Estimates and sample sizes Chapter 6

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Estimates and sample sizes Chapter 6

• Prof. Felix Apfaltrer
• fapfaltrer_at_bmcc.cuny.edu
• OfficeN763
• Phone [phone]
• Office hours
• Tue, Thu 10am-1130 am

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Inferential Statistics
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Inferential Statistics
Population?

• 1. Involves
• Estimation
• Hypothesis Testing
• 2. Purpose
• Make Decisions about Population Characteristics

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Inference Process
Estimates tests
Population
Sample statistic (X)
Sample
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Point Estimates
Estimation
Point
Confidence
Estimate
Interval
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Point Estimates 2
Estimate Population
with Sample
Parameter...
Statistic
Mean
?
?
x

Proportion
p

p
2
2
Variance

s
?
Differences
?
- ?
?
x
-
?
x
1
2
1
2
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Samples and estimation

• Assumptions
• Samples should be simple random samples
• Binomial distribution requirements satisfied
• Normal distribution approximation ok.

Notation p population proportion p x/n
sample proportion of x successes in a sample of
size n q 1 - p sample proportion of
failures in a sample of size n
Definition A point estimate is a single value
used to estimate a population parameter.

• Example (survey about photo-cop)
• 829 Minnesotans surveyed
• 51 opposed to cameras used for issuing traffic
  tickets
• Point estimate p0.51

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Example Point Estimate
A study found the body temperatures of 106
healthy adults. The sample mean was 98.2 degrees
and the sample standard deviation was 0.62
degrees. Find the point estimate of the
population mean ? of all body temperatures.
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Confidence Intervals
Estimation
Point
Confidence
Estimate
Interval
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Confidence Interval Definition
Definition A confidence interval is a range
(or an interval) of values used to estimate the
true value of the population parameter. The
confidence level gives us the success rate of the
procedure used to construct the confidence
interval.
Confidence interval
Confidence limit (upper)
Confidence limit (lower)
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Confidence intervals (cont)

• Definition
• A confidence interval (CI) is a range of values
  used to estimate the true value of a population
  parameter.

• Confidence level associated to confidence
  interval.
• success rate for estimate to be in interval
• given as probability 1- ?
• For example confidence level of 0.95, ?0.05
• confidence level of 0.99, ?0.01

• Example (survey about photo-cop)
• 829 Minnesotans surveyed
• 51 opposed to cameras used for issuing traffic
  tickets
• The 0.95 confidence interval estimate of the
  population proportion p against photo-cop
  is0.476ltplt0.544

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Confidence intervals(cont)

• Definition
• A critical values are the numbers
• -z?/2 and z?/2 that separate the areas ?/2 on
  the left and right tails from the center area 1-
  ?.

• Example (survey about photo-cop)
• For confidence level of 0.95, ?0.05, so
  ? 0.025
• In table A-2, -z?/2 - 1.96
• and z?/2 1.96
• short visit from Navy.
• Calculate P(Xgt308 days).
• What does this suggest?
• Premature if below 4. Find length.
• Q11 SAT ?998 ? 202
• College requires 1100 minimum.
• Find percentage satisfying requirement.
• Find 40 percentile. Why does college
• not ask top 40?

Confidence level ? z?/2 90 0.10
1.645 95 0.05 1.96 99
0.01 2.575
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Level of Confidence
The confidence level is often expressed as
probability 1 - ?, where ? is the complement of
the confidence level. For a 0.95(95) confidence
level, ? 0.05. For a 0.99(99) confidence
level, ? 0.01.
Margin of Error
is the maximum likely difference observed
between sample mean x and population mean µ, and
is denoted by E.
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Confidence Interval (or Interval Estimate) for
Population Mean µ when ? is known
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Procedure for Constructing a Confidence Interval
for µwhen ? is known
1. Verify that the required assumptions are met.
2. Find the critical value z??2 that corresponds
to the desired degree of confidence.
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Round-Off Rule for Confidence Intervals Used to
Estimate µ
1. When using the original set of data, round
the confidence interval limits to one more
decimal place than used in original set of data.
2. When the original set of data is unknown and
only the summary statistics (n,x,s) are used,
round the confidence interval limits to the same
number of decimal places used for the sample mean.
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Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval for µ.
n 106 x 98.20o s 0.62o ? 0.05 ??/2
0.025 z ?/ 2 1.96
98.08o lt ? lt 98.32o
98.20o 0.12 lt ? lt 98.20o 0.12
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Sample Size for Estimating Mean ?
When finding the sample size n, if the Formula
above does not result in a whole number, always
increase the value of n to the next larger whole
number.
If ? is unknown a) ? ? range/4, or b)
calculate the sample standard deviation s and use
it in place of ?, or c) Estimate the value of ?
by using the results of some other study that was
done earlier.
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Example Assume that we want to estimate the
mean IQ score for the population of statistics
professors. How many statistics professors must
be randomly selected for IQ tests if we want 95
confidence that the sample mean is within 2 IQ
points of the population mean? Assume that ?
15, as is found in the general population.
? 0.05 ??/2 0.025 z ?/ 2 1.96 E
2 ? 15
With a simple random sample of only 217
statistics professors, we will be 95 confident
that the sample mean will be within 2 points of
the true population mean ?.
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Many Samples Have Same Interval
?X? ? Z??x
?X
?
?1.65??x ?2.58??x
?-2.58??x ?-1.65??x
?1.96??x
?-1.96??x
90 Samples
95 Samples
99 Samples
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? Not KnownAssumptions
Use Student t distribution
1. The Student t distribution is different for
different sample sizes (see Figure for the cases
n 3 and n 12). 2. The Student t distribution
has the same general symmetric bell shape as the
normal distribution but it reflects the greater
variability (with wider distributions) that is
expected with small samples. 3. The Student t
distribution has a mean of t 0 (just as the
standard normal distribution has a mean of z
0). 4. The standard deviation of the Student t
distribution varies with the sample size and is
greater than 1 (unlike the standard normal
distribution, which has a s 1). 5. As the sample
size n gets larger, the Student t distribution
gets closer to the normal distribution.
1) The sample is a simple random sample. 2)
Either the sample is from a normally
distributed population, or n gt 30.
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If the distribution of a population is
essentially normal, then the distribution of
x - µ
t
s
n
is essentially a Student t Distribution for all
samples of size n, and is used to find
critical values denoted by t?/2.
Degrees of Freedom (df ) corresponds to the
number of sample values that can vary after
certain restrictions have been imposed on all
data values
df n 1 in this section.
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Confidence Interval for the Estimate of EBased
on an Unknown ? and a Small Simple Random Sample
from a Normally Distributed Population
E is the margin of Error and ta/2 has n-1
degrees of freedom.
t?/2 found in Table A-3
t?/2 found in Table A-3
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Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval for µ.
n 106 x 98.20o s 0.62o ? 0.05 ??/2
0.025 t ?/ 2 1.96
98.20o 0.1195 lt ? lt 98.20o 0.1195
98.08o lt ? lt 98.32o
Based on the sample provided, the confidence
interval for the population mean is 98.08o lt ?
lt 98.32o. The interval is the same here as in
above example, but in some other cases, the
difference would be much greater.
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Important Properties of the Student t
Distribution
1. The Student t distribution is different for
different sample sizes (see Figure 6-5 for the
cases n 3 and n 12). 2. The Student t
distribution has the same general symmetric bell
shape as the normal distribution but it reflects
the greater variability (with wider
distributions) that is expected with small
samples. 3. The Student t distribution has a mean
of t 0 (just as the standard normal
distribution has a mean of z 0). 4. The
standard deviation of the Student t distribution
varies with the sample size and is greater than 1
(unlike the standard normal distribution, which
has a ??? 1). 5. As the sample size n gets
larger, the Student t distribution gets closer to
the normal distribution.
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Using the z Normal and t Distribution
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Example Data Set 14 in Appendix B includes the
Flesch ease of reading scores for 12 different
pages randomly selected from J.K. Rowlings Harry
Potter and the Sorcerers Stone. Find the 95
interval estimate of ?, the mean Flesch ease of
reading score. (The 12 pages distribution
appears to be bell-shaped.)
x 80.75 s 4.68 ? 0.05 ?/2 0.025 t?/2
2.201
80.75 2.97355 lt µ lt 80.75 2.97355 77.77645
lt ? lt 83.72355 77.78 lt ? lt 83.72
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Finding the Point Estimate and E from a
Confidence Interval
Point estimate of µ x (upper confidence
limit) (lower confidence limit)
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Margin of Error E (upper confidence limit)
(lower confidence limit) 2