Download From the Generic Syllabus for Chapter 9

From the Generic Syllabus for Chapter 9
The student will be able to:
1.
Determine sample size necessary for a specific confidence interval.
2.
Estimate the value of a population proportion by the point estimate and confidence interval.
3.
Estimate the value of a population mean by the point estimate and confidence interval.
4.
Interpret confidence intervals.
5.
Use technology to calculate confidence intervals and sample sizes.
Section 9.1 – Estimating a Population Proportion
Objectives
1. Obtain a point estimate for the population proportion
2. Construct and interpret a confidence interval for the population proportion
3. Determine the sample size necessary for estimating the population proportion within a
specified margin of error
Objective 1 – Obtain a point estimate for the population proportion
Point Estimate – The value of a statistic that estimates the value of a parameter.
(for example, “I estimate that the proportion of female students enrolled in Statistics is .84”)
The point estimate for a sample proportion is called p-hat and is denoted as p̂ .
The population proportion is called rho and is denoted by  .
x
pˆ 
(x = number of successes, n = # of trials)
p̂ is the point estimate for ρ
n
Objective 2 - Construct and interpret a confidence interval for the population
proportion
Remember from Chapter 7:
z is the z score for which the area under the curve to the right of the z score is 
1
Confidence Interval and Level of Confidence
The most common confidence levels are: 90%, 95%, 99%
Note the relationship between confidence level and alpha when filling out the table below.
Margin of Error (E) – A measure of how accurate the point estimate is.
How are confidence intervals created?
To create confidence intervals, take the point estimate from the sample and add and subtract the
margin of error.
Point Estimate ± Margin of Error
Note, the Margin of Error of the sample proportion is z
2
p̂(1  p̂)
, given a (1   )100% level of
n
confidence.
The margin of error depends on three factors:
 Level of confidence: As the level of confidence increases, the margin of error also increases.
 Sample size: As the size of the random sample increases, the margin of error decreases.
 Standard deviation of the population: The more spread there is in the population, the wider
our interval will be for a given level of confidence.
The value z is called the critical value of the distribution. It represents the number of standard
2
deviations the sample statistic can be from the parameter and still result in an interval that includes the
parameter.
Fill out the following table
Level of Confidence
(1-  )*100%

Area in each tail,

2
Critical value, z
2
90%
95%
99%
Example
Determine z that corresponds to a 96% level of confidence.
2
2
To calculate a confidence interval by hand
Independence requirement - n  0.05N
Normality requirement - np̂(1  p̂)  10
To calculate a confidence interval with the TI 83/84
Example
A survey found 195 of 250 randomly selected Internet users have high-speed Internet access at
home.
a) Calculate a point estimate for the proportion of all internet users who have high-speed
internet access at home.
b) Construct a 90% confidence interval for the proportion of all Internet users who have highspeed Internet access at home.
3
How to Interpret a Confidence Interval
I am __% confident that the true proportion of _____________ is between __ and __.
A 95% level of confidence does NOT tell us that “there is a 95% probability that the parameter lies
between the lower and upper bound”.
A 95% confidence interval means that the method “works” for 95% of all samples. We do not know if
our interval is one of the 95% that contain the parameter or the 5% that does not contain the
parameter.
Put another way, a 95% level of confidence implies that if 100 different confidence intervals are
constructed, each based on a different sample from the same population, then we expect 95 of the
intervals to include the parameter and 5 to not include the parameter.
Example
From the internet survey above, (195 of 250 randomly selected Internet users have high-speed
Internet access at home), write a sentence to interpret the interval.
4
Objective 3 - Determine the sample size necessary for estimating the population
proportion within a specified margin of error
If p̂ is a prior estimate of ρ
 z
ˆ ˆ 2
n = pq
 E





2
where qˆ  1  pˆ
If a prior estimate of ρ is unavailable, presume p̂  0.5 , which results in
 z
n = 0.25  2
 E





2
if this does not result in a whole number, then round UP to the next whole number
The following table will save time if you memorize
Confidence Interval Critical Values, z
2
Level of Confidence
Critical Value, z
2
0.90 or 90%
0.95 or 95%
0.98 or 98%
0.99 or 99%
1.645
1.96
2.33
2.575
Examples
a) An economist wants to know if the proportion of the US population who commute to work by
car-pooling is on the rise due to higher gas prices. What sample size should be obtained if the
economist wants an estimate within 2 percentage points of the true proportion with 95%
confidence?
b) A sociologist wanted to determine the percentage of residents of America that only speak
English at home. What size sample should be obtained if she wishes her estimate to be within
3 percentage points with 90% confidence assuming she uses the 2000 estimate obtained from
the Census 2000 Supplementary Survey of 82.4%?
5
Section 9.2 – Estimating a Population Mean
Objectives
1. Obtain a point estimate for the population mean
2. State properties of Student’s t-distribution
3. Determine t-values
4. Construct and interpret a confidence interval for a population mean
5. Find the sample size needed to estimate the population mean within a given margin or error
Objective 1 – Obtain a point estimate for the population mean
A point estimate is the value of a statistic that estimates the value of a parameter.
The sample mean x is a point estimate of the population mean μ.
Example
Pennies minted after 1982 are made from 97.5% zinc and 2.5% copper. The following data represent
the weights (in grams) of 17 randomly selected pennies minted after 1982.
2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.45 2.49 2.47 2.45 2.46 2.45 2.46 2.47 2.44
2.45
Treat the data as a simple random sample. Estimate the population mean weight of pennies minted
after 1982.
Objective 2 – State the properties of the Student’s t-distribution
6
Use the t-distribution when σ is unknown and EITHER n > 30 or the population is bell-shaped.
7
Objective 4 – Construct and interpret a confidence interval for a population mean
As with the population proportion, the confidence interval will be
Point estimate  margin of error
Constructing a (1 – α)100% Confidence Interval for μ
Provided
 Sample data come from a simple random sample or randomized experiment
 Sample size is small relative to the population size (n ≤ 0.05N)
 The data come from a population that is normally distributed, or the sample size is large
A (1 – α)·100% confidence interval for μ is given by
 s 
Lower bound: x  t  
 n 
2
 s 
Upper bound: x  t  
 n 
2
Where t  is the critical value with n – 1 df (degrees of freedom)
2
TI-83/84 Plus





If necessary, enter raw data into L1
Press STAT, highlight TESTS, and select 8:TInterval
If the data are raw, highlight DATA. Make sure List is set to L1 and Freq to 1. If
summary statistics are known, highlight STATS and enter the summary statistics
Enter the confidence level following C-Level : .
Highlight Calculate: and press Enter
Example
Construct a 99% confidence interval about the population mean weight (in grams) of pennies minted
after 1982. Assume μ = 0.02 grams.
2.46 2.47 2.49 2.48 2.50 2.44 2.46 2.45 2.49 2.47 2.45 2.46 2.45 2.46 2.47
2.44 2.45
8
Write a sentence to interpret the interval.
Example
In a study of the length of time that it takes students to earn a bachelor’s degree, 80 students are
randomly selected and they are found to have a mean of 4.8 years and a standard deviation of 2.2
years. Construct a 95% confidence interval estimate of the mean amount of time it takes all students
to earn a bachelor’s degree. Write a sentence to interpret the interval.
Objective 5 - Find the sample size needed to estimate the population mean within a
given margin or error
Determining the Sample Size n
The sample size required to estimate the population mean, µ, with a level of
confidence (1– α)·100% with a specified margin of error, E, is given by
 z  s 
n 2 
 E 


2
where n is rounded up to the nearest whole number.
Back to the pennies. How large a sample would be required to estimate the mean weight of a penny
manufactured after 1982 within 0.005 grams with 99% confidence? Assume  = 0.02.
9