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 ______.
Determining the Target Parameter
ParameterKey Words of PhraseType of Data
_____Mean; averageQuantitative
_____Proportion; percentage fraction; rateQualitative
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
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, . The hospital randomly samples n = 100 of its patients and finds that the sample mean length of stay is days. Also, suppose it is known that the standard deviation of the length of stay for all hospital patients is days. Use the interval estimator to calculate a confidence interval for the target parameter, .
Conditions Required for a Valid Large-SampleConfidence Interval for µ
- A random sample is selected from the target population.
- The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .
Large-Sample (1 – )% Confidence Interval for µ
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
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: = 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.
- Calculate a 90% confidence interval for the target parameter. Interpret the results.
- 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.
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 t-statistic has (n – 1) ______(df).
Small-SampleConfidence Interval for µ
where ta/2 is based on (n – 1) degrees of freedom.
Conditions Required for a Valid Small-Sample Confidence Interval for µ
- A random sample is selected from the target population.
- The population has a relative frequency distribution that is approximately normal.
Estimation Example Mean (s Unknown)
A random sample of n = 25 has = 50 and s = 8. Set up a 95% confidence interval estimate for m.
Example:
Blood Pressure Increase (Points) for Six Patients1.7 / 3.0 / .8 / 3.4 / 2.7 / 2.1
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 , the mean increase in blood pressure associated with the new drug for all patients in the population.
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?