Ch 8. confidence intervals part I

CONFIDENCE INTERVALSCHAPTER 8 – PART 1

OBJECTIVES

By the end of this set of slides, you should be able to:1. Calculate and interpret intervals for estimating a populations mean and a population

proportion2. Understand what the Student t distribution is, and how to use it3. Discriminate between problems applying the normal and the Student’s t distribution4. Calculate the sample size required to estimate a population mean and a population

proportion given a desired confidence level and margin of error

Confidence Interval for a Mean Student t Distribution Confidence Interval for a Proportion

INFERENTIAL STATISTICS

Inferential Statistics: Using sample data to make generalizations about an unknown population

The sample data helps us to make an estimate of a population parameter We can think of a sample statistic as a “point estimate” of the population parameter Example: Imagine you want to figure out what the population mean Ã of students at

Mendocino College is How can we construct a point estimate of the population mean? We can collect a random sample of students and use the sample mean to estimate the population

mean Ã Question: If somebody else collected a random sample of students and calculated the sample

average , would you expect them to get exactly the same estimate as you did?


INFERENTIAL STATISTICS

We can estimate a population parameter using a sample statistic We can estimate the population mean Ã using the sample mean But every time we collect a different sample we will get a different sample statistic, so how do we deal

with this? We need some way to calculate the amount of error we believe our estimate has We can quantify the amount of error in our estimate by calculating the margin of error

The book refers to Margin of Error as the Error Bound for the Mean, or EBM Once we calculate the margin of error, we can think of estimating the population parameter with an

interval

(point estimate – margin of error, point estimate + margin of error) This interval is what we shall call a Confidence Interval


CONFIDENCE INTERVALS

A confidence interval (CI) is going to look something like

CI = (point estimate – margin of error, point estimate + margin of error) A confidence interval states the range within which a population parameter “probably” lies The intervals within which a population parameter is expected to occur is called a

confidence interval The confidence interval for a mean has a very specific form

CI = ( - margin of error, + margin of error) Question: How do we calculate the margin of error (ME)?


THE DISTRIBUTION OF THE SAMPLE MEAN

Recall: The CLT told us that the distribution of the sample mean was


THE DISTRIBUTION OF THE SAMPLE MEAN

The empirical rule tells us that we should expect to see roughly all of our data between 3 standard deviations above and below the population mean, i.e.,

Or rather,

This looks like a confidence interval! So, the margin of error depends upon the standard deviation of the distribution of the

sample mean and the z score.


MARGIN OF ERROR FOR THE MEAN

If we know the population standard deviation then the margin of error is the following: So to calculate the margin of error all we need is

The population standard deviation Ç The sample size n The level The z score corresponding to is called the critical value Let’s look at the following picture to understand what is


MARGIN OF ERROR FOR THE MEAN The level is a number

between 0 and 1 The level tells us the

percentage of area around the mean, which is

So is the percentage of area in the two tales

Every will correspond to a certain z score we call 0 −𝑧𝛼 /2−𝑧𝛼 /2

1−𝛼𝛼 /2 𝛼 /2

Z


MARGIN OF ERROR FOR THE MEANExample: Let , then is

and

=1.96


CALCULATING THE CRITICAL VALUES

Let . Find the critical value You can use your calculator to do this If you want to calculate the value that corresponds to a certain confidence level, then

follow these steps:1. Push 2nd, then DISTR2. Select invNorm() and then push ENTER3. Then enter the following: invNorm(, 0, 1)

a) If you have the TI-84, area: Question: What is the critical value when Solution: invNorm(0.975, 0, 1) = 1.95996


CONFIDENCE INTERVALS FOR THE MEAN

A confidence interval, in general, looks likePoint Estimate Margin of Error

A confidence interval for the mean Ã (when the population standard deviation is known) is

So we calculate , Ç, and n from the sample data How do we specify the level? will be determined by the confidence level (CL)


CONFIDENCE LEVEL

The confidence level (CL) is a number between 0 and 1 The confidence level will always be The confidence level tells us if we were to take repeated samples and calculate many

confidence intervals based on those samples, then we would expect that CL% of the confidence intervals would be “good estimates” that would contain the true value of the population parameter we are trying to estimate

Example: If our CL = 90%, that means that if we took 10 random samples, calculated 10 averages, and then constructed 10 confidence intervals, then we would expect 9 out of the 10 confidence intervals to contain the true population

If our CL = 90% then or 10%


CONFIDENCE INTERVAL FOR THE MEAN

When the population standard deviation is known, the CL% confidence interval is

The three confidence intervals that are used the most often are the 90%, 95%, and 99% confidence intervals The 90% confidence interval is

The 95% confidence interval is



EXAMPLE 1

The Dean wants to estimate the mean numbers of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hoursa) What is the 95% confidence interval for the average number of hours worked per week by the

students Solution: We know the Standard Deviation (it’s given). If we are considering the 95%

confidence interval, then we use:

Further, we’re given that , thus = =

So, the 95% confidence interval is


EXAMPLE 1

The Dean wants to estimate the mean numbers of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hoursb) What is the 99% confidence interval for the average number of hours worked per week by the

students Solution: We know the Standard Deviation (it’s given). If we are considering the 95%

confidence interval, then we use:

Further, we’re given that , thus = ≈ 1.47

So, the 99% confidence interval is


EXAMPLE 1

The Dean wants to estimate the mean numbers of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hoursa) What is the 95% confidence interval for the average number of hours worked per week by the studentsb) What is the 99% confidence interval for the average number of hours worked per week by the students The 95% confidence interval is


Question: What is the difference in these two answers, and can you explain why this difference seems to fit ‘common sense’ in your own words?


EXAMPLE 2

A soda bottling plant fills cans labeled to contain 12 ounces of soda. The filling machine varies and does not fill each can with exactly 12 ounces. To determine if the filling machine needs adjustment, each day the quality control manager measures the amount of soda per can for a random sample of 50 cans. Experience shows that its filling machines have a known population standard deviation of 0.35 ounces. In today’s sample of 50 cans of soda, the sample average amount of soda per can is 12.1 ounces. Construct and interpret a 90% confidence interval estimate for the true population average amount

of soda contained in all cans filled today at this bottling plant. Use the 90% confidence level.


EXAMPLE 2

A soda bottling plant fills cans labeled to contain 12 ounces of soda. The filling machine varies and does not fill each can with exactly 12 ounces. To determine if the filling machine needs adjustment, each day the quality control manager measures the amount of soda per can for a random sample of 50 cans. Experience shows that its filling machines have a known population standard deviation of 0.35 ounces. In today’s sample of 50 cans of soda, the sample average amount of soda per can is 12.1 ounces. Construct and interpret a 90% confidence interval

estimate for the true population average amount of soda contained in all cans filled today at this bottling plant. Use the 90% confidence level.

The information we are given is:

To find the 90% confidence interval, we’ll use

= =

We are 90% confident that the true population mean weight of soda is between 12.017 ounce and 12.181 ounces.That is, the true mean is within the range of about .08 ounces of the mean, 12.1 ounces.


EXAMPLE 2

Suppose the manager at the soda bottling plant wants to be 95% confident that the sample mean is within 0.05 ounces of the true mean. How many bottles will they need to randomly sample? Experience shows that its filling machines have a known population standard deviation of 0.35 ounces. In today’s sample of 50 cans of soda, the sample average amount of soda per can is 12.1 ounces.

If we rewrite the equation for Margin of Error

We know

Then, solving for n =188.2384

So, the manager would need to do a sample of 189 bottles to be 95% confidence that the sample mean is within 0.05 ounces of the true mean.


Education

Ch 8. confidence intervals part I