AP Statistics

Chapter 22

Inference for

Means

Objectives:

• The t-Distributions

• Degrees of Freedom (df)

• One-Sample t Confidence Interval for Means

• One-Sample t Hypothesis Test for Means

Getting Started

• Now that we know how to create confidence intervals and test

hypotheses about proportions, it’d be nice to be able to do the

same for means.

• Just as we did before, we will base both our confidence interval

and our hypothesis test on the sampling distribution model.

• The Central Limit Theorem told us that the sampling distribution

model for means is Normal with mean μ and standard deviation

SD yn

• All we need is a random sample of quantitative

data.

• And the true population standard deviation, σ.

– Well, that’s a problem…

Getting Started (cont.)

Getting Started (cont.)

• Proportions have a link between the proportion value and the

standard deviation of the sample proportion.

• This is not the case with means—knowing the sample mean tells

us nothing about

• We’ll do the best we can: estimate the population parameter σ

with the sample statistic s.

• Our resulting standard error is

( )SD y

s

SE yn

Getting Started (cont.)

• We now have extra variation in our standard error

from s, the sample standard deviation.

– We need to allow for the extra variation so that it does not

mess up the margin of error and P-value, especially for a

small sample.

• And, the shape of the sampling model changes—the

model is no longer Normal. So, what is the sampling

model?

Gosset’s t

• William S. Gosset, an employee of the Guinness Brewery in

Dublin, Ireland, worked long and hard to find out what the

sampling model was.

• The sampling model that Gosset found has been known as

Student’s t.

• The Student’s t-models form a whole family of related

distributions that depend on a parameter known as degrees of

freedom.

– We often denote degrees of freedom as df, and the model as tdf.

t - Distributions

• When we do not know σ, we substitute the

standard error of for its standard

deviation . The resulting test statistic does

not have a normal distribution, but has a

t – distribution.

ys

n

n

t - Distributions

• Unlike the standard normal distribution, there is

a different t – distribution for each sample size n.

• We specify a particular t – distribution by giving

its degrees of freedom (df).

• df = n-1

• We write the t – distribution with k degrees of

freedom as t(k) for short.

Properties of the

t – Distribution.

• The model of the t – distributions are similar in shape to the

standard normal curve. They are symmetric about zero, single-

peaked, and bell-shaped.

• The spread of the t – distributions is a bit greater than that of the

standard normal distribution.

– The t – distributions have more probability in the tails and less in the

center than does the standard normal.

– This is true because substituting the estimate s for the fixed parameter σintroduces more variation into the statistic.

• As the degrees of freedom k increase, the t(k) model approaches

the N(0,1) curve ever more closely.

– This happens because s estimates σ more accurately as the sample size

increases.

Properties of the

t – Distribution.

• Student’s t-models are unimodal, symmetric, and

bell shaped, just like the Normal.

• But t-models with only a few degrees of freedom

have much fatter tails than the Normal. (That’s

what makes the margin of error bigger.)

Properties of the

t – Distribution

• As the degrees of freedom increase, the t-models look more and more like the Normal.

• In fact, the t-model with infinite degrees of freedom is exactly Normal.

Properties of the

t – Distribution

Finding t-Values By Hand

• The Student’s t-model is different for each value of degrees of freedom.

• Because of this, Statistics books usually have one table of t-model critical values for selected confidence levels.

Finding t-Values By Hand

(cont.)

• The critical value t – use Table T to find, enter

table with probability p or Confidence level C.

The table gives the t* with probability p lying to

its right and probability C lying between –t* and

t*.

Table T - t*

Finding t*

• Suppose you want to construct a 95% confidence

interval for the mean of a population based on an

SRS of size n=12. What critical value t* should

you use?

Solution

• In Table T, consult the

row corresponding to

df = 11. Move across

that row to the entry

that is directly above

95% CI on the bottom

of the chart. The

desired critical value is

t*=2.201.

Finding t-Values By

Hand (cont.)

• Alternatively, we could use technology to find tcritical values for any number of degrees of freedom and any confidence level you need.

• The TI-84 calculator contains an inverse t

function that works much the same as invNorm

function. t* = invT (area to the left of t*, df).

A Confidence Interval for Means?

A practical sampling distribution model for means

When the conditions are met, the standardized sample

mean

follows a Student’s t-model with n – 1 degrees of

freedom.

We estimate the standard error with

y

tSE y

s

SE yn

A Confidence Interval for

Means? (cont.)

• When Gosset corrected the model for the extra

uncertainty, the margin of error got bigger.

– Your confidence intervals will be just a bit wider and

your P-values just a bit larger than they were with

the Normal model.

• By using the t-model, you’ve compensated for

the extra variability in precisely the right way.

• When the conditions are met, we are ready to find the confidence interval for the population mean, μ.

• The confidence interval is

where the standard error of the mean is

• The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size.

1ny t SE y

1*n

t

s

SE yn

A Confidence Interval for

Means? (cont.)

One-sample t-interval for the mean

Assumptions and

Conditions

• Gosset found the t-model by simulation.

• Years later, when Sir Ronald A. Fisher showed

mathematically that Gosset was right, he needed

to make some assumptions to make the proof

work.

• We will use these assumptions when working

with Student’s t.

Assumptions and Conditions

(cont.)

• Independence Assumption:

– The data values should be independent.

– Randomization Condition: The data arise from a random

sample or suitably randomized experiment. Randomly

sampled data (particularly from an SRS) are ideal.

– 10% Condition: When a sample is drawn without

replacement, the sample should be no more than 10% of the

population.

Assumptions and Conditions

(cont.)

• Normal Population Assumption:

– We can never be certain that the data are from a

population that follows a Normal model, but we can

check the

– Nearly Normal Condition: The data come from a

distribution that is unimodal and symmetric.

• Check this condition by making a histogram or

Normal probability plot.

Assumptions and Conditions

(cont.)

– Nearly Normal Condition:

• The smaller the sample size (n < 15 or so), the more closely the data should follow a Normal model.

• For moderate sample sizes (n between 15 and 40 or so), the t works well as long as the data are unimodal and reasonably symmetric.

• For larger sample sizes, the t methods are safe to use unless the data are extremely skewed.

One Sample t Confidence

Interval for Means

• The confidence interval for the population

mean is:

• The critical value t* depends on the particular

confidence level, C, that you specify and on the

number of degrees of freedom, n-1.

1ny t SE y

s

SE yn

Example: t Confidence

Interval

• Some AP Statistics students are concerned about safety near an

elementary school. Though there is a 25 MPH SCHOOL ZONE

sign nearby, most drivers seem to go much faster than that, even

when the warning sign flashes. The students randomly selected

20 flashing zone times during the school year, noted the speeds

of the cars passing the school during the flashing zone times, and

recorded the averages. They found that the overall average speed

during the flashing school zone times for the 20 time periods was

24.6 mph with a standard deviation of 7.24 mph. Find a 90%

confidence interval for the average speed of all vehicles passing

the school during those hours. A histogram of their data follows.

Solution

• Choose method

– One-sample t confidence interval for the mean.

• Why? – because σ is unknown.

• Check conditions

– Randomization – Yes (it is a random selection of days from the year).

– Population Size (10n ≤ N) – Yes (it is reasonable to assume that at least 200 flashing zone times

exist during the school year since the sign flashes both morning and afternoon).

– Sample Size – Yes (n = 20, the distribution shows roughly symmetric and unimodal, no strong

skew or outliers).

• Construct the confidence interval

– df = n-1 = 20-1 = 19 and the 90% t* = 1.729

–

– 90% CI: (21.8,27.4)

• Conclusion

– We are 90% confident that the true mean speed of cars passing the school when the warning sign

is flashing is between 21.8 and 27.4 mph.

Your Turn:

• A SRS of 75 male adults living in a particular

suburb was taken to study the amount of time

they spent per week doing rigorous exercises. It

indicated a mean of 73 minutes with a standard

deviation of 21 minutes. Find the 95%

confidence interval of the mean foe all males in

the suburb.

Solution

• Check conditions

– Randomization – Yes data from SRS.

– Population Size (10n ≤ N) – Yes (it is reasonable to assume that at least 750 males live in

that suburb).

– Sample Size – Yes (n is large, n>40).

• Construct the confidence interval

– df = n-1 = 75-1 = 74 and the 95% t* = 1.990

– 95% CI: (68.17, 77.83)

• Conclusion

– We are 95% confident that the true mean time of rigorous exercise for males in the suburb is

between 68.17 min. and 77.83 min.

* 2173 1.990 73 4.83

75

sy t

n

Ti-83/84 Calculator

• One-sample t Confidence

Interval

– STAT/TESTS/TInterval

• Stats

– :

– Sx:

– n:

– C-Level:

– Calculate:

• One-sample t Confidence

Interval

– STAT/TESTS/TInterval

• Data

– List:

– Freq:

– C-Level:

– Calculate

x

Example: Use TI-84

• Environmentalists, government officials, and vehicle manufacturers are all

interested in studying the auto exhaust emissions produced by motor vehicles.

The major pollutants in auto exhaust from gasoline engines are hydrocarbons,

monoxide, and nitrogen oxides (NOX). The table gives the NOX levels (in

grams per mile) for a random sample of light-duty engines of the same type.

• Construct a 95% confidence interval for the mean amount of NOX emitted by

light-duty engines of this type.

1.28 1.17 1.16 1.08 0.60 1.32 1.24 0.71 0.49 1.38 1.20 0.78

0.95 2.20 1.78 1.83 1.26 1.73 1.31 1.80 1,15 0.97 1.12 0.72

1.31 1.45 1.22 1.32 1.47 1.44 0.51 1.49 1.33 0.86 0.57 1.79

2.27 1.87 2.94 1.16 1.45 1.51 1.47 1.06 2.01 1.39

Solution

• 95% Confidence Interval (1.185, 1.473)

• We are 95% confident that the true mean level of

NOX emitted by this type of light-duty engine is

between 1.185 and 1.473 grams/mile.

More Cautions About Interpreting

Confidence Intervals

• Remember that interpretation of your confidence interval is key.

• What NOT to say:

– “90% of all the vehicles on Triphammer Road drive at a speed between 29.5 and 32.5 mph.”

• The confidence interval is about the mean not the individual values.

– “We are 90% confident that a randomly selected vehiclewill have a speed between 29.5 and 32.5 mph.”

• Again, the confidence interval is about the mean not the individual values.

More Cautions About Interpreting

Confidence Intervals (cont.)

• What NOT to say:

– “The mean speed of the vehicles is 31.0 mph 90% of the time.”

• The true mean does not vary—it’s the confidence interval that would be different had we gotten a different sample.

– “90% of all samples will have mean speeds between 29.5 and 32.5 mph.”

• The interval we calculate does not set a standard for every other interval—it is no more (or less) likely to be correct than any other interval.

More Cautions About Interpreting

Confidence Intervals (cont.)

• DO SAY:

– “90% of intervals that could be found in this way would cover the true value.”

– Or make it more personal and say, “I am 90% confident that the true mean is between 29.5 and 32.5 mph.”

Make a Picture, Make a Picture,

Make a Picture

• Pictures tell us far more about our data set than a list of the data ever could.

• The only reasonable way to check the Nearly Normal Condition is with graphs of the data.

– Make a histogram of the data and verify that its distribution is unimodal and symmetric with no outliers.

– You may also want to make a Normal probability plot to see that it’s reasonably straight.

A Test for the Mean

• The conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval.

• We test the hypothesis H0: = 0 using the statistic

• The standard error of the sample mean is

• When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t model with n – 1 df. We use that model to obtain a P-value.

0

1n

yt

SE y

s

SE yn

One-sample t-test for the mean

Check Conditions

• Randomization Condition – the data come from a

random sample or a randomized experiment.

• Population-size Condition – the sample size is less

than 10% of the population.

• Nearly Normal Condition – the data come from a

unimodal, symmetric, bell-shaped distribution. This can

be verified by constructing a histogram or a normal

probability plot of the data.

Check Conditions

• If the underlying population is approximately normal,

the t-statistic is appropriate regardless of sample size, n.

• For n ≥ 40, the t-statistic is appropriate provided that

there are no outliers.

• For 15 ≤ n < 40, the t-statistic is appropriate provided

that there are no outliers or strong skew.

• For n < 15, the t-statistic is appropriate provided that

the distribution of the data is approximately normal.

One-Sample t Test

Statistic for Means

• The one-sample t statistic is:

• Has the t-distribution with n-1 degrees of

freedom.

y

tSE y

s

SE yn

Example: One-Sample t

Hypothesis Test for Mean

• A recent report on the evening news stated that teens watch an average of 13

hours of TV per week. A teacher at Central High School believes that the

students in her school actually watch more than 13 hours per week. She

randomly selects 25 students from the school and directs them to record their

TV hours for one week. The 25 students reported the following number of

hours: 5 5 6 18 23

13 0 18 20 5

23 11 0 25 22

13 24 6 14 20

23 20 11 22 11

• Is there enough evidence at the 5% significance level to support the teacher’s

claim?

Solution:

• Choose method

– One-sample t hypothesis test

• Check conditions

– Randomization – Yes (stated the students were randomly selected).

– Population Size – Yes (reasonable to think there are more than 250 students in the school).

– Nearly Normal – Yes (n=30, 15<n<40 and no outliers or strong skew).

• Hypothesis

– H0: μ = 13

– Ha: μ > 13

• Calculate the t test statistic:

• Calculate the p-value: df = 29, p-value = P(t > .83) = .2077

• Conclusion

– The p-value of .2077 is not significant at the 5% significance level ( .2077 > .05). Therefore, we fail to reject the

null hypothesis. There is not enough evidence to conclude that the true mean hours of TV viewing per week for

students in this school is greater than 13 hours.

Your Turn:

• The belief is that the mean number of hours per

week of part-time work of high school seniors in

a city is 10.6 hours. Data from a SRS of 50

seniors indicated that their mean number of

hours of part-time work was 12.5 with a standard

deviation of 1.3 hours. Test whether these data

cast doubt on the current belief (α=.05).

Solution

• Check conditions

– Randomization – Yes (stated from SRS).

– Population Size – Yes (reasonable to think there are more than 500 High School seniors students in the

city).

– Sample Size – Yes (no outliers or strong skew).

• Hypothesis

– H0: μ = 10.6

– Ha: μ ≠ 10.6

• Calculate the t test statistic:

• Calculate the p-value: df = 49, p-value = 2P(t > 10.33) = 6.77x10-14

• Conclusion

– The p-value of 6.77x10-14 is significant at the 5% significance level (6.77x10-14 < .05). Therefore, we

reject the null hypothesis. There is enough evidence at the 5% significance level to conclude that the true

mean hours of part-time work by High School seniors in this city is greater than 10.6 hours per week.

12.5 10.610.33

1.3

50

yt

s

n

Significance and

Importance

• Remember that “statistically significant” does

not mean “actually important” or “meaningful.”

– Because of this, it’s always a good idea when we test

a hypothesis to check the confidence interval and

think about likely values for the mean.

Intervals and Tests

• Confidence intervals and hypothesis tests are

built from the same calculations.

– In fact, they are complementary ways of looking at

the same question.

– The confidence interval contains all the null

hypothesis values we can’t reject with these data.

Intervals and Tests

(cont.)

• More precisely, a level C confidence interval contains all of the plausible null hypothesis values that would not be rejected by a two-sided hypothesis text at alpha level 1 – C.

– So a 95% confidence interval matches a 0.05 level two-sided test for these data.

• Confidence intervals are naturally two-sided, so they match exactly with two-sided hypothesis tests.

– When the hypothesis is one sided, the corresponding alpha level is (1 – C)/2.

Sample Size

• To find the sample size needed for a particular confidence level with a particular margin of error (ME), solve this equation for n:

• The problem with using the equation above is that we don’t know most of the values. We can overcome this:

– We can use s from a small pilot study.

– We can use z* in place of the necessary t value.

1n

sME t

n

Sample Size (cont.)

• Sample size calculations are never exact.

– The margin of error you find after collecting the data won’t match exactly the one you used to find n.

• The sample size formula depends on quantities you won’t have until you collect the data, but using it is an important first step.

• Before you collect data, it’s always a good idea to know whether the sample size is large enough to give you a good chance of being able to tell you what you want to know.

Degrees of Freedom

• If only we knew the true population mean, µ, we would find the sample standard deviation as

• But, we use instead of µ, though, and that causes a problem.

• When we use to calculate s, our standard deviation estimate would be too small.

• The amazing mathematical fact is that we can compensate for the smaller sum exactly by dividing by n – 1 which we call the degrees of freedom.

2

.( )y

sn

y

y y

2

instead of y 2

What Can Go Wrong?

• Don’t confuse proportions and means.

Ways to Not Be Normal:

• Beware of multimodality.

– The Nearly Normal Condition clearly fails if a histogram of

the data has two or more modes.

• Beware of skewed data.

– If the data are very skewed, try re-expressing the variable.

• Set outliers aside—but remember to report on these

outliers individually.

What Can Go Wrong?

(cont.)

…And of Course:

• Watch out for bias—we can never overcome the

problems of a biased sample.

• Make sure data are independent.

– Check for random sampling and the 10% Condition.

• Make sure that data are from an appropriately

randomized sample.

What Can Go Wrong?

(cont.)

…And of Course, again:

• Interpret your confidence interval correctly.

– Many statements that sound tempting are, in fact, misinterpretations of a confidence interval for a mean.

– A confidence interval is about the mean of the population, not about the means of samples, individuals in samples, or individuals in the population.

What have we learned?

• Statistical inference for means relies on the same concepts as for proportions—only the mechanics and the model have changed.

– What we say about a population mean is inferred from the data.

– Student’s t family based on degrees of freedom.

– Ruler for measuring variability is SE.

– Find ME based on that ruler and a student’s t model.

– Use that ruler to test hypotheses about the population mean.

What have we learned?

• The reasoning of inference, the need to verify

that the appropriate assumptions are met, and the

proper interpretation of confidence intervals and

P-values all remain the same regardless of

whether we are investigating means or

proportions.