Estimating with Confidence:
Suppose I want to
know how often teenagers go to the movies.
Specifically, I want to know how many times per month a typical teenager
(ages 13 through 17) goes to the movies.
Suppose I take an
SRS of 100 teenagers and calculate the sample mean to be _{}.
The sample mean is
an unbiased estimator of the unknown population mean _{}, so I would estimate the population mean to be approximately
2.1. However, a different sample would have
given a different sample mean, so I must consider the amount of variation in
the sampling model for _{}.
▪
The
sampling model for _{} is approximately
normal.
▪
The
mean of the sampling model is _{}.
▪
The
standard deviation of the sampling model is _{} assuming the
population size is at least 10n.
_{} _{} _{}
Suppose we know
that the population standard deviation is _{}. Then the standard
deviation for the sampling model is _{}
Then 95% of our
samples will produce a statistic _{} that is between _{} and _{}.
Therefore in 95% of
our samples, the interval between _{} and _{} will contain the
parameter _{}.
The margin of error is 0.10.
For our sample of
100 teenagers, _{}. Because the margin
of error is 0.10, then we are 95% confident that the true population mean lies
somewhere in the interval _{}, or [2.0, 2.2].
The interval [2.0,
2.2] is a 95% confidence interval
because we are 95% confident that the unknown _{} lies between 2.0 and
2.2.
Answers to
Exercises:
#10.1
a) 44% to 50%
b) The
statistic 47% came from a small sample; it is a good estimate, but we do not
expect it to be exactly correct.
c) In
95% of all samples, the true value will be within 3% of the statistic.
#10.2
a) The
mean of the sampling model is 280 and the standard deviation is _{}, assuming the population is at least 10,000 people.
b) N(280,
1.9)
274.3 276.2
278.1 280 281.9
283.8 285.7
c) 2
standard deviations, or 3.8 points
e) 95%
#10.3
NO! 95% is not a probability or a
proportion; it is a confidence level. We
are 95% confident that the unknown parameter
_{} lies between 267.8 and
276.2. Also, the parameter is a mean,
not an individual score.
Start with sample data. Compute
an interval that has probability C of containing the true value of the
parameter. This is called a level C confidence interval.
How do we construct confidence intervals?
Since the sampling model of the sample mean _{} is approximately
normal, we can use normal calculations to construct confidence intervals.
For a 95% confidence interval, we want the interval corresponding to the
middle 95% of the normal curve.
For a 90% confidence interval, we want the interval corresponding to the
middle 90% of the normal curve.
And so on…
If we are using the standard normal curve, we want to find the interval
using zvalues.
Suppose we want to find a 90% confidence interval for a standard normal
curve. If the middle 90% lies within our
interval, then the remaining 10% lies outside our interval. Because the curve is symmetric, there is 5%
below the interval and 5% above the interval.
Find the zvalues with area 5% below and 5% above.
These zvalues are denoted _{}. Because they come
from the standard normal curve, they are centered at mean 0.
_{} is called the upper p critical value, with
probability p lying to its right under the standard normal curve.
To find p, we find the complement of C and divide it in half, or find _{}.
For a 95% confidence interval, we want the zvalues with upper p
critical value 2.5%.
For a 99% confidence interval, we want the zvalues with upper p
critical value 0.5%.
Remember that zvalues tell us how many standard
deviations we are above or below the mean.
To construct a 95% confidence interval, we want to find the values 1.96
standard deviation below the mean and 1.96 standard deviations above the mean,
or _{}.
Using our sample data, this is _{}, assuming the population is at least 10n.
In general, to construct a level C confidence interval using our sample
data, we want to find _{}.
The estimate for _{} is _{}.
The margin of error is _{}. Note that the margin
of error is a positive number. It
is not an interval.
We would like high confidence
and a small margin of error.
A higher confidence level means 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.
So how do we reduce the margin
of 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.
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.
_{}
CAUTION!!
These methods only apply to certain situations. In order to construct a level C confidence
interval using the formula _{}, for example, the data must be an SRS and we must know the
population standard deviation. Also, we
want to eliminate (if possible) any outliers.
The margin of error only covers random sampling errors. Things like undercoverage, nonresponse, and
poor sampling designs can cause additional errors.
Tests of Significance:
Example 10.7
Diet colas use
artificial sweeteners to avoid sugar.
Colas with artificial sweeteners gradually lose sweetness over
time. Manufacturers therefore test new
colas for loss of sweetness before marketing them. Trained tasters sip the cola along with
drinks of standard sweetness and score the cola on a “sweetness score” of 1 to
10. The cola is then stored for a period
of time, then each taster scores the stored cola. This is a matched pairs experiment. The reported data is the difference in
tasters’ scores. The bigger the
difference, the bigger the loss in sweetness.
2.0 0.4 0.7 2.0 0.4
2.2 1.3 1.2 1.1 2.3
The sample mean _{} indicates a small loss
of sweetness.
Consider that a
different sample of tasters would have resulted in different scores, and that
some variation in scores is expected due to chance.
Does the data provide good evidence that the
cola lost sweetness in storage?
To answer that
question, we will perform a significance
test.
1. Identify
the parameter.
The
parameter of interest is , the mean loss in sweetness.
2. State the null hypothesis.
There
is no effect or change in the population.
This is the statement we are trying to find evidence against.
The
cola does not lose sweetness.
_{}
State the alternative hypothesis.
There
is an effect or change in the population.
This
is the statement we are trying to find evidence for.
The cola does lose sweetness.
_{}
3. Calculate a statistic to estimate the
parameter.
Is
the value of the statistic far from the value of the parameter? If so, reject the null hypothesis. If not, accept the null hypothesis.
4. Calculate the Pvalue.
If it is small, your result is statistically significant.
Suppose the
individual tasters’ scores vary according to a normal distribution with mean _{} and _{}. We want to test the
null hypothesis so we assume _{}.
So the sampling model for _{} is approximately
normal with mean _{} and standard deviation
_{}
N(0, 0.316)
0.9 0.6
0.3 0 0.3
0.6 0.9
Our sample mean, _{}, was 1.02. Assuming
that the null hypothesis is true, what is the probability of getting a result
at least that large?
Normalcdf ( 1.02, 1E99, 0, 0.316 ) = 0.0006
The probability to the right of _{} is called the Pvalue.
The Pvalue is 0.0006,
meaning that we would only expect to get this result in 6 out of 10,000
samples. This is very unlikely, so we
will reject the null hypothesis in favor of the alternative hypothesis and
conclude that the cola actually did lose sweetness.
If the Pvalue is small we say that our result is statistically significant.
The smaller the Pvalue, the stronger the evidence provided by the data.
How small is small enough?
Compare the Pvalue to the value of the significance level This
value is usually predetermined.
If the Pvalue is as small or
smaller than , we say that
the data are statistically significant
at level .
Hypotheses can be onesided or twosided.
▪
_{} is a onesided
hypothesis because we are only looking at one direction, greater than.
▪
_{} is a twosided
hypothesis because we are looking at two directions, greater than and less
than.
Inference as Design:
If we use the
results of a significance test to make a decision, then we either reject the
null hypothesis in favor of the alternative hypothesis, or we accept the null
hypothesis. This is called acceptance sampling.
We hope that our
decision will be correct, but it is possible that we make the wrong
decision. There are two ways to make a
wrong decision:
¯
We can
reject the null hypothesis when in fact it is true. This is called a Type I Error.
¯
We can
accept (fail to reject) the null hypothesis when in fact it is false. This is called a Type II Error.


Truth
about the population 



H_{0}
is true 
H_{0}
is false 
Decision
based on sample 
Reject H_{0} 
Type I Error p = 
Correct Decision p = 1 – 
Accept H_{0} 
Correct Decision 
Type II Error p = 
We are interested
in knowing the probability of making a Type I Error and the probability of
making a Type II Error.
A Type I Error occurs if we reject the null hypothesis when it is in
fact true. When do we reject the null hypothesis? When we assume that it is true and find that
the statistic of interest falls in the rejection region. The probability that the statistic falls in
the rejection region is the area of the shaded region, or .
OneSided Test
Therefore the
probability of a Type I Error is equal to the significance level of a fixed level test. The probability that the test will reject the
null hypothesis H_{0} when in fact H_{0 }is true is .
A Type II Error
occurs if we accept (or fail to reject) the null hypothesis when it is in fact
false. When do we accept (or fail to reject) the null hypothesis? When we assume that it is true and find that
the statistic of interest falls outside the rejection region.
However, the
probability that the statistic falls outside the rejection region is NOT the
area of the unshaded region. Think about it… If the null hypothesis
is in fact false, then the picture is NOT CORRECT… it is off center.
TwoSided Test
To calculate the
probability of a Type II Error, we must find the probability that the statistic
falls outside the rejection region (the unshaded area) given that the mean is
some other specified value.
lower upper
critical critical
value value
lower upper
critical critical
value value
The probability of
a Type II Error tells us the probability of accepting the null hypothesis when
it is actually false.
The complement of
this would be the probability of not
accepting (in other words rejecting)
the null hypothesis when it is actually false.
To calculate the
probability of rejecting the null hypothesis when it is actually false,
compute 1 – P(Type II Error),
or (1 – . This is called
the power of a significance test.