Chapter 8 (Inference) Flashcards
What are confidence intervals?
They facilite the estimation of population parameters, address questions like “what is mean body temperature?”
What are hypothesis tests?
They enable us to compare a population parameter to a specified value. Address questions such as “is the mean body temperature 98.6?”
What is a (1-alpha)100% confidence interval? (CI)
For an unknown parameter it Contains a set of plausible values of the parameter that are consistent with the data.
What is a confidence interval composed of?
A point estimate +- the margin or error
What is the a point estimate?
The value of the statistic. Asked in the chosen sample.
What is the margin of error?
It’s based on the standard error and a desired level of confidence.
Critical value * standard error
What is a critical value?
t_{alpha/2}
What is the area to the right of a t_{alpha/2}?
Alpha/2
How to find a confidence interval for the mean
X_bar ± (t_{alpha/2, n-1}) * s/sqrt(n)
What is the difference between the standard deviation and the standard error?
The standard deviation is how much a data set varies
s.d. = sqrt(sigma^2)
The standard error is how much our statistic distribution set varies
s.e. = sigma/sqrt(n)
What is the standard error of p_hat?
s.e.(p_hat) = sqrt((p_hat(1-p_hat)/n)
What is the confidence level?
(1-alpha)100%
Most common values for alpha?
.05, or .01.
Thus CIs are often 95% or 99%
If not given, assume alpha = .05
Why do we accept some probability that the interval doesn’t cover our parameter?
To infer something meaningful about the population.
The interval is _____
The parameter is ______
random,
fixed!!
Which table do we use to infer the proportion?
The normal table!
Which table do we use to infer the mean?
The t-table!
A confidence interval makes a(n) ________,
A hypothesis test makes a(n) _______.
Estimate,
Comparison
Hypothesis testing allows us to do what?
Asses the plausibility of a specific statement of hypothesis.
The null and alternative hypothesis are…?
Complementary statements about the parameter of interests
How to denote the null hypothesis?
H_o
h_naught
How to denote the alternative hypothesis?
H_A
The H_o takes on the _______ statement/portion of the question.
Simple! Which means one value.
The H_A takes on the _______ statement/portion of the question.
Composite, the one with many values!
What is a two sided hypothesis?
Where H_A is simply not H_o
H_o: µ = µ_o
H_A: µ ≠ µ_o
What is a one sided hypothesis?
Where H_A > H_o - H_o: µ = µ_o versus H_A: µ > µ_o or Where H_A < H_o - H_o: µ = µ_o versus H_A: µ < µ_o
What is a test statistic?
A numeric summary of our experimental results that can be used to determine how consistent the data are with the null hypothesis.
When conducting a hypothesis test for a MEAN, what test statistic will we use?
T= sqrt(n)(X_bar - µ_o) ----–-–-–––––––––– s aka T= (X_bar - µ_o) ----–-–-–––––––––– s/sqrt(n)
What is alpha?
The probability that the confidence interval does NOT cover the parameter.
How do we denote the normal distribution of p_hat?
p_hat ~ N (p, (p(1-p))/n)
What is the hypothesis test process?
- State hypotheses about the parameter
- Collect data
- Construct a test statistic
- Compute a p-value
- Draw conclusions (in statistical terms and in context)
What does the test statistic do?
It quantifies how different what we observe is from what we expect.
It takes into account how much we’d expect the value of the statistic to vary by chance.
When we observe an extreme value for our test statistic, we have evidence ________ the null hypothesis
against.
When conducting a hypothesis test for a PROPORTION, what test statistic will we use?
Z = (p_hat - p_o) / sqrt[(p_o *(1-p_o))/n] ~ N(01)
using p_o to compute the s.e. since, under H-o, we assume p_o is the true value of p.
What is a p-value?
the probability of obtaining the data we observed or data more extreme (less consistent with H-o) IF the null hypothesis is true.
For H_A: µ ≠ µ_o, the p-value =
P(|T| > t_{n-1} | H_o is true)
For H_A: µ > µ_o, the p-value =
P(T > t_{n-1} | H_o is true)
For H_A: µ < µ_o, the p-value =
P(T < t_{n-1} | H_o is true)
What does a small p-value tell us?
That our data are unlikely to occur if the null hypothesis is true, thus it provides evidence agains H_o.
In the term “small” p-value, what is small defined by?
the significance level, or alpha, or size of the test.
Unless stated otherwise, we assume alpha = .05
What is the significance level, or alpha?
The size of the test
When should the significance level, or alpha, be specified?
before the test is conducted.
If p < alpha, do we accept/reject the H_o?
We reject H_o, or reject the null, and say the results are statistically significant.
If p ≥ alpha, do we accept/reject the H_o?
We fail to reject the H_o.
If p > alpha but close to it, do we accept/reject the H_o?
We may say that there is marginal evidence against the null
Does statistical significance imply practical significance?
No.
What does a hypothesis test determine?
If there is evidence against the null hypothesis
What does a small p_value indicate?
That the null hypothesis isn’t plausible, and thus there is evidence against the null hypothesis.
What is the confidence interval for proportion?
(p_hat ± (Z_{alpha/2} * sqrt[p_hat(1-p_hat)/n])
What is the confidence interval for the mean?
(X_bar ± ( t_{alpha/2,n-1} * (s/sqrt[n]))
What is the connection between confidence intervals and the hypotheses test?
The confidence interval values are the same values for which we would fail to reject the null hypothesis.
