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Unit 7: Inference for Quantitative Data 📝 - Coggle Diagram
Unit 7: Inference for Quantitative Data
📝
Significance Test for a Population Mean
State
Define the
null hypothesis
w/ respect to pop. mean
Define the
alternative hypothesis
w/ respect to pop. mean
State significance/alpha level
Plan
- refer to steps in CI for a pop. mean
"All conditions met, perform a one-sample t-test for population mean."
Do
By t test statistics
use
tcdf
w/ t test statistics (two-sied alternative hypothesis? multiply p-value by 2)
By calculator: T-Test
Conclude
p-value greater than sig. level
"Since the (p-value) is greater than the (sig. level), we fail to reject the null hypothesis and we do not have convincing evidence for the alternative hypothesis (+ context)."
p-value smaller than sig. level
"Since the (p-value) is less than the (sig. level), we reject the null hypothesis and we have convincing evidence for the alternative hypothesis (+ context)."
interpret p-value
"If the null hypothesis is true, the the probability of getting a sample mean greater than/less than (null-value) is (p-value).
Summary of Important Points
Matched pairs
- when experimental units are paried up based on similar characteristics yet one individual is assigned to carry out one treatment while the other belongs to the control group
t-distribution
Similar to a normal distribution in its bell-shaped curve, symmetry, and single peak
How is it different from z-scores? Used to approximate
quantitative data
(how far sample mean is from population mean)
Spread is greater than z-distribution (why? t-dist. uses sample standard deviation)
Degrees of freedom (df)
- the number of values in calculation of a statistic that is free to vary
df increase? Spread decrease -> approach normal distribution
Constructing a Confidence Interval for the Difference of Two Means
State
- define BOTH pop. means
Plan
- check conditions for BOTH samples
"All conditions met, construct a two-sample t-interval to estimate the difference in two population means."
Do
calculate confidence interval for difference in mu(s)
By calculator: 2-SampTInt
t-critical value
area: same as one-sample
df: CANNOT do n-1, refer to df from calculator
Conclude
- refer to CI for a pop. mean
Significance Test for the Difference in Means
Conclude
fail to reject or reject null: same as one-sample, except add "difference in pop. mean"
interpret p-value
"If the null hypothesis is true, the probability of getting a sample mean
at least (if alt. hyp. is greater than 0)/ at most (if alt. hyp. is less than 0)
is (p-value).
State
null hypothesis: pop. mean 1 - pop. mean 2 (or vice versa) = 0
alt. hypothesis: pop. mean 1 - pop. mean 2
not equal to 0
greater than 0
less than 0
Plan
- check conditions for both samples
"All conditions met, perform a two-sample t-test for difference in population means."
Do
By t test statistics
use
tcdf
w/ t test statistics (two-sied alternative hypothesis? multiply p-value by 2)
By calculator: 2-Samp TTest
Reminder: make sure your x-bar 1 and x-bar 2 values are placed in the right place! The calculater does x-bar 1 minues x-bar 2
Construct a Confidence Interval for a Population Mean
State
Define parameters: Use mu for population mean, NOT x-bar for sample mean!
Write a complete sentence detailing what we will estimate via the four-step plan
Plan
Random: sample procured via random sampling (SRS) or random assignment
Normality:
Central Limit Theorem
population mean is normal? can assume sample distribution is also normal
pop. mean not normal? if sample size (n) greater than or equal to 30 -> sampling distribution approx. normal
n not greater than or equal to 30?
graph sample data
, if (1) no severe skew nor (2) outliers -> sample distribution approx. normal
Independence: 10% condition for random sampling, "what is being measured is independent" for random assignment
"All conditions met, construct a one-sample t-interval to estimate the population mean."
Do
calculating confidence interval for mu
By calculator: TInterval
t-critical value (invT)
df: n-1
area: (1 - confidence level)/2
Conclude
Confidence interval: "We are __% confident that the interval from (lower bound) to (upper bound) captures the true population mean (+ add context)."
Confidence level: "If we take many samples of the same sample size, approx. __% of the confidence intervals constructed will capture the true population mean (+ context)."
