CONFIDENCE INTERVALS
– Vocabulary

Using sample statistics (x bar or p hat) to estimate population parameters ( μ or p)

Point Estimate: is the best estimate for our population parameter. It is the mean (x bar) or proportion (p hat) of the selected sample. We consider the point estimate as a “plausible” value for the population parameter ( μ or p)

Confidence Interval: Is a range of values likely to contain the population mean ( μ ) or population proportion (p). Range of “plausible” values for the population parameter ( μ or p).

Confidence Level (example: 95% confidence interval) refers to the long run success rate of the method. If we repeatedly select samples of the same size n and construct the 95% confidence interval around each point estimate (x bar or p hat) we expect about 95% of the intervals to con

Significance Level

= 1 – CL = 1 - .95 = 0.05 is the long run error rate of the method. For a 95% confidence level, a bout 5% of the CI constructed will “miss” the population parameter

Geometrically,
 is the area of the two tails and it is used to find the critical values

Critical values are the borderline z or t scores that separate sample statistics that are likely to occur (usual scores, with large probability) from those that are unlikely
(unusual, with very small probability)

To find the critical values use

/2 and table A-2 for z or table A-3 for t

Finding critical values – Example for 99% confidence interval:
TABLE 2

= 1 – CL = 1 - .99 = 0.01

/2 = 0.005
Look at 0.005 in table 2 from inside out and find –2.575, then, z

/ 2
= 2.575
TABLE 3
Select the column with two tails of 0.01 and 1 tail of 0.005
Select the row according to DF = n - 1

Margin of Error: is the largest possible sampling error when estimating the population parameter (p or μ) with the sample statistic (p hat or x bar).

Formula for error: E = (critical value) * (standard deviation of the distribution)

Formulas for obtaining Confidence Intervals = point estimate +/- Error
1

In this section, we are going to be estimating a population proportion; p. In order to do that, we’ll use the proportion that we get from a sample.
Notation for Proportions p is the population proportion (percent of those who have the
"quality" under discussion) n is the sample size p (read "p-hat") is the sample proportion (percent of those in the sample who have the "quality" under discussion) p
 x n x is the # of successes (# of those who have the "quality" under discussion) in a sample of size n. q (read "q-hat") is the percent of those in the sample who do not have the "quality" under discussion) q 1 p
Assumptions
1. The sample is a simple random sample. (SRS)
2. The conditions for a binomial distribution are satisfied by the sample. That is: there is a fixed number of trials, the trials are independent, there are two categories of outcomes, and the probabilities remain constant f or each trial. A “trial” would be the examination of each sample element to see which of the two possibilities it is.
3. The normal distribution can be used to approximate the distribution of sample proportions because np ≥ 5 and nq ≥ 5 are both satisfied. (q = 1 – p)
Procedure for Constructing a Confidence Interval for p
1) Verify that the assumptions are satisfied.
2) Find the critical value z

/ 2
(from table A-2).
3) Evaluate the margin of error E. E
 z

/ 2 n
4) Find the interval and write it in one of the following forms p

E ; p
   
E ; ( p

,

E )
Round-Off Rule for Confidence Interval Estimates of p
3 significant digits
Using the TI-83 to Construct Confidence Intervals for p:
STAT>>TESTS choose A:1-propZInt.
2

Finding the Point Estimate and the Margin of Error from a Confidence Interval
Point estimate of p (middle of interval)
.
.int
erval .lim
it

.
.int
erval .lim
it p =
2
Margin of E (1/2 the length of the interval)
.
.int
erval .lim
it

.
.int
erval .lim
it
E=
2
Determining Sample Size:
If we have an approximate idea of what p is : n

( z

/ 2
)
2
E 2 p q
If no estimate of p is known: n

0.25 ( z

/ 2
)
2
E 2
Round-Off Rule for Sample Size, n: Use the computed size if it is a whole number. If it is not a whole number, round it up to the next higher whole number.
3

In this section, we are going to be estimating a population mean; μ. In order to do that, we’ll use the mean that we get from a sample.
Assumptions:
1. The sample is a simple random sample (All samples of the same size have an equal chance of being selected.)
2. The value of the population standard deviation
is known.
3. Either or both of these conditions is satisfied: i) The population is normally distributed, or ii) n > 30 (The sample has more than 30 values)
Procedure for Constructing a Confidence Interval for μ (with Known σ)
1. Verify that the required assumptions are satisfied.
2. Find the critical value z

/ 2
(from table A-2).

3. Evaluate the margin of error E. (E = z 
/2
) n
4. Then using E and the sample mean the confidence interval is: x E
 x E or ( x

,

E )
The two values x

E and x

E are called confidence interval limits
Round-off Rule for Confidence Intervals used to Estimate μ: a) If original data is given: use one more decimal place than original values. b) If you are given summary statistics from a data set, use the same number of decimal places used for the sample mean.
Using the TI-83 to Construct a Confidence Inte rval for Estimating μ
STAT>>TESTS select 7:ZInterval
4

Determining Sample Size Required to Estimate μ n

( z

/ 2
E
*

) 2
What if σ is not known?
to the nearest whole number
1) Use the range rule of thumb: σ ~ range/4
2) We often use s from a pilot test (n > 30)
3) Estimate the value of σ by using the results of some other study that was done earlier.
5

Conditions for Using the Student t Distribution
1.
is unknown, (if
is known we use the methods of 6.3);
2. The sample is a simple random sample
3. Either or both of these conditions is satisfied: i) The population is normally distributed, or ii) n > 30 (The sample has more than 30 values)
Notes:
1. Criteria for deciding whether the population is normally distributed:
Population need not be exactly normal, but it should appear to be somewhat symmetric with one mode and no outliers.
To asses normality, use the last graph in STAT PLOT with data list. If close to a line without significant outliers, the distribution of the data is normal.
2. Sample size n > 30 :
This is a commonly used guideline, but sample sizes of 15 to 30 are adequate if the population appears to have a distribution that is not far from being normal and there are no outliers. For some population with distributions that are extremely far from normal, the sample size might need to be larger than 50 or even 100.
Confidence Interval for the Estimate of μ (With σ Not Known)
1. Verify that the required assumptions are satisfied.
2. Find the critical value t

/ 2
from table A-3 (n -1 degrees of freedom). s
3. Evaluate the margin of error E. (E = t 
/2
) n
4. Then using E and the sample mean the confidence interval is: x E
 x E or ( x

,

E )
Using the TI83 to Construct a Confidence Interval for Estimating μ
STAT>>TESTS select 8:TInterval
6

CONFIDENCE LEVEL AND SAMPLE SIZE VERSUS PRECISION

Select a sample of size n = 100, find x bar, and construct the 95% confidence interval o What happens if we change the confidence level?
ERROR:
Smaller or larger?
LENGTH OF
INTERVAL:
Shorter or longer?
PRECISION
More precise or less precise?
Increase CL to 99%
Decrease CL to
90%

Select a sample of size n = 100, find x bar, and construct the 95% confidence interval o What happens if we change the sample size?
ERROR:
Smaller or larger?
LENGTH OF
INTERVAL:
Shorter or longer?
PRECISION
More precise or less precise?
Increase n to 150
Decrease n to 50
7