Download Week 6 Vocabulary Summary Section 6.1– Confidence Intervals for

Week 6 Vocabulary Summary
Section 6.1– Confidence Intervals for the Mean (Large Samples)
Point Estimate: is a single value estimate for a population parameter.
The most unbiased point estimate of the populations mean µ (mu) is the
mean x xbar (sample mean)
Interval Estimate: is an interval, or range of values, used to estimate a population parameter.
To find an interval estimate, + - the statistic by a margin of error
Level of Confidence: c is the probability that the interval estimate contains the population parameter.
The Central Limit Theorem States: ( page 272)
1. If samples of size n, where n >= 30 are drawn from a population with a mean of µ and a
standard deviation of  sigma., then the sampling distribution of the sample means
approximates a normal distribution. The greater the sample size, the better the
approximation.
2. If the populations itself is normally distributed, the sampling distribution or sample means is
normally distributed for any sample size n.
Sample Mean:  x  
Standard Deviation:  x 

n
The standard deviation of the sampling distribution of the sample means  x is
called the standard error of the mean.
Translation: If n >= 30, use the assumed mean and standard deviation for calculations
For any size sample n, use modifications if the population is normally
distributed.
Critical values: are values that separate sample statistics that are probable from sample
statistics that are improbable, or unusual. Represent by zc. On one side are values that are
probable, on the other improbable.
Most common used:
Level of Confidence
zc
90%
1.645
95%
1.96
99%
2.575
Sampling error: is the difference between the point estimate and the actual parameter value.
When µ is estimated the sampling error is x xbar - µ mu
Margin of Error (maximum error of estimate or error tolerance) : Represent by E.
Given level of confidence C, E is the greatest possible distance between the point estimate and
the value of the parameter it is estimating. E  zc * x  zc *

n
Round-Off Rule: Round off the confidence interval for the population mean to the same number of
decimal places given for the sample mean
STEPS
Construction the Confidence Intervals for the Population Mean IF  sigma IS KNOWN
The steps
1. Find the sample statistics n and xbar x 
x
n
2. Specify  sigma
3. Find the critical value zc that corresponds to the given level of confidence
4. Find the margin of error E  zc *

n
5. Find the left and right endpoints and form the confidence interval.
x  E left endpoint
x  E right endpoint
Interval x  E    x  E
STEPS
Construction the Confidence Intervals for the Population Mean IF sigma IS NOT KNOWN
The steps
1. Find the sample statistics n and xbar x 
x
n
2. If n >= 30, find the sample standard deviation s 
estimate for  sigma
3. Find the critical value
 ( x  x)
n 1
2
and use it as an
zc that corresponds to the given level of confidence
4. Find the margin of error E  zc *

n
5. Find the left and right endpoints and form the confidence interval
x  E left endpoint
x  E right endpoint
Interval x  E    x  E
STEPS
In reverse: find the minimum sample size n, given c-confidence and a margin of error E
If  sigma is unknown, estimate it using s, provided you have a preliminary sample with at least
30 members
Formula is derived from the Margin of error formula above.
E  zc *

n
E * n  zc * 
n
zc * 
E
 z * 
n c

 E 
2
Section 6.2– Confidence Intervals for the Mean (Small Samples n < 30)
t-distribution: If the distribution of the random variable x is approximately normal, then
x
t
s
n
Properties of a t-distribution:
1. Bell-shaped and symmetric about the mean
2. Family of curves, each determined by a parameter called “degrees of freedom”
3. Total area under a t-curve = 1 (100%)
4. The mean, mode, median, are centered on the curve at 0.
5. As the “degrees of freedom” increases, the t-distribution approaches the normal distribution.
After 30df the t-distribution is very close to the standard normal z-distribution
df – “degrees of freedom”: defined as the number of free choices left after a sample statistic is
calculated. d . f .  n  1
See page 325
STEPS
Construction the Confidence Intervals for the Mean: t-distribution
The steps
1.
 x , and s   ( x  x)
Find the sample statistics n, x 
n
2
n 1
2. Find the “degrees of freedom” d.f.= n - 1
3. Find tc from table #5
4. Find the margin of error E  tc *
s
n
5. Find the left and right endpoints and form the confidence interval
x  E left endpoint
x  E right endpoint
Interval x  E    x  E
Section 6.3– Confidence Intervals for Population Proportion p
Population Proportion: is the proportion of success (concepts learned about the Binomial distribution)
Point Estimate: for the population proportion p.
x
, x is the number of successes in the sample and n is number in the sample.
n
qˆ  1  pˆ , is the proportion of failures.
pˆ 
c-confidence Interval for the population proportion p:
pˆ  E  p  pˆ  E , where E  zc
ˆˆ
pq
n
The probability that the confidence interval contains p is c
STEPS
Construction the Confidence Intervals for the Population Proportion
The steps
1. Find the sample statistics n and x.
2. Find the point estimate pˆ 
x
n
3. Verify that the sampling distribution of p̂ can be approximated by the normal
distribution
4. Find the critical value
zc that corresponds to the given level of confidence
5. Find the margin of error E  zc
ˆˆ
pq
n
6. Find the left and right endpoints and form the confidence interval.
p̂  E left endpoint
p̂  E right endpoint
Interval pˆ  E  p  pˆ  E
Find the minimum sample size to estimate p WITH OR WITHOUT preliminary estimate available
The steps
x
, if one cannot be found use 0.50 for p-hat
n
2. Verify that the sampling distribution of p̂ can be approximated by the normal
1. Find the point estimate pˆ 
distribution
3. Find the critical value
zc that corresponds to the given level of confidence
z 
ˆ ˆ c 
4. Find the sample size n  pq
E
2