Estimating with Confidence

1
Estimating with Confidence

• Lesson 10.1
• Statistical inference provides methods for
  drawing conclusions about a population from a
  sample

2
Law of Large Numbers Draw independent
observations at random from any population with
finite mean µ. Decide how accurately you would
like to estimate µ. As the number of observations
drawn increases, the mean of the observed
values eventually approaches the mean µ of the
population as closely as you specified and then
stays that close. (p.413)
3
A good example of law of large numbers for
proportions is that of tossing a coin to see the
proportion of heads that you toss. The more
times you toss the coin, the closer your
proportion should approach 0.5. With a few
tosses, the proportion will vary a great deal.
4

• Recall the essential facts about the sampling
  distributions of
• The central limit theorem tells us that the mean
  of 500 scores has a distribution that is close
  to normal.
• The mean of this normal sampling distribution is
  the same as the unknown mean µ of the entire
  population.
• The standard deviation of for an SRS of 500
  students is , where s is the
  standard deviation of individual SAT Math scores
  among all California high school seniors.

5
If the standard deviation of SAT math scores for
all California seniors is s 100, then the
standard deviation for is
.
The sampling distribution of the mean score x of
an SRS of 500 California seniors on the SAT Math
test.
6

• The 68-95-99.7 rule says that in 95 of all
  samples, the mean score for the sample will
  be within two standard deviations (2x4.5) of the
  population mean score ?. So the mean of 500
  scores will be within 9 points of ? in 95 of all
  samples.
• Whenever is within 9 points of the unknown ?,
  ? is within 9 points of the observed . This
  happens in 95 of all samples.
• So in 95 of all samples, the unknown ? lies
  between - 9, and 9.

7
The interval between is called a 95
confidence interval for ?.
For the SRS with , we are 95 confident
that the unknown ? is contained in the interval
452 and 470, or our SRS was one of the few
samples which is not within 9 points of the
true ?. Only 5 of all samples give such
inaccurate results.
8

• A level C confidence interval for a parameter has
  two parts
• An interval calculated from the data, usually of
  the form
• A confidence level C, which gives the
  probability that the interval will capture the
  true parameter value in repeated samples.
• The previous example had a C.95 confidence level

9
Note you have no way of knowing whether your
sample is one of the 95 for which the interval
captures ? or one of the unlucky 5. The
statement that we are 95 confident that the
unknown ? lies between 452 and 470 is shorthand
for saying We have arrived at these numbers by a
method that gives correct results 95 of the
time. To sum it up Confidence interval gives
us a range of values that should contain the true
(unknown) value of the population parameter with
a high level of confidence.