n, smaller m. Quadrupling the sample size, divides margin of error in half.. Let’s try the inverse.


.𝟏=𝒎=𝟗 .𝟒𝟑𝟕 .𝟓

√𝟓𝟎𝟎𝟎𝟎𝟎



Larger n, smaller m. Quadrupling the sample size, divides margin of error in half.. Let’s try the inverse. If we desire to double the m (m=.20) we need one-fourth the sample size..


.𝟏=𝒎=𝟗 .𝟒𝟑𝟕 .𝟓

√𝟓𝟎𝟎𝟎𝟎𝟎

𝑚=9.43∗7.5

√¿¿¿


Poll: Would you expect that ________ a) more or b) fewer passengers would need to be sampled to estimate the mean weight of carry-on luggage within a margin of error of 0.2 lbs. with 95% confidence?Larger n, smaller m. Quadrupling the sample size, divides margin of error in half.. Let’s try the inverse. If we desire to double the m (m=.20) we need one-fourth the sample size..





𝑚=9.43∗7.5

√¿¿¿



Taking one-25th of 125,000 is about 5000.


𝑚=9.43∗7.5

√¿¿¿9.43∗

7.5

√(125000 )=.2=𝑚=1.96∗

7.5

√(125000𝑐2 )


▸ Explicitly determine the sample size.


𝑛=( 𝑧∗𝜎𝑚 )

2





2

=( 1.96∗7.5.2 )

2



Resulting in a much smaller sample size.



2

=( 1.96∗7.5.2 )

2

=5402.25⇒5403

Example: The Justice System

16.5 Errors in Significance Testing

Jury VerdictTruth about the Defendant

Innocent Guilty

Guilty

Not Guilty




Innocent Guilty

Guilty Correct decision

Not Guilty Correct decision




Innocent Guilty

Guilty Error Correct decision

Not Guilty Correct decision Error

Power, Type I Error, and Type II Error

Decision based on data

Truth about a hypothesis

Ho is true Ha is true

Reject Ho

Fail to reject Ho Correct decision






Reject Ho Correct Decision

Fail to reject Ho Correct decision






Reject Ho Type I Error Correct Decision

Fail to reject Ho Correct decision Type II Error




▸ Type I Error – the maximum allowable “error” of a falsely rejected H0

(also the significance level, a).

▸ Type II Error – the probability of not rejecting H0, when we should

have rejected it.

▸ Power – the probability that the test will reject H0 when the alternative

value of the parameter is true.

Note: Increasing the sample size increases the power of a significance test.

▸ Effect size – the departure from a null hypothesis that results in

practical significance.

Example: Coffee consumption


▸ Recall the coffee consumption example from last chapter with standard

deviation of 9.2 oz. A random sample of 48 people drank an average of

26.31 oz. of coffee daily. A significance test of the mean being different

from our original estimate is conducted. Provide examples of , a b, and

power.

Example: Coffee Consumption


▸ Recall the coffee consumption example from last chapter with standard

deviation of 9.2 oz. A random sample of 48 people drank an average of

26.31 oz. of coffee daily. A significance test of the mean being different

from our original estimate is conducted. Provide examples of , a b, and

power.

We reject H0 ( = “20”), when it was actually true.

We fail to reject H0, when actually the true mean m was not “20.”

Power: Probability of rejecting H0 for a specific value of ≠ 20.

Five-Minute Summary

▸ List at least 3 concepts that had the most impact on your knowledge of

inference in practice.

___________ _____________

_______________

Chapter 16: Inference in Practice STAT 1450. Connecting Chapter 16 to our Current Knowledge of...

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