Download Inferences about means

INFERENCES
ABOUT MEANS
CHAPTER 23
When dealing with quantitative data we are talking about means
instead of proportions. We should always start our analysis with
a histogram of the data.
The Central Limit Theorem still holds and our sampling
distribution will still be approximately normal as long as our
sample size is large enough. The standard deviation of our
𝝈
sample is :
𝒏
We can’t calculate SE from quantitative data like we could for
𝒔
proportions, so we estimate it with:
.
𝒏
We can’t use the Normal model for this, since the sample size
affects the model, so instead we will use the Student’s t-model,
which requires naming the degrees of freedom (df) which is n-1.
T models are unimodal, symmetric, and bell shaped, but have
fatter tails than the normal model. The more degrees of freedom,
the more a t model approaches a normal model.
Sampling Distribution Model for Means:
Instead of calculating z as our sample statistic, we use t.
𝒚−𝝁
𝒕=
𝑺𝑬(𝒚)
With n-1 degrees of freedom.
𝒔
𝑺𝑬 𝒚 =
𝒏
One-sample t-interval:
𝒚 ± 𝒕∗ 𝒏−𝟏 ∗ 𝑺𝑬 𝒚
Assumptions & Conditions
Independence: Randomization and 10%
Normal Population Assumption:
• Nearly Normal condition – you MUST make a histogram of the
data. If n<15 if must be clearly normal. Between 15 and 40 the
data needs to be unimodal and reasonably symmetric. Above 40
t model is good unless the data is extremely skewed.
Hypothesis testing for means
One sample t-test for the mean:
𝒕𝒏−𝟏
𝒚 − 𝝁𝟎
=
𝑺𝑬(𝒚)
A coffee machine dispenses coffee into paper cups. You’re
supposed to get 10 ounces of coffee, but the amount varies slightly
from cup to cup. Here are the amounts measured in a random
sample of 20 cups. Is there evidence that the machine is
shortchanging customers?
9.9
9.7
10.0
10.1
9.9
9.6
9.8
9.8
10.0
9.5
9.7
10.1
9.9
9.6
10.2
9.8
10.0
9.9
9.5
9.9
A company has set a goal of developing a battery that lasts over 5
hours (300 minutes) in continuous use. A first test of 12 of these
batteries measured the following lifespans (in minutes): 321, 295,
332, 351, 281, 336, 311, 253, 270, 326, 311, and 288.
• Is there evidence that the company has met its goal?