Download Estimating Population Mean NOTES

Section 8.3: Estimating a Population Mean
AP Statistics
Name: ____________________________________________
Date: ___________________________ Period: __________
To calculate the confidence interval for a population mean and 𝜎 is known, it follows the same procedure as calculating a
confidence interval for a population proportion with just 2 minor differences.
Conditions/Assumptions
a. Random?
b. Normal?
c. Independence? Check 10 % condition
When 𝛔 Is Known (A Very Rare Case)
The formula still has the general form for a confidence interval.
𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄 ± (𝒄𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒗𝒂𝒍𝒖𝒆)(𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒆𝒓𝒓𝒐𝒓 𝒐𝒇 𝒔𝒕𝒂𝒕𝒊𝒔𝒕𝒊𝒄)
We don’t use this very often because if we don’t know 𝜇, then we don’t know 𝜎. But we can still use the one-sample z
interval to estimate the sample size to achieve a specified margin of error.
Choosing a Sample Size:
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Example: To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly.
The scale readings are normally distributed with unknown mean. In previous studies, the standard deviation of scales
readings has been about 0.0002 grams. How many measurements must be averaged to get a margin of error of 0.0001 with
98% confidence?
When 𝝈 is NOT known – Introduction to t-Distributions
When we seek to capture the population mean 𝜇, by using a sample mean 𝑥̅ with a CI, chances are we do NOT know the
population standard deviation 𝜎. In reality, 𝜎 is not known because if you knew 𝜎, you would also know 𝜇 and have no
need to estimate it.
1) What do we use to estimate 𝜇? ________________________
2) What do we use to estimate 𝜎? _________________________
3) Rather than use 𝜎𝑥̅ =
𝜎
, what will the new formula be? __________________________________
√𝑛
When we use “s” in place of “𝜎”, the statistic that results does not have a normal distribution. It has a new distribution
called a _________________________________________________________.
t-statistics are used and interpreted the same way as a z-statistic, but there is a different t distribution for each sample.
We specify a t-distribution by giving its ______________________________________________________ which is
__________________________________________.
So this → 𝑥̅ ± 𝑧 ∗
𝜎
√𝑛
turns into this →
Describing the t-distribution:
The density curves of the t distribution are similar in shape to the ___________________________________________________.
S: As the degrees of freedom _______________________, the t-distribution approaches the ___________________________ curve ever
more closely. Why does this happen?
O: There should be few outliers, if any.
C: They are centered at ______________________, just like normal curves.
S: The spread of the t distributions is a bit greater than that
of the standard normal distribution. Why?
Table B (in the back of the book) and the back of your formula sheet gives critical values for the t distributions.
On your calculator, “invT” is found below “normalcdf” and “invNorm.”
Examples: What critical value t* from Table B satisfies each of the following conditions?
a. The t distribution with 10 degrees of freedom and has probability 0.01 to the right of t *.
b. The t distribution with 28 degrees of freedom and has probability 0.90 to the left of t *.
c. The one-sample t statistic from a sample of 42 observations that has probability 0.05 to the right of 𝑡 ∗.
d. The one-sample t statistic from a SRS of 980 observations and has probability 0.975 to the left of 𝑡 ∗.
e. A 95% CI based on a SRS of 15 observations.
f.
A 99% CI from a sample size of 110 observations.
g. A 90% CI based on 30 observations.
Constructing Confidence Intervals when 𝝈 is unknown:
1. Random – does it state it?
2. Normal (3 options)
3. Independence – 10% condition
Formula:
Examples:
1) Environmentalists, government officials, and vehicle manufacturers are all interested in studying the auto
exhaust emissions produced by motor vehicles. The major pollutants in auto exhaust from gasoline engines are
hydrocarbons, carbon monoxide, and nitrogen oxides (NOX). Researchers collected data on the NOX levels (in
grams/mile) for a random sample of 40 light-duty engines of the same type. The mean NOX reading was 1.2675
and the standard deviation was 0.3332. Construct and interpret a 95% confidence interval for the mean amount
of NOX emitted by light-duty engines of this type.
2) A company manufactures portable music devices—called “mBoxes”—for the listening pleasure of on-the-go
teens. The mBox uses batteries that are advertised to provide, on average, 25 hours of continuous use. The
students in Mr. Jones’s statistics class—looking for any excuse to listen to music while doing their homework—
decide to use their statistics to test this advertising claim. To do this, new batteries are installed in eight
randomly selected mBoxes, and they are used only in Mr. Jones’s class until the batteries run down. Here are the
results for the lifetimes of the batteries (in hours):
15
22
26
25
21
27
18
22
Construct and interpret a 90% CI for the mean number of hours (µ) that the batteries can be expected to last.