Slide1 l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 112

Chapter 10: Estimating With Confidence PowerPoint PPT Presentation


  • 107 Views
  • Uploaded on
  • Presentation posted in: General

Chapter 10: Estimating With Confidence. 10.1 – Confidence Intervals: The basics . Statistical Inference:. Using sample data to draw conclusions about a population. Note:. Each sample may vary, but the population parameter doesn’t!. Sampling Distribution:.

Download Presentation

Chapter 10: Estimating With Confidence

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Slide1 l.jpg

Chapter 10: Estimating With Confidence


Slide2 l.jpg

10.1 – Confidence Intervals: The basics


Slide3 l.jpg

Statistical Inference:

Using sample data to draw conclusions about a population

Note:

Each sample may vary, but the population parameter doesn’t!


Slide4 l.jpg

Sampling Distribution:

  • If population is approximately normal, so is

  • the sample distribution

  • If population is skewed, the sample distribution

  • is approximately normal if n30 by the central

  • limit theorem

  • If given sample data, look at the distribution to

  • assess normality if needed. (Normal Prob. Plot)


Slide5 l.jpg

Confidence Interval:

  • Uses the sample distribution to predict population parameter

  • It is an interval of numbers above and below the sample statistic


Slide6 l.jpg

Confidence Level:

The probability the interval will capture the true parameter value in repeated samples

Critical Value:

The probability p lying to its right under the standard Normal curve. ( Z* )


Slide7 l.jpg

Margin of Error:

  • How accurate our estimate is based on the variability of the sample distribution. We add and subtract this from our estimate.

estimate  margin of error

Caution! Margin of error is only from random sampling errors. This does not include errors in collecting the data!


Slide8 l.jpg

Most Common Critical Values

Confidence Level (C)Upper tail prob.Z* Value

90%

 1.645

0.05

0.05

0.05

0.90

Z=?

Z=?


Slide10 l.jpg

Most Common Critical Values

Confidence Level (C)Upper tail prob.Z* Value

95%

 1.96

0.025

0.025

0.025

0.95

Z=?

Z=?


Slide11 l.jpg

Most Common Critical Values

Confidence Level (C)Upper tail prob.Z* Value

99%

 2.576

0.005

0.005

0.005

0.99

Z=?

Z=?


Slide12 l.jpg

Calculator Tip:

Critical Values

2nd Dist – invNorm( (1 + C)/2 )

OR: Look at the T-Tables for the most common ones!

(You will learn more about them later)


Slide13 l.jpg

Confidence Interval for a Population mean ( known)

(Z-Interval)

estimate  margin of error

estimate  critical value  standard error


Slide14 l.jpg

Properties of Confidence Intervals

  • The interval is always centered around the statistic

  • The higher the confidence level, the wider the interval becomes

  • If you increase n, then the margin of error decreases


Slide15 l.jpg

Calculator Tip:

Z-Interval

Stat – Tests – ZInterval

Data: If given actual values

Stats: If given summary of values


Slide16 l.jpg

Interpreting a Confidence Interval:

What you will say:

I am C% confident that the true parameter is captured in the interval

What it means:

If we took many, many, SRS from a population and calculated a confidence interval for each sample, C% of the confidence intervals will contain the true mean


Slide17 l.jpg

CAUTION!

Never Say:

The interval will capture the true mean C% of the time.

It either does or does not!


Slide19 l.jpg

Conditions for a Z-Interval:

(should say)

  • SRS

(CLT or population approx normal)

2. Normality

(Population 10x sample size)

3. Independence


Slide20 l.jpg

Steps to Construct ANY Confidence Interval:

PANIC

P:

Parameter of Interest (what are you looking for?)

A:

Assumptions (what are the conditions?)

N:

Name the type of interval (what type of data do we have?)

I:

Interval (Finally! You can calculate!)

C:

Conclusion in context (I am ___% confident the true parameter lies between ________ and _________)


Slide21 l.jpg

Example #1

Serum Cholesterol-Dr. Paul Oswick wants to estimate the true mean serum HDL cholesterol for all of his 20-29 year old female patients. He randomly selects 30 patients and computes the sample mean to be 50.67. Assume from past records, the population standard deviation for the serum HDL cholesterol for 20-29 year old female patients is =13.4.

  • Construct a 95% confidence interval for the mean serum HDL cholesterol for all of Dr. Oswick’s 20-29 year old female patients.

P:

The true mean serum HDL cholesterol for all of Dr. Oswick’s 20-29 year old female patients.


Slide22 l.jpg

A:

SRS:

Says randomly selected

Normality:

Approximately normal by the

CLT (n 30)

I am assuming that Dr. Oswick has 300 patients or more.

Independence:

N:

One sample Z-Interval


Slide23 l.jpg

I:


Slide24 l.jpg

C:

I am 95% confident the true mean serum HDL cholesterol for all of Dr. Oswick’s 20-29 year old female patients is between 45.875 and 55.465


Slide25 l.jpg

Example #1

Serum Cholesterol-Dr. Paul Oswick wants to estimate the true mean serum HDL cholesterol for all of his 20-29 year old female patients. He randomly selects 30 patients and computes the sample mean to be 50.67. Assume from past records, the population standard deviation for the serum HDL cholesterol for 20-29 year old female patients is =13.4.

b. If the US National Center for Health Statistics reports the mean serum HDL cholesterol for females between 20-29 years old to be  = 53, do Dr. Oswick’s patients appear to have a different serum level compared to the general population? Explain.

No,

53 is contained in the interval.


Slide26 l.jpg

Example #1

Serum Cholesterol-Dr. Paul Oswick wants to estimate the true mean serum HDL cholesterol for all of his 20-29 year old female patients. He randomly selects 30 patients and computes the sample mean to be 50.67. Assume from past records, the population standard deviation for the serum HDL cholesterol for 20-29 year old female patients is =13.4.

c. What two things could you do to decrease your margin of error?

Increase n

Lower confidence level


Slide27 l.jpg

Example #2

Suppose your class is investigating the weights of Snickers 1-ounce Fun-Size candy bars to see if customers are getting full value for their money. Assume that the weights are Normally distributed with standard deviation = 0.005 ounces. Several candy bars are randomly selected and weighed with sensitive balances borrowed from the physics lab. The weights are

0.95 1.020.980.971.051.010.981.00

ounces. Determine a 90% confidence interval for the true mean, µ. Can you say that the bars weigh 1oz on average?

P:

The true mean weight of Snickers 1-oz Fun-size candy bars


Slide28 l.jpg

A:

Says randomly selected

SRS:

Normality:

Approximately normal because the population is approximately normal

I am assuming that Snickers

has 80 bars or more in the 1-oz size

Independence:

N:

One sample Z-Interval


Slide29 l.jpg

I:


Slide30 l.jpg

C:

I am 90% confident the true mean weight of Snickers 1-oz Fun-size candy barsis between .9921 and .9979 ounces. I am not confident that the candy bars weigh as advertised at the 90% level.


Slide31 l.jpg

Choosing a Sample Size for a specific margin of error

Note: Always round up! You can’t have part of a person! Ex: 163.2 rounds up to 164.


Slide32 l.jpg

Example #3

A statistician calculates a 95% confidence interval for the mean income of the depositors at Bank of America, located in a poverty stricken area. The confidence interval is $18,201 to $21,799.

  • What is the sample mean income?


Slide33 l.jpg

Example #3

A statistician calculates a 95% confidence interval for the mean income of the depositors at Bank of America, located in a poverty stricken area. The confidence interval is $18,201 to $21,799.

b. What is the margin of error?

m

m = 21,799 – 20,000

m = 1,799


Slide34 l.jpg

Example #4

A researcher wishes to estimate the mean number of miles on four-year-old Saturn SCI’s. How many cars should be in a sample in order to estimate the mean number of miles within a margin of error of  1000 miles with 99% confidence assuming =19,700.


Slide35 l.jpg

10.2 – Estimating a Population Mean


Slide36 l.jpg

In the 10.1 we made an unrealistic assumption that the population standard deviation was known and could be used to calculate confidence intervals.


Slide37 l.jpg

Standard Error:

When the standard deviation of a statistic is estimated from the data


Slide38 l.jpg

When we know  we can use the Z-table to make a confidence interval. But, when we don’t know it, then we have to use something else!


Slide39 l.jpg

Properties of the t-distribution:

  • σ is unknown

  • Degrees of Freedom = n – 1

  • More variable than the normal distribution (it has fatter tails than the normal curve)

  • Approaches the normal distribution when the degrees of freedom are large (sample size is large).

  • Area is found to the right of the t-value


Slide40 l.jpg

Properties of the t-distribution:

  • If n < 15, if population is approx normal, then so is the sample distribution. If the data are clearly non-Normal or if outliers are present, don’t use!

  • If n > 15, sample distribution is normal, except if population has outliers or strong skewness

  • If n  30, sample distribution is normal, even if population has outliers or strong skewness


Slide44 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

10

1.093


Slide46 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

0.15

10

1.093


Slide47 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

0.85

10

1.093


Slide48 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

23

0.685


Slide50 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

0.25

23

0.685


Slide51 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

0.25

23

-0.685


Slide52 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

10

0.70

1.093


Slide54 l.jpg

Example #1

Determine the degrees of freedom and use the t-table to find probabilities for each of the following:

10

0.1

0.70

1.093

.25 – .15 = 0.1


Slide55 l.jpg

Calculator Tip:

Finding P(t)

2nd – Dist – tcdf( lower bound, upper bound, degrees of freedom)


Slide56 l.jpg

One-Sample t-interval:


Slide57 l.jpg

Calculator Tip:

One sample t-Interval

Stat – Tests – TInterval

Data: If given actual values

Stats: If given summary of values


Slide58 l.jpg

Conditions for a t-interval:

  • SRS

(should say)

(population approx normal and n<15, or moderate size (15≤ n < 30) with moderate skewness or outliers, or large sample size n ≥ 30)

2. Normality

3. Independence

(Population 10x sample size)


Slide59 l.jpg

Robustness:

The probability calculations remain fairly accurate when a condition for use of the procedure is violated

The t-distribution is robust for large n values, mostly because as n increases, the t-distribution approaches the Z-distribution. And by the CLT, it is approx normal.


Slide60 l.jpg

Example #2

Practice finding t*

9


Slide62 l.jpg

Example #2

Practice finding t*

9

3.250

19


Slide64 l.jpg

Example #2

Practice finding t*

9

3.250

19

1.729

39


Slide66 l.jpg

Example #2

Practice finding t*

9

3.250

19

1.729

39

2.042

29


Slide68 l.jpg

Example #2

Practice finding t*

9

3.250

19

1.729

39

2.042

29

2.756


Slide69 l.jpg

Example #3

As part of your work in an environmental awareness group, you want to estimate the mean waste generated by American adults. In a random sample of 20 American adults, you find that the mean waste generated per person per day is 4.3 pounds with a standard deviation of 1.2 pounds. Calculate a 99% confidence interval for  and explain it’s meaning to someone who doesn’t know statistics.

P:

The true mean waste generated per person per day.


Slide70 l.jpg

A:

Says randomly selected

SRS:

Normality:

15<n<30. We must assume the population doesn’t have strong skewness. Proceeding with caution!

It is safe to assume that there are more than 200 Americans that create waste.

Independence:

N:

One Sample t-interval


Slide71 l.jpg

I:

df =

20 – 1 =

19


Slide73 l.jpg

I:

df = 20 – 1 = 19


Slide74 l.jpg

C:

I am 99% confident the true mean waste generated per person per day is between 3.5323 and 5.0677 pounds.


Slide75 l.jpg

Matched Pairs t-procedures:

Subjects are matched according to characteristics that affect the response, and then one member is randomly assigned to treatment 1 and the other to treatment 2. Recall that twin studies provide a natural pairing. Before and after studies are examples of matched pairs designs, but they require careful interpretation because random assignment is not used.

Apply the one-sample t procedures to the differences


Slide76 l.jpg

Confidence Intervals for Matched Pairs


Slide77 l.jpg

Example #4

Archaeologists use the chemical composition of clay found in pottery artifacts to determine whether different sites were populated by the same ancient people. They collected five random samples from each of two sites in Great Britain and measured the percentage of aluminum oxide in each. Based on these data, do you think the same people used these two kiln sites? Use a 95% confidence interval for the difference in aluminum oxide content of pottery made at the sites and assume the population distribution is approximately normal. Can you say there is no difference between the sites?

1.7

3.2

1.3

-2.5

.6


Slide78 l.jpg

P:

μn = New Forrest percentage of aluminum oxide

μa = Ashley Trails percentage of aluminum oxide

μd = μn - μa = Difference in aluminum oxide levels

The true mean difference in aluminum oxide levels between the New Forrest and Ashley Trails.


Slide79 l.jpg

A:

Says randomly selected

SRS:

Normality:

Says population is approx normal

It is safe to assume that there are more than 50 samples available

Independence:

N:

Matched Pairs t-interval


Slide80 l.jpg

I:

df =

5 – 1 =

4


Slide82 l.jpg

I:

df = 20 – 1 = 19


Slide83 l.jpg

C:

I am 95% confident the true mean difference in aluminum oxide levels between the New Forrest and Ashley Trails is between –1.754 and 3.4743.

Can you say there is no difference between the sites?

zero is in the confidence interval, so it is safe to say there is no difference.

Yes,


Slide84 l.jpg

Example #5

The National Endowment for the Humanities sponsors summer institutes to improve the skills of high school language teachers. One institute hosted 20 Spanish teachers for four weeks. At the beginning of the period, the teachers took the Modern Language Association’s listening test of understanding of spoken Spanish. After four weeks of immersion in Spanish in and out of class, they took the listening test again. (The actual spoken Spanish in the two tests was different, so that simply taking the first test should not improve the score on the second test.) Below is the pretest and posttest scores. Give a 90% confidence interval for the mean increase in listening score due to attending the summer institute. Can you say the program was successful?


Slide86 l.jpg

P:

μB = Pretest score

μA = Posttest score

μd = μB - μA = Difference in test scores

The true mean difference in test scores between the Pretest and Posttest


Slide87 l.jpg

A:

We must assume the 20 teachers are randomly selected

SRS:

Normality:


Slide89 l.jpg

A:

We must assume the 20 teachers are randomly selected

SRS:

Normality:

15<n<30 and distribution is approximately normal, so safe to assume

It is safe to assume that there are more than 200 Spanish teachers

Independence:

N:

Matched Pairs t-interval


Slide90 l.jpg

19

20 – 1 =

I:

df =


Slide92 l.jpg

I:

df = 20 – 1 = 19


Slide93 l.jpg

C:

I am 90% confident the true mean difference in test scores between the Pretest and Posttest

is between –2.689 and –0.2115.

Can you say the program was successful?

zero is not in the confidence interval, so the pretest score is lower than the posttest score.

Yes,


Slide94 l.jpg

10.3 – Estimating a Population Proportion


Slide95 l.jpg

Properties of :


Slide96 l.jpg

Confidence Interval for a Population Proportion:

Notice! We use Z* and not t*


Slide97 l.jpg

Conditions for a p-interval:

  • SRS

(should say)

2. Normality

3. Independence

(Population 10x sample size)


Slide98 l.jpg

Calculator Tip:

One sample p-Interval

Stat – Tests – 1–PropZInt

x = # of successes in the sample


Slide99 l.jpg

Example #1

A news release by the IRS reported 90% of all Americans fill out their tax forms correctly. A random sample of 1500 returns revealed that 1200 of them were correctly filled out. Calculate a 92% confidence interval for the proportion of Americans who correctly fill out their tax forms. Is the IRS correct in their report?

P:

The true percent of Americans who fill out their tax forms correctly


Slide100 l.jpg

A:

Says randomly selected

SRS:

Normality:

0.80

Yes, safe to assume an approximately normally distribution

It is safe to assume that there are more than 15,000 people who file their taxes

Independence:


Slide101 l.jpg

N:

One Sample Proportion Interval

I:

Z* = ?


Slide102 l.jpg

?


Slide103 l.jpg

Confidence Level (C)Upper tail prob.Z* Value

92%

0.04

0.04

0.04

0.92

Z=?

Z=?


Slide105 l.jpg

Confidence Level (C)Upper tail prob.Z* Value

92%

 1.75

0.04

0.04

0.04

0.92

Z=?

Z=?


Slide106 l.jpg

N:

One Sample Proportion Interval

I:


Slide107 l.jpg

C:

I am 92% confident the true percent of Americans who fill out their tax forms correctly is between 78.19% and 81.8%

Is the IRS correct in their report?

No,

90% is not in the interval!


Slide108 l.jpg

Sample size for a Desired Margin of Error

If we want the margin of error in a level C confidence interval for p to be m, then we need n subjects in the sample, where:

p* =

An estimate for

n 


Slide109 l.jpg

Note: If p is unknown use the most conservative value of p = 0.5. Since n is the sample size, it must be a whole number!!! Round up!


Slide110 l.jpg

Example #2

You wish to estimate with 95% confidence; the proportion of computers that need repairs or have problems by the time the product is three years old. Your estimate must be accurate within 3.5% of the true proportion.

a. Find the sample size needed if a prior study found that 19% of computers needed repairs or had problems by the time the product as three years old.


Slide111 l.jpg

Example #2

You wish to estimate with 95% confidence; the proportion of computers that need repairs or have problems by the time the product is three years old. Your estimate must be accurate within 3.5% of the true proportion.

b. If no preliminary estimate is available, find the most conservative sample size required.


Slide112 l.jpg

Example #2

You wish to estimate with 95% confidence; the proportion of computers that need repairs or have problems by the time the product is three years old. Your estimate must be accurate within 3.5% of the true proportion.

c. Compare the results from a and b.

Using 0.5 makes the sample size very large, ensuring that enough people will be surveyed.


  • Login