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Chapter 10. Introduction to Inference and Confidence Intervals 1/31 & 2/1. Hello!. Preferred Intelligence and Learning Modality Activity. Mr. Force’s Expectations. Expectations Be on time to class Be prepared – have homework every class Be respectful to your peers and to myself

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Chapter 10

Chapter 10

Introduction to Inference and Confidence Intervals

1/31 & 2/1


Hello

Hello!

  • Preferred Intelligence and Learning Modality Activity


Mr force s expectations

Mr. Force’s Expectations

  • Expectations

    • Be on time to class

    • Be prepared – have homework every class

    • Be respectful to your peers and to myself

    • Work hard the entire period – bell to bell

    • To listen when it’s time to listen, and talk when it’s time to talk

  • Chapter 10!!


Chapter 101

Chapter 10

  • What is statistical inference?

  • What is a confidence interval?

  • Today, we will look at an introduction to statistical inference, and then we will look at confidence intervals – what they mean and how we construct one.


Notes

Notes

  • We are not satisfied with only knowing information about a sample. We wish to infer from the sample data some conclusion about a wider population that the sample represents.

  • Statistical Inference– provides methods for drawing conclusions about a population from sample data.

    • Confidence Intervals & Tests of Significance.


Confidence intervals

Confidence Intervals

  • A level C confidence interval for a parameter has two parts:

    • Interval calculated by: Estimate margin of error

    • Confidence level C– gives the probability that the interval will capture the true parameter value in repeated samples.


Confidence intervals for a population mean

Confidence Intervals for a population mean

  • Remember, it is essential that the sampling distribution of the sample mean s approximately normal.

  • Whenis the construction of a CI for appropriate?

    • The data comesfrom an SRS of the population of interest

    • The sampling distribution of isapproximately normal.


How to find z

How to find z*

  • For example, the find an 80% confidence interval, we must catch the central 80% of the normal sample distribution of .

  • Hence, z* is the point with area 0.1 to it’s right and 0.9 to its left under the standard normal curve.

    • Use Table A to find the point with area 0.9 to its left.

    • z*=1.28  What does this mean?

      • It means that there is 00.8 under the standard normal curve between -1.28 and 1.28.


Common z

Common z*

  • In order to catch the central C probability we have (1-C) to the right or (1/C)/2 on either side.

  • C is between –z* and z*

  • Most commonly used z*

  • Z* is called the upper p critical value, where p is the probability lying to the right of z*


Confidence interval construction

Confidence Interval Construction

  • Our confidence interval is given by:

  • The estimate of unknown parameter is and for it is . The margin of error (MOE) is

means

proportions


Inference toolbox confidence intervals

Inference Toolbox: Confidence Intervals

To construct a confidence interval:

Step 1: Identify the population of interest and the parameter you want to draw conclusions about.

Step 2: Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

Step 3: If the conditions are met, carry out the inference procedure.

CI=estimate +/- margin of error

Step 4: Interpret your results in the context of the problem.


Group activity

Group Activity

  • When you flip a fair coin, it is equally likely to lands heads or tails. Is the same true for flipping Hershey kisses? In this activity, you will toss a Hershey kiss several times and observe whether it lands on its base or with its point up.


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Group activity contd

Group Activity, contd.

  • What does this mean?

    • We are 95% confident that the interval 1.96 will contain the true proportion p.

    • From A day, we have that the average 95% confidence interval is (.2574,.4960)

  • This activity showed how sample statistics can be used to estimate unknown population parameters. This is one of the two types of statistical inference in this chapter.


How do confidence intervals behave

How do confidence intervals behave?

  • We would like HIGH confidence and a SMALL margin of error.

  • How do we make smaller?

    • When z* is smaller

    • When is smaller

    • Increasing n


Margin of error

Margin of Error

  • The margin of error does not include:

    • the effects of under-coverage

    • nonresponse, or

    • other practical difficulties.


Choosing the sample size

Choosing the sample size

  • We just saw that we can have high confidence level with small margin of error if we take many observations.

  • How many observations is enough?

    • We set m=

  • Hence, to determine how large of a sample size to use, determine the specified margin of error. Then solve for n.


Example

example

  • Say that we want a confidence level of 95% with a margin of error at most 5. We are given that

  • We set up the equation:

    • We have that z*=1.96.

    • Since we want a margin of error of at most 5, we set

    • We now have 284.125

    • So, we choose n to be 285


Cautions

Cautions

  • Data must be an SRS from the population.

  • We must know the sample standard deviation

  • The margin of error in a confidence interval only covers random sampling errors.

  • There are many more on page 553.


What ci s do not say

What CI’s DO not say

  • For example, a confidence interval does say that we are 95% confident that the mean SAT Math score for all California high school seniors lies between 452 and 470.

    • These numbers were calculated by a method that gives correct results in 95% of all possible samples

  • WE CANNOT SAY: the probability is 95% that the true mean falls between 452 and 470.

  • The true mean is or is not between 452&470

  • The probability calculations of standard statistical inference describe how often the method gives correct answers.


Example 10 2

Example 10.2

  • You want to estimate the mean SAT Math score for more than 350,000 high school seniors in California, however only 49% of these seniors took the test. We instead give the test to a SRS of 500 students.

    • =461

  • We take very many SRS of the population. From last chapter, we know that:

    • The CLT tells us that the mean x̄ of 500 scores is close to normal

    • The mean of this normal sampling distribution is the same as the unknown mean of the entire population

    • The standard deviation of x̄ for an SRS of 500 is given by where is the s.d. of the population.


Chapter 10

  • Now, suppose we know =100. Then,

    • =4.5.

  • Knowing this, what can we say about the unknown population mean ?

    • We can say that in 95% of all samples the interval x̄ - 1.96(4.5) and x̄ + 1.96(4.5) contains theunknown

  • Now, how do we construct a confidence interval for this example?

    • We know that x̄ =461. Then we can say that the interval 452.18 and 469.82 contains

    • We are 95% confident that the interval 452.18 and 469.82 contains the unknown

      This is a 95% confidence interval for


Individual activity

Individual Activity

The local farmers association in Cass County wants to estimate the mean number of bushels of corn produced per acre in the county. A simple random sample of 13 one-acre plots produced the following results:

  • (in number of bushels per acre): 98, 103, 95, 99, 92, 106, 101, 91, 99, 101, 97, 95, 98.

  • Assume that the population is distributed normally and that

  • Construct a 95% and a 99% confidence interval for the mean number of bushels per acre in the entire county.

  • The local association has been advertising that the mean yield per acre is 100 bushels. Do you think it is justified in this claim?

    Use the 4 steps in the CI Inference Toolbox to complete this activity.


  • Chapter 10

    • Step 1:

      • The population of interest is that number of bushels of corn in Cass County. The parameter that we are interested in is that mean number of bushels of corn produced per acre.

    • Step 2:

      • We are to use confidence interval. The two conditions are:

        • The sample is an SRS from the population of interest

          • Yes, the problem states that this is true

        • The sampling distribution of is distributed normally because it comes from a normally distributed population


    Chapter 10

    • Step 3: Carry out inference procedure

      • CI: estimate MOE

        • 95%:

        • 99%:

    • Step 4:

      • We are 95% confident that our interval will include the true mean of bushels of corn per acre is between, and since 100 is not in the interval we say that the association is not justified in advertising the true mean as 100.

      • They are justified if we look at the 99% confidence interval, however. Why?


    Homework

    Homework

    • Read and take notes on §10.2

    • Do: #24,26,30,32,35


    1 a confidence intervals

    1A Confidence Intervals

    • (.156504,.431496)

    • (.2989,.589)

    • (.299,.3313)

    • (.1955,.49108)

    • (.211,.489)

    • (.138,.396)

    • (.2561,.3959)

    • (.2157,.4955)

    • (.276,.564)

    • (.388,.679)

    • (.2999,.5896)

    • (.115,.365)

    • (.6116,.6326)

    • (.27594,.56406)

    • (.2407,.5148)

    • (.32196,.61204)


    4a confidence intervals

    4A Confidence Intervals

    • (.1488,.4112)

    • (.334,.376)

    • (.1958,.47)

    • (.281,.431)

    • (.6018,.8643)

    • (.1866,.4694)

    • (.364,.656)

    • (.158,.42)

    • (.306,.36)

    • (.41096,.70104)

    • (.152,.388)

    • (.156,.419)

    • (.236,.483)

    • (.238,.522)

    • (.115,.365)

    • (.364,.658)

    • (.221,.259)

    • (.257,.543)

    • (.157,.421)

    • (.157,.421)

    • (.156504,.431496)

    • (.275,.579)

    • (.236,.518)

    • (.2161,.4959)

    • (.193,.467)

    • (.343,.635)

    • (.157,.421)


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