Chapter 11 Section 1
Inference for the Mean of a Population
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 _____ and standard deviation _______.
PROBLEM:
If is unknown, then we cannot calculate the standard deviation for the sampling model.
Chapter 11 Section 1
SOLUTION:
We will use _____ (the standard deviation of the sample) to
estimate . Then the __________ __________ of the sample mean is _____. In order to standardize , we subtract its mean and
divide
by its standard deviation.
has the normal distribution N( 0, 1).
s
x
x
Chapter 11 Section 1
If we replace with _____, then the statistic has more variation and no longer has a
normal distribution so we cannot call it z. It has a new distribution called the
_________________________.
The One-Sample t Statistic and the t Distributions (pg. 618)
Draw an ______ of size n from a population that has the normal distribution with
mean _____ and standard deviation _____. The one-sample t statistic
has the ____________________ with _______ degrees of freedom.
t is a standardized value. Like z, t tells us how many standardized units _____ is
Chapter 11 Section 1
When we describe a t distribution we must identify its
__________________________ because there is a different t statistic for
each sample size. The degrees of freedom for the one-sample t statistic is
____________. The degrees of freedom (df) come from the fact that we're
estimating the value of a parameter with a statistic (sx for ). We did not
have to do that with the standard normal distribution (Z), since it only works
with the parameters themselves (no estimating). Whenever we start using
parameter estimates in calculations for other parameters, we have to use
degrees of freedom (if we want unbiased estimators). That's why we divide
by n – 1 in the formula for standard deviation of a sample—we estimated
μX with , and lost one degree of freedom.
s
xChapter 11 Section 1
The t distribution is _________________ about zero and is bell-shaped, but
there is __________ variation so the spread is greater. As n gets larger, the
shape of the t distribution becomes more and more normal (since the value of sx gets closer to ). So for large sample sizes, you could just
use the standard normal distribution.
If the distribution of the population is ______________, then the t statistic
has a t distribution. If the population isn't normal, then this isn't true, BUT it
might be close. We’ll discuss this in more detail later.
Chapter 11 Section 1
Using the t distribution
Finding the area under the t distribution is a bit different from the standard normal ________________.
* Determine the required degrees of freedom.
* Determine the area in the right tail of the distribution.
* In the t table, go down to the row that corresponds to the degrees of freedom, and then go
across to the column that corresponds to the area in the right tail.
Example #1: What critical value t* from Table C satisfies each of the following conditions:
(a) The t distribution with 6 degrees of freedom has probability of 0.025 to the right of t*.
(b) The t distribution with 20 degrees of freedom has probability of 0.80 to the left of t*.
We can construct a confidence interval using the t distribution in the same way we constructed confidence intervals for the z distribution.
Formula:
Remember, the t Table uses the area to the __________ of t*.
Example #2: Using the t table, estimate:
(a) the critical value of t for a 90% confidence interval with df = 17.
(b) the critical value of t for a 98% confidence interval with df = 80.
Example #3: Consider a vending machine that is supposed to
dispense 8 ounces of soft drink. A random sample of 20 cups taken over a one-week period (by Catina Tree) contained the following amounts in ounces:
8.1 7.7 7.9 8.0 7.7 7.8 7.9 8.07.6
7.9 8.0 7.9 7.6 7.5 8.1 7.8 7.8
7.9 8.2 7.5
(a)Estimate the true mean amount dispensed by the machine, using a 95 percent confidence interval.
Example #3: Consider a vending machine that is supposed to dispense 8
ounces of soft drink. A random sample of 20 cups taken over a one-week period (by Catina Tree) contained the following amounts in ounces:
8.1 7.7 7.9 8.0 7.7 7.8 7.9 8.07.6
7.9 8.0 7.9 7.6 7.5 8.1 7.8 7.8
7.9 8.2 7.5
(b) Is there significant evidence to conclude that the machine is dispensing less than 8 ounces?
One-sample t procedures are exactly correct only when the population is
_______________. We assume that the population is
National Fuelsaver Corporation manufactures the Platinum Gasaver, a device they claim “may increase gas mileage by 22%.” Here are the percent changes in gas mileage for 15 identical vehicles, as presented in one of the company's advertisements:
48.3 46.9 46.8 44.6 40.2 38.5 34.6 33.7 28.7 28.7 24.8 10.8 10.4 6.9 -12.4
(b) Explain what “90% confidence” means in this setting.
Chapter 11 Section 1
In a ________________________________, subjects are matched in pairs
and each treatment is given to one subject in each pair. To compare
the responses to the two treatments in a matched pairs design, apply
the ___________________ procedures to the observed differences.
One sample t procedures have two conditions – the sample must
come from an _______, and the population must have a
___________________________. A confidence interval or significance test
Chapter 11 Section 1
The t procedures are strongly influenced by _____________. Always check the data first! If there are __________ and the sample size is
__________, the results will not be reliable. The t procedures are __________ when there are no outliers, especially when the
distribution
Chapter 11 Section 1
When to use t procedures:
If the sample size is less than __________, only use t procedures if the data are close to normal. If the data are clearly nonnormal or if outliers are present, do not use t.
If the sample size is at least __________, only use t procedures if there are no outliers.
If the sample size is at least __________, you may use t procedures, even if the data is skewed.
Example #1: A random sample of 36 U.S. millionaires were asked about their age. The results are shown below.
Describe the shape of the histogram. Can t procedures be used in this situation?
If t procedures can be used, follow the steps and find the confidence interval.
Example #2: According to Nielsen Media Research, during 1998, the average person watched 7 hours and 43 minutes of prime-time television during a week. A random sample of 40 women yielded the following results (in minutes).
Example #1: Workers at a factory asked their supervisor to provide music during their shift. The supervisor wanted to know whether the music really helped improve the worker’s performance. Because all the workers were assigned to different rooms at random, the supervisor randomly selected one room. For one week, he provided music in this room, and for one
week he provided none. He flipped a coin to determine which week to provide music. Afterward, he recorded the productivity of the workers, using the average number of items assembled per day:
Estimate the difference between the workers’ performance in the
Example #1: Workers at a factory asked their supervisor to provide music during their shift. The supervisor wanted to know whether the music really helped improve the worker’s performance. Because all the workers were assigned to different rooms at random, the supervisor randomly selected one room. For one week, he provided music in this room, and for one
week he provided none. He flipped a coin to determine which week to provide music. Afterward, he recorded the productivity of the workers, using the average number of items assembled per day: