When σ is Unknown When σ is Unknown The One – Sample Interval For a The One – Sample Interval For a Population Mean Population Mean

Target Goal:Target Goal:I can construct and interpret a CI for a population I can construct and interpret a CI for a population mean when σ is unknown.mean when σ is unknown.I can carry out the 4 step process for confidence I can carry out the 4 step process for confidence intervals.intervals.

8.3b

h.w: pg. 518: 57, 59, 63 (4 step, show work. Do not say “given in the stem”.)

Inference for the Mean of a Inference for the Mean of a PopulationPopulation

• If our data comes from a simple random sample (SRS) and the sample size is sufficiently large, then we know that the sampling distribution of the sample means is approximately normal with

• mean μ and

• standard deviation n

PROBLEM:PROBLEM:

• If σ is unknown, then we cannot calculate the standard deviation for the sampling model.

• We must estimate the value of σ in order to use the methods of inference that we have learned.

SOLUTION:SOLUTION:

• We will use s (the standard deviation of the sample) to estimate σ.

• Then the standard error of the sample mean is (referred to as SE or SEM).

s

n

• Recall: when we know σ, we base confidence intervals and tests for μ on the one sample z statistic.

•

has the normal distribution N( 0, 1)

/

xz

n

PROBLEM:PROBLEM:

• When we do not know σ, we replace for .

• The statistic that results has more variation and no longer has a normal distribution, so we cannot call it z.

• It has a new distribution called the t distribution .

s nn

One-Sample t StatisticOne-Sample t Statistic

has the t distribution with n-1 degrees of freedom.

t is a standardized value.

• Like z, t tells us how many standardized units is from the mean μ.

/

xt

s n

x

• When we describe a t distribution we must identify its degrees of freedom because there is a different t statistic for each sample size.

• The degrees of freedom (df) for the one-sample t statistic is (n – 1).

• The t distribution is symmetric about zero and is bell-shaped, but there is more variation so the spread is greater.

As the degrees of freedom increase, the t distribution gets closer to the normal distribution, since s gets closer to σ.

There is more area in the tails of t distributions.

As df increases, the distribution approaches “normal”.

t curve for 2 df

z curve

Why is the z curve taller than the t curve for 2 df?

Ex. Using the “t-table”Ex. Using the “t-table”

• Table B is used to find critical values t* with known probability to its right!

What critical value t* would you use for a t dist with 18 df , having a probability 0.90 to the left of t*?

.90 corresponds with upper tail probability of .10 so,

t* = 1.330

Try: invT(.90, 18)

• Using Table B to Find Critical t* Values

Suppose you want to construct a 95% confidence interval for the mean µ of a Normal population based on an

SRS of size n = 12. What critical t* should you use?

Estim

atin

g a P

opu

lation M

ea

nE

stima

ting

a Po

pulation

Me

an

In Table B, we consult the row corresponding to df = n – 1 = 11.

The desired critical value is t * = 2.201.

We move across that row to the entry that is directly above 95% confidence level.

Upper-tail probability p

df .05 .025 .02 .01

10 1.812 2.228 2.359 2.764

11 1.796 2.201 2.328 2.718

12 1.782 2.179 2.303 2.681

z* 1.645 1.960 2.054 2.326

90% 95% 96% 98%

Confidence level C

t Confidence Intervals and t Confidence Intervals and TestsTests

• We can construct a confidence interval using the t distribution in the same way we constructed confidence intervals for the z distribution.

• A level C confidence interval for μ when σ is not known is:

• Remember, the t Table uses the area to the RIGHT of t*.

• t* is the upper (1-C)/2 critical value for the t(n-1) distribution

*s

x tn

Ex. Auto PollutionEx. Auto Pollution (C.I. for one sample t-test). (C.I. for one sample t-test).

• Read as class bottom page 509

Construct a 95% C.I. Construct a 95% C.I. for the for the mean amount of NOX emitted.mean amount of NOX emitted.

Step 1: State - Identify the population of interest and the parameter you want to draw a conclusion about.

• We want to estimate the true mean amount µ of NOX emitted by all light duty engines of this type at a 95% confidence level.

Step 2. Step 2.

• Choose the appropriate inference procedure. Plan- Since σ is not known, we should construct a one-sample t interval for µ if the conditions are met.

Verify the conditions. (Plot data when possible.)

Plot data: Plot data:

(statplot,data L1,data axis: X)If the data are normally distributed, the normal

probability plot will be roughly linear..

• Random: The data come from a “random sample” of 40 engines from the population of all light duty engines of this type.

• Normal: We don’t know whether the population is normal but because the sample size, n = 40 , is large (at least 30), the CLT tells us the distribution is approximately normal.

• Independent: We are sampling without replacement, so we need to check the 10% condition; we must assume that there at least 10(40) = 400 light duty engines of this type.

Step 3.Step 3. Carry out the inference procedure.Carry out the inference procedure.DO -DO -

• Given = , df =

• There is no row for 39,use the more conservative df = 30 which is

• t* = 2.042

(this gives a higher critical value and wider c.i).

• The 95% Confidence interval for μ is

x 1.2675 40 -1 = 39.

*s

x tn

.3332

1.2675 2.04240

1.2675 0.1076

= (1.1599, 1.3751)

0.3332 / ,xs g ml

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

• We are 95% confident that the true mean level of NOX emitted by all light duty engines is between 1.1599 grams/mile and 1.3751 grams/mile.

• Since the entire interval exceeds 1.0, it appears that this type of engine violates EPA limits.

•

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