Chapter 10 Flashcards
Various patients have been treated for musculoskeletal injuries using either Acetaminophen or Ibuprofen, and each respondent reported showing significant improvement or slight improvement.
Assuming all requirements are met, which hypothesis test would be most appropriate to test the claim that the type of drug and the type of relief the patient received are related?
F-Test for the ratio of two variances
ANOVA
χ2 Test for Independence
χ2 Goodness of Fit Test
χ2 Test for Independence
This claim is about two variables being related, which means that the test for independence is most appropriate.
Hypotheses:
Ho: Type of drug and type of relief are independent.
Ha: Type of drug and type of relief are dependent.
When do you use
F-Test for the ratio of two variances
Purpose: Compares two population variances (assumed equal under 𝐻0 ).
Null Hypothesis (H0): Population variances are equal.
Alternative Hypothesis (H1): Population variances are not equal.
Conditions:
Samples must be random.
Samples must be independent.
Each population must have a normal distribution.
Formula: F = (sample variance 1) / (sample variance 2)
Degrees of Freedom:
Numerator: df = n1 - 1 (for sample variance 1)
Denominator: df = n2 - 1 (for sample variance 2)
When do you use ANOVA
When do you use χ2 Test for Independence
When do you use χ2 Goodness of Fit Test
Purpose: Tests if a frequency distribution matches an expected distribution (compares proportions).
Null hypothesis (H₀): The frequency distribution fits the expected distribution.
Alternative hypothesis (H₁): The frequency distribution does not fit the expected distribution.
Test type: Two-sided, but primarily concerned with the right side.
Conditions for Chi-Square Goodness-of-Fit Test:
The observed frequencies must come from a random sample.
Each expected frequency must be ≥ 5 for all categories.
Formula:
𝜒^2=∑(𝑂−𝐸)^2/E
Where:
O = Observed frequency
E = Expected frequency
An urban planner claims that the number of accidents on a certain city street is uniform by the day of the week.
Which hypothesis test would be most appropriate to test the planner’s claim, assuming all requirements are met?
χ2 Goodness of Fit Test
These are the hypotheses:
Ho: The outcomes occur in a uniform distribution (all categories have the same frequency).
Ha: The outcomes do not occur in a uniform distribution
A state school administrator claims that the standard deviations of U.S. history assessment test scores for eighth-grade students are the same in Districts 1 and 2. A sample of 10 test scores from District 1 has a standard deviation of 30.9 points, while a sample of 12 test scores from district 2 had a standard deviation of 34.5 points.
Using a 5% level of significance, which of the following information would be used to find the critical value from the F-tables?
dfN=11,dfD=9,α=0.025
Since the claim is that the standard deviations (and variances) are the same, that means that the alternative hypothesis is Ha: σ21σ22≠1which suggests a two-tailed test.
Therefore, use the table with α=0.025
(half of 0.05).
Since district 2, based on 12 scores, has the larger sample standard deviation, dfN=11
Since district 1, based on 10 scores, has the smaller sample standard deviation, dfD=9
Thus, the critical value is found using dfN=11,dfD=9,α=0.025
t or f for chi-square goodness of fit test
df = n - 1, where n is the sample size.
The degrees of freedom in a chi-square goodness-of-fit test are k - 1, where k is the number of categories.
A plant manager wants to test the claim that the new machines they use help to make production times more consistent than the old machines did.
Assuming all requirements are met, which hypothesis test is most appropriate to use to test the manager’s claim?
F-Test for the Ratio of Two Variances
Which of the following is the sample size requirement for either of the chi-square tests?
The sample size must be at least 30.
Each outcome must have an expected count that is at least 5.
Each outcome must have an observed count that is at least 5.
Each count must have an expected count of at least 5
A restaurant owner claims that their average revenues are the same at each of their 4 locations.
Which hypothesis test is most appropriate to use to test this claim, assuming all requirements are met?
ANOVA is used to test whether at least 3 population means are equal to each other, or if at least one is different from the others.
The hypotheses for this claim:
Ho: μ1=μ2=μ3=μ4
Ha: At least one of the means is different from the others.
A manufacturer claims that the distribution of the six colors of candies across all large packages is uniform.
If the null hypothesis is rejected, how is this decision interpreted?
ANOVA
χ2 Test for Independence
F-Test for the Ratio of Two Variances
χ2 Goodness of Fit Test
How to tell the difference between ANOVA
χ2 Test for Independence
F-Test for the Ratio of Two Variances
χ2 Goodness of Fit Test
ANOVA: Compares means of 3+ groups.
χ² Independence: Tests if two categorical variables are related.
F-Test: Compares two variances.
χ² Goodness-of-Fit: Checks if data fits an expected distribution.
The four hypothesis tests covered in this chapter are listed to the right.
Which of these are always right-tailed tests? Select all that apply.
F-Test for the ratio of two variances
χ2 Test for Independence
ANOVA
χ2 Goodness-of-Fit Test
χ2 Test for Independence
ANOVA
χ2 Goodness-of-Fit Test
Find the expected frequency, Upper Ei, for the given values of n and pi.
n=150, pi=0.1
Ei = npi
Finding the Expected Frequency in a Contingency Table
Formula:
E=
(RowTotal)×(ColumnTotal)/grand total
Steps:
Find the row total for the cell.
Find the column total for the cell.
Multiply these two values together.
Divide the result by the grand total (sum of all observations).
Purpose:
Ensures expected values reflect the data proportions.
Assumes independence between row and column variables.
Example:
If a cell is in a row with a total of 50, a column with a total of 40, and the grand total is 200, then:
𝐸=50×40/200= 10
The characteristic “Has d.f. = (r - 1)(c - 1)” applies to the…..
Requires data to be from random sample
Has d.f.=k-1
Use this formula: Ei=npi
Uses the following formula to find the expected frequency:
Er,c =
(RowTotal)×(ColumnTotal)/ (SampleSize)
The characteristic “Has d.f. = (r - 1)(c - 1)” applies to the chi-square independence test
independence and goodness of fit
Has d.f.=k-1
Chi-square goodness-of-fit test
goodness-of-fit test
INdependence
If the test statistic for the chi-square independence test is large, you will, in most cases, reject the null hypothesi
true!
When the differences between the observed frequencies and expected frequencies are large, the chi-square test statistic is also large. A large chi-square test statistic is evidence for rejecting the null hypothesis.
Null dependent vs independent
The alternative hypothesis always represents the idea that there is an association (dependence) between the variables.
The null hypothesis always represents the idea that there is no association (independence) between the variables.
Explain how to find the critical value for an F-test.
- Specify the level of significance, alpha.
- Determine the degrees of freedom for the numerator (
d.f.n ) and denominator (
d.f.D ). - Find the critical value of F using technology or the F-distribution table.
List five properties of the F-distribution.
- All values of F are greater than or equal to 0. F values can never be negative.
- The degrees of freedom corresponding to the variance in the numerator, denoted by Dfn and the degrees of freedom corresponding to the variance in the denominator, denoted by
d.f. d - The F-distribution is
positively skewed and therefore the distribution is not
symmetric. - The total area under each F-distribution curve is equal to
- For all F-distributions, the mean value of F is approximately equal to 1
List the three conditions that must be met in order to use a two-sample F-test.
- The samples must be randomly selected.
- The samples must be independent.
- Each population must have a normal distribution.
value of dfn and dfd=…
The value of d.f.N is equal to n 1 -1, and the value of d.f. D is equal to n 2 -1, where n 1 and n 2 represent the sample sizes of the numerator and denominator (respectively).
Anova null vs alt
null= all pop means =
Alt= one isnt=
