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Ch9 Estimation and Confidence Intervals, Adjust the margin of error in…
Ch9 Estimation and Confidence Intervals
1 Point Estimate-Ratio（比例尺度）
1-1 Mean（母體平均數的點估計）
(P288/P271)
The mean of this sample is a point estimate of the population mean. (P271/P287)
The sample mean xbar.
(P288/P272)
1-2 Standard deviation (母體標準差的點估計）
(P288/P272)
s, the sample standard deviation, is a point estimate of σ, the population standard deviation. (P288/P272)
1-3 Confidence Intervals（信賴區間）
(P288/272)
Purpose: While expecting the point estimate to be close to the population parameter, we would like to measure how close it really is. (P288/P272)
Calculation. (P288/P272)
1-3-1 Use sample data to estimate μ with x bar and the population standard deviation (σ) is
known.
(P288/272)
Assume
: a
Know
the value of the population standard deviation
. b Based on
the central limit theorem**. (P288/272)
Known
: the sampling distribution of the sample mean is normally distributed with a mean of
μ
and a standard deviation
σ/square root n
(the standard error). (P289/272)
(1)
Find Z value
(P289/P273)
(9-1) (P291/P274)
(2)
A Computer Simulation
(P292/P277)
How do we determine a confidence interval?The width of the interval is determined by two factors: (P290/P274)
(1) the level of confidence. (P290/P274)
(2) the size of the standard error of the mean(P290/P274)
The size of the standard error is affected by two values. (P290/P274)
1 The variability in the population, usually estimated by the sample standard deviation of the population, s. (P290/P274, P316/P299)
2 The number of observations in the sample, n.(P290/P274, P316/P299)
1-3-2 Use sample data to estimate μ with x bar and the population standard deviation (σ) is
unknown.
(P296/P281)
2 Substitute
the sample standard deviation (s)
for the population standard deviation (σ). (P288/P272, P298/P281)
(1)
Find t value
(P289/P273)
(9-2) (P298/P282)
3 Use the t-distribution.
(P298/P281, P298/P282)
The t-distribution is a continuous probability distribution, with many similar characteristics to the z-distribution. (P298/P281)
4 More spread out and flatter at the center than the standard normal distribution.
Chart 9-1 (P298/P281)
As the sample size increases, however, the t-distribution approaches the standard normal distribution because the errors in using s to estimate σ decrease with larger samples.
1 more item...
(P298/P282)
1 A continuous distribution.
2 Bell shaped and symmetrical.
3 Not one t-distribution, but rather a family of t-distributions, depending on the number of degrees of freedom. (P316/P299)
(P298/P281)
s is the sample standard deviation and the point estimate of the population standard deviation, σ. (P298/P281)
1
Assume
the sampled population is normal or approximately normal.
may be questionable for
*small sample sizes
(P298/P282)
1-5 Small Conclusion (P299/P282~283)
If we know the population standard deviation, then we use z. If we do not know the population standard deviation, then we must use t. Chart 9–3 summarizes the decision-making process.
(P299/P283)
1-4 Sample size
(P309/P293)
1-4-1 For a
finite
population
Adjust formula (9–2) as follows:(P309/P294)
1-4-2 Sample Size to Estimate a Population Mean (P321/P296)
Three factors that determine the sample size:(P316/P300)
2 The desired level of confidence.
1 The margin of error, E.
3 The variation in the population.
(9-5)(P321/P296)
n is the size of the sample.z is the standard normal z-value corresponding to the desired level of confidence. σ is the population standard deviation.E is the maximum allowable error.(P321/P296)
The result of this calculation is not always a whole number. When the outcome is not a whole number, the usual practice is to round up any fractional result to the next whole number. (無條件進位法)(P321/P296)
0 Intro (P287/P270)
Focus on sample statistics.
That means use sample information to estimate the values of unknown population parameters.
0-1 Point Estimates 點估計 (P287/P270)
Definition: A point estimate is a single (point) value computed from sample information.
(P287/P270)
EX: the sample mean and sample proportion, estimate the population mean and population proportion. (P287/P270)
A point estimate is a single statistical value used to estimate a population parameter. (P288/P271)
The statistic, computed from sample information, that estimates a population parameter.
(P288/P271)
0-2
Confidence Interval （信賴區間) The level of confidence (信賴水準)
(P271/P287, P288/P272)
0-2-1 Definition: Use the sampling distribution to determine a level of confidence that the population mean is within the interval.
(P271/P287)
A range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence (信賴水準).
(P288/P272)
Definition: A confidence interval is a range of values within which the population parameter is expected to occur.
(P316/299)
Review: Ch8 An interval estimate of the population mean. (P271/P287)
0-2-2 Sample Size
. (P271/P287)
Sample size is crucial in knowing the distribution of a sample statistic is
normal
. (P271/P287)
Sample size is also important in controlling the size of
the standard error
and
the accuracy of an interval estimate
. (P271/P287)
Larger
sample sizes are related to smaller standard errors and
increased accuracy
of an estimate. (P271/P287)
Interval estimates will be
narrower
. (P271/P287)
What if the sampled population is not very large? (P309/P293)
Need to make some adjustments in the way we compute the standard error of the sample means and the standard error of the sample proportions. (P309/P293)
For a finite population, where
the total number of objects or individuals is N
and
the number of objects or individuals in the sample is n
, we need to adjust the margin of error in the confidence interval formulas.(P309/P293)
Why is it necessary to apply a factor, and what is its effect? Logically, if the sample is a substantial percentage of the population, the estimate of the population parameter is more precise.
A population that has a fixed upper bound is finite.(P309/P293)
How to choose an appropriate sample size? (P311/P295)
Based on three factors
2 The level of confidence desired, for example, 95%.
In working with confidence intervals, logically choose relatively high levels of confidence such as 95% and 99%.
Larger sample sizes (and more time and money to collect the sample) correspond with higher levels of confidence.
Use a z-statistic.
3 The variation or dispersion of the population being studied. (the population standard deviation)
If the population is widely dispersed, a large sample is required to get a good estimate.
if the population is concentrated (homogeneous), the required sample size to get a good estimate will be smaller.
Not know the population standard deviation, how to estimate the population standard deviation...(P312/P296)
2 Use a comparable study.
Use this approach when there is an estimate of the standard deviation from another study.
3 Use a range-based approach.
To use this approach, we need to know or have an estimate of the largest and smallest values in the population.
1 Conduct a pilot study.
The most common method.
1 The margin of error the researcher will tolerate.
The margin of error. It is designated as E and is the amount that is added and subtracted to the sample mean (or sample proportion) to determine the endpoints of the confidence interval.
The margin of error is the amount of error we are willing to tolerate in estimating a population parameter.
A trade-off between the margin of error and sample size.
2 more items...
2 Nominal (名目尺度)
2-1 Proportion (比例)
(P305/P289)
The fraction, ratio, or percent indicating the part of the sample or the population having a particular trait of interest.
(P305/P289, P316/P300)
p, a sample proportion, is a point estimate of π, the population proportion. (P288/P272)
(9-3) (P305/P289)
Let p represent the sample proportion, x the number of “successes,” and n the number of items sampled.
A sample proportion, p, is found by x, the number of successes, divided by n, the number of observations. ( P316/P300)
2-2 The standard error of the sample proportion
(P306/P290)
2-3 Confidence Intervals（信賴區間）
(P288/272)
Requirements:
1 The binomial conditions have been met.
(P305/P289)
b. There are only two possible outcomes.
c. The probability of a success remains the same from one trial to the next.
a The sample data are the number of successes in n trials.
d. The trials are independent.
2
The values nπ and n(1 − π) should both be greater than or equal to 5
. (P305/P290)
Calculation
(P305/P290)
(9-4) (P306/P290)
The margin of error for the confidence interval.(信賴區間的誤差範圍)
(P306/P290)
2-4 Sample Size
2-4-1 For a
finite
population
Adjust formula (9–4) as follows:(P309/P294)
2-4-2 Sample Size to Estimate a Population Proportion. (P313/P296)
Three factors that determine the sample size:(P316/P300)
2 The desired level of confidence.
3 A value for π to calculate the variation in the population.(The variation or dispersion of the population being studied.)
1 The margin of error, E.
(9-6)(P313/P297)
n is the size of the sample.z is the standard normal z-value corresponding to the desired level of confidence.π is the population proportion.E is the maximum allowable error.
If a reliable value cannot be determined with a pilot study or found in a comparable study, then a value of
.50
is used for π. (P313/P298)
π(1 − π) has the largest value using 0.50 and, therefore, without a good estimate of the population proportion,
using 0.50 as an estimate of π overstates the sample size
.
Using a larger sample size will not hurt the estimate of the population proportion.
Adjust the margin of error in formulas：Finite-population correction factor. （有限母體校正因子）(P309/P293)
