Chapter 9
9.1
9.2
9.3
9.5
9.6
MIDTERM
Constructing a confidence interval for a mean. x̄±z*σ/√n
for Margin of error E, you need a sample size of
N=(z*σ/E)^2
Conditions: Check if sample is an SRS and that 10 times it size is less than the population
Run a significance test for two means:
1) Check conditions (SRS, n < .10 of population)
2) Set up null and alternate hypotheses
3) calculate summary statistic: z=x̄-µ0/(σ/√n)
compare the observed statistic to the critical value z which is determined via the preassigned significance level α of the test. if z>z, do not reject null. Then use the summary statistic z to calculate the p-value and compare the p-value to the α level. if p<α, you have evidence to suggest the sample result is statistically significant
when the population standard deviation σ is unknown, we replace σ with sample standard deviation s, and run a t-test instead of a z-test.
t= (x̄-µ0)/(s/√n)
find t* using t and DF, DF=n-1
Samples have been randomly and independently selected from two different populations
The two populations are approximately normally distributed or the sample sizes are large enough
The standard deviations σ1 and σ2 are unknown. Use t-tests and t-procedures.
If they are known use z-procedures.
A confidence interval for the difference µ1-µ2 between the means of two populations has the form (x̄1-x̄2)±t*√(s1^2/n1)(s2^2/n2)
The test statistic for a test of significance is
t=(x̄1-x̄2)/√(s1^2/n1)(s2^2/n2)
Matched pair design comparison- take two samples and make matched pairs with each of their individual cases to block any external factors. Make a list L3 = L1-L2 then run all your tests using that list, if σ is known use z-procedures. if σ is unknown use t-procedures
Chapter 10
10.1
X^2 is not a normal distribution and is always skewed rightwards. DF=# of outcomes - 1.
X^2 = Σ[((O-E)^2)/E].
For X^2 GOF:
1) Check conditions: Each outcome falls into exactly one of a fixed number of categories, You can make hypothesized proportions, SRS, E is 5 or greater in each category
2) State Hypothesis:
Ho: Proportions = model
Ha : At least one proportion not equal to model
3) compute test statistic, Find P-value:
X^2 = Σ[((O-E)^2)/E], then use X^2-cdf, use DF and tail probability (α)
4) reject null hypothesis and write a conclusion about the significance of your evidence
10.2
X^2 Homogeneity test checks whether it is reasonable to believe that when several different populations are broken down into the same categories, they have the same proportions of members in each category.
X^2 homogeneity test:
1) Check conditions:
Independent SRS of two or more populations
Each outcome falls into one of several categories
2) Hypothesis:
Ho: The proportion that falls into each category is the same for every population
Ha: For at least one category it is not the case that each population has the same proportion in that category
3) compute test statistic.
X^2 = Σ[((O-E)^2)/E]
DF=(r-1)(c-1) = (# of populations - 1)(# of categories - 1)
or
use the matrices and X^2 functions for matrices in calculator
X^2 * use table (DF and α)
4) reject null and conclude the significance of your evidence
10.3
X^2 test for independence: Check to see whether one trait has an affect on the chance proportions or population proportions of people with another trait. EG are survival rate on the titanic and gender of passengers independent
1) check conditions: SRS, outcomes can be classified into one of several categories on both variables, E is known, E is greater than or equal to 5
2) Hypothesis: Ho: Two variables are independent
Ha: two variables are not independent
3) X^2 test (independence)
X^2 = Σ[((O-E)^2)/E]
DF=(r-1)(c-1) = (# of populations - 1)(# of categories - 1)
TI-84; 2nd x^-1 --> matrix
Ti-Inspire --> matrix --> Create
4) reject null and conclude the significance of your evidence
11.2
Lin reg t-test:
1) check conditions: You have a random sample, The relationship between variables is linear, residuals have equal standard deviations across all values of x, residuals are normal at each fixed x
2) State Hypothesis:
Ho : b1 = 0 ; no linear relationship
Ha : b1 /= 0; linear relationship
3) run linear reg t-test and input L1-->x, L2 --> y, t* = invt (area, DF)