Download use when n 30 or σ known with a normally distributed population

Probability & Statistics
Name:
Confidence Intervals for the Mean (Large Samples)
Section 6.1
Date:________________Period:
______
(Use when n  30 or σ known with a normally distributed population)
Point estimate:
Example: A random sample of 32 textbook prices (rounded to the nearest dollar) is taken from a local
college bookstore. Find a point estimate for the population mean, .
Interval estimate:
Level of Confidence
Common Levels of Confidence
Level of Confidence
90%
95%
99%
For instance, if the level of confidence is 90%, this means that we are 90% confident that the
interval contains the population mean μ.
sampling error:
When μ is estimated, the sampling error is the difference μ –. . Since μ is usually unknown, the
maximum value for the error can be calculated using the level of confidence.
Given a level of confidence, the margin of error (sometimes called the maximum error of estimate or
error tolerance) E is:
Example:
A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample
is x = 74.22, and the sample standard deviation is s = 23.44. Use a 95% confidence level to find the
margin of error for the mean price of all textbooks in the bookstore.
Example:
1. If CI: -2.17 <  < 2.57, find x and E.
2.
Find the critical value Zc, for 96% degree of confidence.
Confidence Interval
Example:
A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample
is x = 74.22, the sample standard deviation is s = 23.44, and the margin of error is E = 8.12. Construct a
95% confidence interval for the mean price of all textbooks in the bookstore.
Constructing a Confidence Interval for a Population Mean
Example:
A random sample of 25 students had a grade point average with a mean of 2.86. Past studies have shown
that the standard deviation is 0.15 and the population is normally distributed. Construct a 90%
confidence interval for the population mean grade point average.
Practice:
39 packages are randomly selected from packages received by a parcel service. The sample has a mean
weight of 10.0 pounds and a standard deviation of 1.6 pounds. What is the 95% confidence interval for
the true mean weight of all packages received by the parcel service?
Sample Size
For the same sample statistics, as the level of confidence increases, the confidence interval __________.
As the confidence interval widens, the precision of the estimate ___________.
One way to improve the precision of an estimate without decreasing the level of confidence is to
___________the sample size. But how large a sample size is needed to guarantee a certain level of
confidence for a given margin of error?
Example:
You want to estimate the mean price of all the textbooks in the college bookstore. How many books
must be included in your sample if you want to be 99% confident that the sample mean is within $5 of
the population mean?