Download 5.1-5.3 Guided Notes - Pendleton County Schools

STA220- 5.1-5.3 Guided Notes
The unknown population parameter (e.g., mean or proportion) that we are
interested in estimating is called the _______________________________________.
Parameter
_____
_____
Determining the Target Parameter
Key Words of Phrase Type of Data
Mean; average
Proportion; percentage fraction; rate
Quantitative
Qualitative
A ___________________________________ of a population parameter is a rule or formula that
tells us how to use the sample data to calculate a single number that can be used as
an estimate of the target parameter.
An interval estimator (or __________________________________________) is a formula that
tells us how to use the sample data to calculate an interval that estimates the target
parameter.
Confidence Interval for a Population Mean: Normal (z) Statistic
Confidence Interval
According to the Central Limit Theorem, the sampling distribution of the sample
mean is approximately normal for large samples. Let us calculate the interval
estimator:
That is, we form an interval from 1.96 standard deviations below the sample mean
x ± 1.96s x = x ±
1.96s
n
to 1.96 standard deviations above the mean. Prior to drawing the sample, what are
the chances that this interval will enclose µ, the population mean?
The _______________________________________________________ is the probability that a
randomly selected confidence interval encloses the population parameter - that is,
the relative frequency with which similarly constructed intervals enclose the
population parameter when the estimator is used repeatedly a very large number of
times. The confidence level is the confidence coefficient expressed as a percentage.
Example:
Consider the large hospital that wants to estimate the average length of stay of its
patients, 𝜇. The hospital randomly samples n = 100 of its patients and finds that the
sample mean length of stay is 𝑥̅ = 4.5 days. Also, suppose it is known that the
standard deviation of the length of stay for all hospital patients is 𝜎 = 4 days. Use
the interval estimator 𝑥̅ ± 1.96𝜎𝑥̅ to calculate a confidence interval for the target
parameter, 𝜇.
Conditions Required for a Valid Large-Sample Confidence Interval for µ
1. A random sample is selected from the target population.
2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this
condition guarantees that the sampling distribution of 𝑥̅ is approximately
normal. Also, for large n, s will be a good estimator of .
Large-Sample (1 – )% Confidence Interval for µ
𝝈
̅ ± (𝒛𝜶 /𝟐)𝝈𝒙̅ = 𝒙
̅ ± 𝒛𝜶\𝟐 ( )
𝒙
√𝒏
where z/2 is the z-value with an area /2 to its right and the parameter  is the
standard deviation of the sampled population, and n is the sample size.
Note: When  is unknown and n is large (n ≥ 30),
the confidence interval is approximately equal to
𝒔
̅ ± 𝒛𝜶\𝟐 ( )
𝒙
√𝒏
Example:
Many middle schools have initiated a program that provides every student with a
free laptop (notebook) computer. Student usage of laptops at a middle school that
participates in the initiative was investigated in American Secondary Education (fall
2009). In a sample of 106 students, the researchers reported the following statistics
on how many minutes per day each student used his or her laptop for taking
notes: 𝑥̅ = 13.2 and s = 19.5. Now the researchers want to estimate the average
amount of time per day laptops are used for taking notes for all middle school
students across the country.
a. Calculate a 90% confidence interval for the target parameter. Interpret the
results.
b. Explain what the phrase “90% confidence” implies in part a.
Example:
You’re a Q/C inspector for Gallo. The s for 2-liter bottles is .05 liters. A random
sample of 100 bottles showed x = 1.99 liters. What is the 90% confidence interval
estimate of the true mean amount in 2-liter bottles?
Confidence Interval for a Population Mean: Student’s t-Statistic
Small Sample  Unknown
t-statistic is used when the sample size is small.
(𝑥̅ – 𝜇)
𝑡 =
(𝑠/√𝑛)
in which the sample standard deviation, s, replaces the population standard
deviation, .
The actual amount of variability in the sampling distribution of t depends on the
sample size n. A convenient way of expressing this dependence is to say that the tstatistic has (n – 1) __________________________________ (df).
Small-Sample Confidence Interval for µ
̅ ± 𝒛𝜶\𝟐 (
𝒙
𝒔
)
𝒏
√
where ta/2 is based on (n – 1) degrees of freedom.
Conditions Required for a Valid Small-Sample Confidence Interval for µ
1. A random sample is selected from the target population.
2. The population has a relative frequency distribution that is approximately
normal.
Estimation Example Mean (s Unknown)
A random sample of n = 25 has 𝑥̅ = 50 and s = 8. Set up a 95% confidence interval
estimate for m.
Example:
Consider the pharmaceutical company that desires an estimate of the mean
increase in blood pressure of patients who take a new drug. The blood pressure
increases (measured in points) for the n = 6 patients in the human testing phase are
shown below. Use this information to construct a 95% confidence interval for 𝜇, the
mean increase in blood pressure associated with the new drug for all patients in the
population.
Blood Pressure Increase (Points)
for Six Patients
1.7 3.0 .8
3.4 2.7 2.1
Example:
You’re a time study analyst in manufacturing. You’ve recorded the following task
times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence interval estimate of the population mean task time?