Download Chapter 10 Estimating with Confidence

Chapter 10 Estimating with Confidence
Vocabulary:
Confidence interval
Interval
A level C confidence interval
t distribution
Test of significance
Z*
Robust
significance level
Confidence level
test statistic
z distribution
Upper p critical value
margin of error
Margin of error
Critical value
z test statistic
paired t procedures
standard error
degrees of freedom
Calculator Skills:
Z interval
Z-test
T interval
10-1
Confidence Intervals: The Basics (617-642)
1.
In statistics, what is meant by a 95% confidence interval?
95% of samples will capture the true mean, parameter
2.
Sketch and label a 95% confidence interval for the standard normal curve.
3.
In a sampling distribution of x , why is the interval of numbers between x
called a 95% confidence interval?
95% of Normal population is within 2 standard deviations of the mean
4.
Define a level C confidence interval.
5.
Sketch and label a 90% confidence interval for the standard normal curve.
6.
What does z* represent?
7.
What is the value of z* for a 90% confidence interval? Include a sketch.
8.
What is the value of z* for a 95% confidence interval? Include a sketch.
 2s
Estimate ± margin of error
The critical value of z or t. ( # of standard deviations)
1.645
1.96
9.
What is the value of z* for a 99% confidence interval? Include a sketch. 2.576
10.
What is meant by the upper p critical value of the standard normal distribution?
Area to the right of a positive z* value
11.
Explain how to find a level C confidence interval for an SRS of size n having unknown mean  and known standard
deviation .
The formula to find the confidence interval is
  
X  z* 

 n
12.
What is meant by a margin of error?
13.
Why is it best to have high confidence and a small margin of error?
It will give a closer approximation to the true mean.
14.
What happens to the margin of error as z* decreases? Does this result in a higher or lower confidence level?
The margin of error is the formula:
  
z* 

 n
It means how far “off” you are from the true mean, parameter.
As z* decreases, the margin of error decreases and lowers confidence.
15.
What happens to the margin of error as  decreases?
As σ decreases, the margin of error decreases and lowers confidence.
16.
What happens to the margin of error as n increases? By how many times must the sample size increase in order to
cut the margin of error in half?
As n increases, the margin of error decreases. The sample size must increase by 4 times to decrease the
margin of error in half.
17.
The formula used to determine the sample size that will yield a confidence interval for a population mean with a
specified margin of error m is
  
z *
m
 n
   m


 n  z*
z *  m n
z *
 n
m
2
 z * 

 n
 m 
z*

n
 m.
Solve for n.
10-2
Estimating a Population Mean (642 - 662)
1.
Under what assumptions is s a reasonable estimate of σ?
1.
If it is an SRS.
2.
If the samples are independent. N ≥10n
3.
If the sampling distribution is normal.
np ≥ 10 and n(1-p) ≥ 10
2.
In general, what is meant by the standard error of a statistic?
3.
What is the standard deviation of the sample mean
4.
What is the standard error of the sample mean
5.
Describe the similarities between a standard normal distribution and a t distribution?
x?
x?
It is the standard deviation of a statistic.
It is
It is

n
.
s
.
n
The distributions have similar shape. They are symmetric, bell-shaped and unimodal.
6.
Describe the differences between a standard normal distribution and a t distribution?
The spread of the t-distribution is larger.
7.
How do you calculate the degrees of freedom for a t distribution?
Degrees of Freedom is found by subtracting one from the sample size. The formula is: df = n – 1.
8.
What happens to the t distribution as the degrees of freedom increase?
As n increases, the degrees of freedom increases and the t distribution becomes more normal.
9.
How would you construct a level C confidence interval for μ if σ is unknown?
 s 
X  t* 

 n
10.
The z-table gives the area under the standard normal curve to the left of z. What does the t-table give?
It gives the area to the right.
11.
In a matched pairs t procedure, what is μ, the parameter of interest?
The parameter of interest is the mean difference in responses of two treatments.
For example: difference  after  before
12.
Samples from normal distributions have very few outliers. If your data contains outliers, what does this suggest?
It suggests that the parent population is non-normal.
10-3
13.
If the size of the SRS is less than 15, when can we use t procedures on the data?
If the data are clearly normal or no obvious outliers.
14.
If the size of the SRS is at least 15, when can we use t procedures on the data?
We can use t procedures unless there are outliers or strong skewness.
15.
If the size of the SRS is at least 30 or 40, when can we use t procedures on the data?
If the data is at least 30 or 40 then t distribution can be used.
Estimating a Population Proportion (663- 677)
1.
In statistics, what is meant by a sample proportion?
The statistic that estimates the parameter, p, of the sample proportion.
2.
Give the mean and standard deviation for the sampling distribution of
The mean is p.
3.
 p  p
p , SE is:
p
 p 
p (1  p)
.
n
p̂ ?
SEp 
p(1  p)
n
What assumptions must be met in order to use z procedures for inference about a proportion?
1.
2.
3.
5.
The standard deviation is
How do you calculate the standard error of
The standard error of
4.
.
p̂ ?
If it is an SRS.
If the samples are independent. N ≥10n
If the sampling distribution is normal.
np ≥ 10 and n(1-p) ≥ 10
Describe how to construct a level C confidence interval for a population proportion.
 
 
p  z *  p(1  p ) 


n


6.
What formula is used to determine the sample size necessary for a given margin of error?
The margin of error formula is
n

m  z * 

p *(1  p*) 
 . Solving this for n, the sample size, is
n

( z*)2 p* (1  p*)
. Where p* is an estimated value of the sample proportion.
2
m