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College Prep Stats
Chapter 7
Important Info Sheet
7.2
A point estimator is a statistic that provides an estimate of a population parameter.
Point estimators that we will be using are x and pˆ .
A point estimate is the value of that statistic from a sample. Ideally, a point estimate is our “best guess” at the value of an unknown
parameter.
𝑥
The sample proportion pˆ is the best point estimate of the population proportion p.
𝑝̂ =
𝑛
A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population
parameter. A confidence interval is sometimes abbreviated as CI.
Interpreting a Confidence Interval
“We are _____% confident that the interval from _____ to _____ actually does contain the true value of the population proportion p.”
The underlined portion should
be written in the context of the
problem.
Critical Values for a Population Proportion p
When finding a critical value, use the table below, or the calculator command: invNorm(area to the left, , )
Example: If the given confidence level is 86%, α = 1 – 0.86 = 0.14. Therefore, α/2 = 0.07. To find the correct critical value, find the
area to the left (confidence level + α/2) = 0.86 + 0.07 = 0.93. Use the following computation in your technology: invNorn(0.93, 0, 1)
Margin of Error for Proportions
E  z 2
ˆˆ
pq
n
E  margin of error
z /2  z*  critical value
pˆ  proportion of successes
qˆ  proportion of failures
n  sample size
Confidence Interval for Estimating a Population Proportion p
p = population proportion
pˆ = sample proportion
n = number of sample values
E = margin of error
z/2 (z*) = z score separating an area of /2 in the right tail of the standard normal
distribution
3 Different Ways to Write a Confidence Interval for the Estimate of a Population Proportion p
pˆ  E  p  pˆ  E
pˆ  E
 pˆ  E, pˆ  E 
Sample Size for Estimating Proportion p
When an estimate of p̂ is known:
When an estimate of p̂ is unknown:
n
 z / 2 
E
2
ˆˆ
pq
n
2
 z / 2 
2
E
0.25
2
Round-Off Rule for Determining Sample Size
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.
Finding the Point Estimate and E from a Confidence Interval
point estimate of p :
pˆ 
 upper confidence limit    lower confidence limit 
2
Margin of Error:
E
 upper confidence limit    lower confidence limit 
2
7.3: The sample mean x is the best point estimate of the population mean µ.
Critical Values for a Population Mean µ when σ is Known
See the section for Critical Values for a Population Proportion, p
Margin of Error for Means (with  Known)
E  z / 2

n
3 Different Ways to Write a Confidence Interval for Estimating a Population Mean µ (with  Known)
xE    xE
xE
 x  E, x  E 
Confidence Interval for Estimating a Population Mean (with  Known)
 = population mean
 = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
z/2 = (z*) = z score separating an area of a/2 in the right tail of the standard normal distribution
Finding a Sample Size for Estimating a Population Mean
 = population mean
σ = population standard deviation
x = sample mean
E = desired margin of error
z/2 = (z*) = z score separating an area of a/2 in the right tail of the standard normal distribution
  z *   
n

 E 
2
7.4
degrees of freedom = n – 1, for the student t distribution
Choosing the Appropriate Distribution
How do we know when to use zα/2 or tα/2 (z* or t*)?
*If you are working with a categorical variable (estimating a population proportion, p) always use zα/2 (z*).
*If you are working with a quantitative variable (estimating a population mean, µ) and you DO know σ, use zα/2 (z*).
*If you are working with a quantitative variable (estimating a population mean, µ) and you DO NOT know σ, use tα/2 (t*).
**Remember that the population distribution must be normal or n must be large for quantitative variables.**
Critical Values for a Population Mean µ when σ is Not Known
When finding a critical value, use the following calculator command: invT(area to the left, df)
Example: If the given confidence level is 86%, with a sample size of 28, the degrees of freedom will be n – 1, so df = 27.
α = 1 – 0.86 = 0.14. Therefore, α/2 = 0.07.
To find the correct critical value, find the area to the left (confidence level + α/2) = 0.86 + 0.07 = 0.93. Use the following computation
in your technology: invT(0.93, 27)
Margin of Error E for Estimate of µ (With σ Not Known)
s
, where tα/2 has n – 1 degrees of freedom. NOTE: tα/2 = t*
E  t / 2
n
Notation
 = population mean
x = sample mean
s = sample standard deviation
n = number of sample values
E = margin of error
t/2 = t* = critical t value separating an area of /2 in the right tail of the t distribution
3 Different Ways to Write a Confidence Interval for the Estimate of a Population Mean μ (With σ Not Known)
xE    xE
xE
 x  E, x  E 
Finding the Point Estimate and E from a Confidence Interval
point estimate of  :
x
 upper confidence limit    lower confidence limit 
2
Margin of Error:
E
 upper confidence limit    lower confidence limit 
2