Foundations of Hypothesis Testing
What does it mean when results are statistically significant?
It means the observed results are unlikely under the assumption that the null hypothesis is true → we reject H₀
Which program command path do we use in StatCrunch?
Stat → T Stats → One Sample → with summary
What does a p-value represent in words?
The probability of obtaining a sample mean as extreme or more extreme than observed, if H₀ were true.
What is a Type I error?
Rejecting H₀ when it is true.
In the medical analogy, what does the null hypothesis (H₀) represent?
That the patient is healthy (no disease / no effect).
What does “H₀” represent?
The null hypothesis, a statement of no change or no difference (status quo).
What four inputs does StatCrunch require?
Sample mean, sample standard deviation, sample size, and mean under H₀.
If p = 0.045 and α = 0.05, what do we conclude?
Reject H₀ – the result is statistically significant.
What is a Type II error?
Failing to reject H₀ when it is false.
What does the alternative hypothesis (Hₐ) represent
That the patient has the disease (there is an effect).
What does “Hₐ” represent?
The alternative hypothesis, a claim that contradicts H₀ and represents a change or effect.
What does the “t-stat” value next to the p-value represent?
The computed t-test statistic measuring how far the sample mean is from μ₀ in standard error units
Interpret p = 0.35 for α = 0.05.
Fail to reject H₀ – insufficient evidence to support Hₐ.
When you fail to reject H₀, what phrase should appear in your conclusion?
“There is insufficient evidence at the α = __ level to conclude that …”
What is a Type I Error in the medical analogy?
A false positive, the test says the patient is sick when they are actually healthy (rejecting a true H₀).
What is α = 0.05 called?
The level of significance – probability of rejecting H₀ when H₀ is actually true (Type I error rate).
When do we use a two-tailed test?
When the alternative hypothesis states that the population mean is different from μ₀ (≠ sign).
In the SAT example, p = 0.035 means what?
There’s a 3.5 % chance of observing a sample mean of 485 or lower if the true population mean is 501
When you reject H₀, what phrase should appear?
“There is sufficient evidence at the α = __ level to conclude that …”
What is a Type II Error in the medical analogy?
A false negative, the test says the patient is healthy when they are actually sick (failing to reject a false H₀).
If the p-value is less than α, what decision do we make?
Reject the null hypothesis → there is sufficient evidence to support Hₐ.
What is the test statistic formula for a one-sample t-test?
t = (x̄ - μ₀) / (s / √n)
What type of error occurs if H₀ is actually true but we reject it?
Type I error.
In the commute-time study, what type of error would occur if the mean was still 27.5 min but we rejected H₀?
Type I error – we claimed a reduction when none existed.
How does the level of significance (α) relate to the test’s “settings” in the medical analogy?
It’s like the sensitivity threshold; lowering α means you require stronger proof before calling someone “sick,” reducing false positives (Type I errors) but increasing the risk of false negatives (Type II errors).