Statistics Class 16
3/26/2012
Estimating a Population mean: 𝜎 known
We are going to use the point estimate 𝑥, to estimate the
population mean using confidence intervals, when 𝜎is known.
Estimating a Population mean: 𝜎 known
We are going to use the point estimate 𝑥, to estimate the
population mean using confidence intervals, when 𝜎is known.
• Usually you don’t know 𝜎 if you don’t know the population
mean.
Estimating a Population mean: 𝜎 known
We are going to use the point estimate 𝑥, to estimate the
population mean using confidence intervals, when 𝜎is known.
• Usually you don’t know 𝜎 if you don’t know the population
mean.
• We require either normality or 𝑛 > 30 for the most part.
We actually require a loose normality that is that there are
not too many outliers and the data sort of has a bell shape.
Estimating a Population mean: 𝜎 known
Constructing a Confidence Interval for 𝜇 (with 𝜎 known)
1. Verify that the requirements are satisfied.
Estimating a Population mean: 𝜎 known
Constructing a Confidence Interval for 𝜇 (with 𝜎 known)
1. Verify that the requirements are satisfied. (Sample is
simple random, 𝜎 is known, and distribution is either
normal or 𝑛 > 30. )
2. Find the critical value associated with the desired
confidence level.
Estimating a Population mean: 𝜎 known
Constructing a Confidence Interval for 𝜇 (with 𝜎 known)
1. Verify that the requirements are satisfied. (Sample is
simple random, 𝜎 is known, and distribution is either
normal or 𝑛 > 30. )
2. Find the critical value associated with the desired
confidence level.
3. Evaluate the margin of error 𝐸 = 𝑧𝛼/2 ∙ 𝜎 𝑛
Estimating a Population mean: 𝜎 known
Constructing a Confidence Interval for 𝜇 (with 𝜎 known)
1. Verify that the requirements are satisfied. (Sample is
simple random, 𝜎 is known, and distribution is either
normal or 𝑛 > 30. )
2. Find the critical value associated with the desired
confidence level.
3. Evaluate the margin of error 𝐸 = 𝑧𝛼/2 ∙ 𝜎 𝑛
4. Construct 𝑥 − 𝐸, 𝑥 + 𝐸
Estimating a Population mean: 𝜎 known
Constructing a Confidence Interval for 𝜇 (with 𝜎 known)
5. Round
a.
b.
If using the original data set, round to one more decimal than the
original set
If using summary statistics round to the same place as the sample
mean.
Or
Zinterval
Estimating a Population mean: 𝜎 known
A simple random sample of 40 salaries of NCAA football coaches
has a mean of $415,953. Assume 𝜎 = $463,364.
• Find the best point estimate of the mean salary of all NCAA
football coaches.
• Construct a 95% confidence interval estimate of the mean
salary of NCAA football coach.
• Does the confidence interval contain the actual population of
mean of $474,477?
Estimating a Population mean: 𝜎 known
Determining the sample size Required to Estimate 𝜇.
The sample size 𝑛 required to estimate 𝜇 is given by the
formula:
𝑧𝑎/2 𝜎 2
𝑛=
𝐸
Round up to the next larger whole number.
Estimating a Population mean: 𝜎 known
How many integrated circuits must be randomly selected and
tested for time to failure in order to estimate the mean time to
failure? We want a 95% confidence that the sample mean is
within 2 hr. of the population mean, and the population
standard deviation is known to be 18.6 hours.
Estimating a Population mean: 𝜎 Not known
In this section we learn to estimate the population mean
when 𝜎 is not known. Since 𝜎 is typically unknown in real life
this is a useful method as it is practical and realistic.
The sample mean 𝒙 is the best point estimate of the
population mean 𝝁.
When 𝜎 is not know we use a Student t distribution to find our
critical values.
Estimating a Population mean: 𝜎 Not known
If a population is normally distributed, then the distribution
of
𝑥−𝜇
𝑡= 𝑠
𝑛
is a Student t distribution for all samples of size n.
Estimating a Population mean: 𝜎 Not known
If a population is normally distributed, then the distribution
of
𝑥−𝜇
𝑡= 𝑠
𝑛
is a Student t distribution for all samples of size n.
The number of degrees of freedom for a collection of sample
data is the number of sample values that can vary after
certain restrictions have been imposed on all data values. The
number of degrees of freedom is often abbreviated as df.
Estimating a Population mean: 𝜎 Not known
For us the number of degrees of freedom is simply the sample
size minus 1.
Degrees of freedom = 𝒏 − 𝟏
Estimating a Population mean: 𝜎 Not known
For us the number of degrees of freedom is simply the sample
size minus 1.
Degrees of freedom = 𝒏 − 𝟏
Find the critical t value corresponding to a confidence level of
95% for a sample of size 𝑛 = 7.
Estimating a Population mean: 𝜎 Not known
Constructing a Confidence Interval for 𝜇 (with 𝜎 unknown)
1. Verify requirements are satisfied(Simple random sample,
normal dist. or 𝑛 > 30).
Estimating a Population mean: 𝜎 Not known
Constructing a Confidence Interval for 𝜇 (with 𝜎 unknown)
1. Verify requirements are satisfied(Simple random sample,
normal dist. or 𝑛 > 30).
2. Using 𝑛 − 1 degrees of freedom find the critical value 𝑡𝑎/2
corresponding to the desired confidence level.
Estimating a Population mean: 𝜎 Not known
Constructing a Confidence Interval for 𝜇 (with 𝜎 unknown)
1. Verify requirements are satisfied(Simple random sample,
normal dist. or 𝑛 > 30).
2. Using 𝑛 − 1 degrees of freedom find the critical value 𝑡𝑎/2
corresponding to the desired confidence level.
3. Evaluate the margin of error 𝐸 = 𝑡𝛼/2 ∙ 𝑠/ 𝑛.
Estimating a Population mean: 𝜎 Not known
Constructing a Confidence Interval for 𝜇 (with 𝜎 unknown)
1. Verify requirements are satisfied(Simple random sample,
normal dist. or 𝑛 > 30).
2. Using 𝑛 − 1 degrees of freedom find the critical value 𝑡𝑎/2
corresponding to the desired confidence level.
3. Evaluate the margin of error 𝐸 = 𝑡𝛼/2 ∙ 𝑠/ 𝑛.
4. Find 𝑥 − 𝐸, 𝑥 + 𝐸 .
Estimating a Population mean: 𝜎 Not known
Constructing a Confidence Interval for 𝜇 (with 𝜎 unknown)
1. Verify requirements are satisfied(Simple random sample,
normal dist. or 𝑛 > 30).
2. Using 𝑛 − 1 degrees of freedom find the critical value 𝑡𝑎/2
corresponding to the desired confidence level.
3. Evaluate the margin of error 𝐸 = 𝑡𝛼/2 ∙ 𝑠/ 𝑛.
4. Find 𝑥 − 𝐸, 𝑥 + 𝐸 .
5. Round to one extra decimal place if using original data. If
using summary statistics round to the same number of
decimal places as the sample mean.
Estimating a Population mean: 𝜎 Not known
Or use Tinterval
Estimating a Population mean: 𝜎 Not known
In a test of the Atkins weight loss program, 40 individuals
participated in a randomized trial with overweight adults. After
12 months, the mean weight loss was found to be 2.1 lb, with a
standard deviation of 4.8 lb
• What is the best point estimate of the mean weight loss of all
overweight adults who follow the Atkins program?
• Construct a 99% confidence interval estimate of the mean
weight loss for all such subjects.
• Does the Atkins program appear to be effective? Is it
practical?
Estimating a Population mean: 𝜎 Not known
Important properties of the t distribution.
1. The Student t distribution is different for different sample
sizes.
Estimating a Population mean: 𝜎 Not known
Important properties of the t distribution.
1. The Student t distribution is different for different sample
sizes.
2. The Student t distribution has the same general
symmetric bell shape as the standard normal distribution,
but reflects the greater variability that is expected with
small samples.
Estimating a Population mean: 𝜎 Not known
Important properties of the t distribution.
1. The Student t distribution is different for different sample
sizes.
2. The Student t distribution has the same general
symmetric bell shape as the standard normal distribution,
but reflects the greater variability that is expected with
small samples.
3. The Student t distribution has a mean of t=0
Estimating a Population mean: 𝜎 Not known
Important properties of the t distribution.
1. The Student t distribution is different for different sample
sizes.
2. The Student t distribution has the same general
symmetric bell shape as the standard normal distribution,
but reflects the greater variability that is expected with
small samples.
3. The Student t distribution has a mean of t=0
4. The standard deviation of a t distribution varies with the
sample size but is always larger than 1.
Estimating a Population mean: 𝜎 Not known
Important properties of the t distribution.
1. The Student t distribution is different for different sample
sizes.
2. The Student t distribution has the same general
symmetric bell shape as the standard normal distribution,
but reflects the greater variability that is expected with
small samples.
3. The Student t distribution has a mean of t=0
4. The standard deviation of a t distribution varies with the
sample size but is always larger than 1.
5. As n gets large the t distribution becomes more like the
standard normal distribution.
Estimating a Population mean: 𝜎 Not known
Method
Conditions
Use z distribution
𝜎 known and normal dist. Pop.
or
𝜎 and n > 30
Use t distribution
𝜎 unknown and normal dist. Pop.
or
𝜎 unknown and n > 30
Other method
If n≤ 30 and Population not normal
Estimating a Population mean: 𝜎 Not known
Use the given data to decide whether to use
ZInterval or Tinterval
𝑛 = 9, 𝑥=75, s=15 and population has a normal dist.
𝑛 = 5, 𝑥 = 20, 𝑠 = 2 and the population is very skewed.
𝑛 = 12, 𝑥 = 98.6, 𝜎 = 0.6 and the population is normal.
𝑛 = 75, 𝑥 = 98.6, 𝜎 = 0.6 and the population is very
skewed
• 𝑛 = 75, 𝑥 = 98.6, 𝑠 = 0.6 and the population is very
skewed.
•
•
•
•
Estimating a Population mean: 𝜎 Not known
Listed below are 12 lengths (in minutes) of randomly
selected movies.
110 96 125 94 132 120 136 154 149 94 119 132
• Construct a 99% confidence interval estimate of the
mean length of all movies.
• Assuming that it takes 30 min to empty a theater after a
movie, clean it, allow time for the next audience to enter,
and show previews, what is the minimum time that a
theater manager should plan between start times of
movies, assuming that this time will be sufficient for
typical movies?
Quiz 13
When Mendel conducted his famous genetics experiments
with peas, one sample of offspring consisted of 428 green
peas and 152 yellow peas.
a. Find a 95% confidence interval estimate of the percentage
of yellow peas.
b. Based on his theory of genetics, Mendel expected that
25% of the offspring peas would be yellow. Given that the
percentage of offspring yellow peas is not 25%, do the
results contradict Mendel’s Theory? Why or why not.
Homework!!
• 7-3: 1-9 odd, 13,15, 21-27 odd.
• 7-4: 1-12, 17-29 odd.