6.4: Confidence Intervals for Variance and Standard Deviation
The Chi-Square Distribution
2
The point estimate for is
The point estimate for is
2
2
s is the most unbiased estimate for
You can use the chi-square distribution to construct a confidence
Chapter 6 Case Study:
_
Shoulder Heights of Appalachian Black Bears
Name:
Appalachian Bear Rescue (ABR) is a not-for-profit organization located near the Great Smokey
Mountains National Park. ABRs programs include the rehabilitation of orphaned and injure
7.1: Introduction to Hypothesis Testing
Hypothesis test
For example: An automobile manufacturer advertises that its new hybrid car has a
mean mileage of 50 miles per gallon. To test this claim, a sample would be taken. If the
sample mean differs enough fr
7.1B: Hypothesis Testing
Level of significance
Your maximum allowable probability of making a type I error.
By setting the level of significance at a small value, you are saying that you want the
probability of rejecting a true null hypothesis to be smal
7.2B
Rejection Regions and Critical Values
Rejection region (or critical region)
Finding Critical Values in a Normal Distribution
1.
2.
3.
a. left-tailed,
b. right-tailed,
c. two-tailed,
4.
Example 1: Find the critical value and rejection region for a two
7.2: Hypothesis Testing for the Mean (Large Samples)
Decision Rule Based on P-value
To use a P-value to make a conclusion in a hypothesis test, compare the P-value with .
1.
2.
Example 1: The P-value for a hypothesis test is P = 0.0237. What is your deci
7.3: Hypothesis Testing for the Mean (Small Samples)
Finding Critical Values in a t-Distribution
1. Identify the level of significance, .
2. Identify the degrees of freedom d.f. = n 1.
3. Find the critical value(s) using Table 5 in Appendix B in the row w
7.4: Hypothesis Testing for Proportions
z-Test for a Population Proportion
A statistical test for a population proportion.
Can be used when a binomial distribution is given such that np 5 and nq 5.
The test statistic is the sample proportion
The stand
7.5: Hypothesis Testing for Variance and Standard Deviation
2
Finding Critical Values for the -Test
1. Specify the level of significance, .
2. Determine the degrees of freedom d.f. = n 1.
3. The critical values for the 2-distribution are found in Table 6
8.1: Testing the Difference Between Means (Large Independent Samples)
Two Sample Hypothesis Test
Compares two parameters from two populations.
Sampling methods:
Independent Samples
Dependent Samples (paired or matched samples)
Example 1: Classify the pa
8.2: Testing the Difference Between Means (Small Independent Samples)
Two Sample t-Test for the Difference Between Means
If samples of size less than 30 are taken from normally-distributed populations, a t-test may be used to
test the difference between t
8.3: Testing the Difference Between Means (Dependent Samples)
t-Test for the Difference Between Means
To perform a two-sample hypothesis test with dependent samples, the difference between
each data pair is first found:
d = x1 x2 Difference between entrie
8.4: Testing the Difference Between Proportions
Two-Sample z-Test for Proportions
Used to test the difference between two population proportions, p and p .
1
2
Three conditions are required to conduct the test.
1.
2.
3.
If these conditions are met, then t
9.1A: Correlations
Correlation
A relationship between two variables.
The data can be represented by ordered pairs ( x, y)
x is the
y is the
A scatter plot can be used to determine whether a linear (straight line) correlation exists
between two variabl
9.1B: Correlations Continued
Example 1: Finding the Correlation Coefficient
Calculate the correlation coefficient for the advertising expenditures and company sales
data. What can you conclude?
Advertising
Expenses,
($1000), x
2.4
1.6
2
2.6
1.4
1.6
2
2.2
9.1C: Correlations & Causations
Hypothesis Testing for a Population Correlation Coefficient
A hypothesis test can also be used to determine whether the sample correlation
coefficient r provides enough evidence to conclude that the population correlation
9.2: Linear Regression
Regression Lines
After verifying that the linear correlation between two variables is significant, next we
determine the equation of the line that best models the data (regression line).
Can be used to predict the value of y for a
9.3: Measures of Regression and Prediction Intervals
Variation About a Regression Line
Three types of variation about a regression line
To find the total variation, you must first calculate
The total deviation
The explained deviation
The unexplained dev
10.1A: Goodness of Fit
Multinomial Experiment
Example: A radio station claims that the distribution of music preferences for listeners in the
broadcast region is as shown below.
Distribution
Classical
Country
Gospel
of
4%
36%
11%
Music
Oldies
Pop
Rock
Pre
10.1B: Chi-Square Goodness-of-Fit Test
Chi-Square Goodness-of-Fit Test
For the chi-square goodness-of-fit test to be used, the following must be true.
1.
2.
If these conditions are satisfied, then the sampling distribution for the goodness-of-fit test is
10.2: Contingency Tables & Independence
r c contingency table
Shows the observed frequencies for two variables.
The observed frequencies are arranged in r rows and c columns.
The intersection of a row and a column is called a
Example:
The contingency
10.4: Analysis of Variance
One-way analysis of variance
Hypotheses:
H0: 1 = 2 = 3 = k (all population means are equal)
Ha: At least one of the means is different from the others.
In a one-way ANOVA test, the following must be true.
1.
2.
3.
One-Way ANOVA
10.3: Comparing Two Variances
F-Distribution
Properties of the F-Distribution
1. The F-distribution is a family of curves each of which is determined by two types of degrees of
freedom:
The degrees of freedom corresponding to the variance in the numerato
6.1: Confidence Intervals for the Mean (Large Samples)
Point Estimate
Example 1: Market researchers use the number of sentences per advertisement as a measure
of readability for magazine advertisements. The following represents a random sample of the
numb
6.1 Homework
Name: _
1. Find the critical value that corresponds to a 94% confidence level.
For 2 - 5, Find the Margin of Error
2. Determine the sampling error if the grade point averages for 10 randomly selected students
from a class of 125 students has
6.2 Confidence Intervals for the Mean (Small Samples)
The t-Distribution
When the population standard deviation is unknown, the sample size is less than 30,
and the random variable x is approximately normally distributed, it follows a tdistribution.
Crit
6.3: Confidence Intervals for Population Proportions
Point Estimate for Population p
Population Proportion
Point Estimate for p
Point Estimate for q, the proportion of failures
Example 1: In a survey of 1219 U.S. adults, 354 said that their favorite sport