Estimating a Population MeanEstimating a Population MeanWhen σ is Known: The One – Sample When σ is Known: The One – Sample z Interval For a Population Mean z Interval For a Population Mean

Target Goal:Target Goal:I can reduce the margin of error. I can reduce the margin of error. I can construct and interpret a CI for a I can construct and interpret a CI for a population mean when σ is known.population mean when σ is known.

8.3a

h.w: pg 498: 49 – 52, 55

Now let’s review confidence Now let’s review confidence intervals to estimate the intervals to estimate the mean mean of a population. of a population.

( critical value)n

x z

Confidence intervals for Confidence intervals for when when is known is known

Review: the general formula for a confidence interval for a population mean when . . .

1) x is the sample mean from a random sample,2) the sample size n is large (n > 30), and3) , the population standard deviation, is

known

is These are the properties of the sampling distribution of x.

Is this typically known?This confidence interval is appropriate even when n is small, as long as it is reasonable to

think that the population distribution is normal in shape.

Point estimate

Standard deviation of the statistic

Bound on error of estimation

Cosmic radiation levels rise with increasing altitude, promoting researchers to consider how pilots and flight crews might be affected by increased exposure to cosmic radiation. A study reported a mean annual cosmic radiation dose of 219 mrems for a sample of flight personnel of Xinjiang Airlines. Suppose this mean is based on a random sample of 100 flight crew members. Let = 35 mrems.

Calculate and interpret a 95% confidence interval for the actual mean annual cosmic radiation exposure for Xinjiang flight crew members.

First, state population and parameter of interest.

StateWe want to estimate parameter μ, the mean cosmic radiation for all crew members of Xinjiang Airlines. .

A study reported a mean annual cosmic radiation dose of 219 mrems for a sample of flight personnel of Xinjiang Airlines. Suppose this mean is based on a random sample of 100 flight crew members. Let = 35 mrems.

Plan: Since we know standard deviation, we should use a one-sample z confidence interval for the population to estimate μ.

1)SRS: Data is from a random sample of crew members

2)Normal: Sample size n is large (n > 30)

3) Independent: population all crew members > (10) 100

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

Cosmic Radiation Continued . . .

Let x = 219 mrems

n = 100 flight crew members

= 35 mrems.

What does this mean in context?35

219 1.96 (212.14,225.86)100

We are 95% confident that the actual mean annual cosmic radiation exposure for all Xinjiang flight crew members is between 212.14 mrems and 225.86 mrems.

What would happen to the width of this interval if the confidence level was

90% instead of 95%?

( critical value)x zn

Step 3: D0

Step 4: Conclude

Confidence IntervalsConfidence Intervals

• We would like high confidence and a small margin of error.

• A higher confidence level means that a higher percentage of all samples produce a statistic close to the true value of the parameter. Therefore we want a high level of confidence.

• A smaller margin of error allows us to get closer to the true value of the parameter, so we want a small margin of error.

m zn

So how do we reduce the margin of So how do we reduce the margin of error?error?

• Lower the confidence level (by decreasing the value of z*)

• Lower the standard deviation• Increase the sample size. To cut the margin of

error in half, increase the sample size by four times the previous size.

• You can have high confidence and a small margin of error if you choose the right sample size.

m zn

Ex: Changing the Confidence Ex: Changing the Confidence IntervalInterval

Video screen tension - recall: 90% confidence gave us m = 15.8, CI = (290.5, 322.1),

= 1.645 , stand dev. = 43, = 306.3.

• We want 99% confidence for μ:

Calculate new *z

x*z

.99

2ndVARS:Invnorm(.995) =

2.575

• C I =

• CI =

.005

How much in each tail?

.005

*z

*x zn

43306.3 2.575

2306. .

03 24 8

90% vs. 99% Confidence 90% vs. 99% Confidence

• CI90 = (290.5, 322.1)

• CI99 = (281.5 , 331.1)

• A higher confidence level means that a higher percentage of all samples produce a statistic close to the true value of the parameter.

Changing a Sample SizeChanging a Sample Size

To determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m,

• set the expression for the margin of error to be less than or equal to m and solve for n.

z mn

Ex: Determining Sample Size nEx: Determining Sample Size n

• We want a mean screen tension to be accurate to with in +- 5 mV with 95% confidence.

• How large must n be given σ = 43 ?

For 95%confidence, find

2ndVARS:Invnorm(.975) =

= 1.96

*z

*z

Set m to be at most 5:Set m to be at most 5:

m ≤ 5

5zn

431.96 5

n , solve for n! (round)

284.125n Take n = 285

CAUTION!!CAUTION!!• These methods only apply to certain situations. • In order to construct a level C confidence interval

using the formula ,

the data must be an SRS and we must know the population standard

deviation. eliminate (if possible) any outliers

x zn

• The margin of error only covers random sampling errors.

• Things like under coverage, non-response, and poor sampling designs can cause additional errors.

95% Confident: 95% Confident: What does it mean?What does it mean?

E.g.. We are 95% confident that the mean SAT Math score for all California high school seniors lies between 452 and 470.

Add to notes:• We can say: these numbers were calculated by a

method that gives correct results in 95% off all possible examples.

• Or, we are 95% confident that the true mean SAT score lies in the interval between 452 to 470.

• We can not say: the probability is 95% that the true mean falls between 452 and 470.

(Because the true mean is either in or not in the interval.)

Read pg. 550 - 556Read pg. 550 - 556