One sample T hypothesis test for Mean μ

Underlying machinery: the sampling distribution of x̄, the (estimated) standard error of the mean (SEM)≈ s/√n, the t distribution.

Null hypothesis H₀: already "known", [current] belief, established, default, status quo, old, pre-existing, current practice, well-known, consensus, reference, "true until proven otherwise", original, working assumption, nothing new, boring. The parameter μ equals some number a; there is no significant difference between μ and x̄.
Alternative hypothesis H_A: new, exciting, hoped/wished, changed, different, research, challenger, the conjecture. Either the parameter μ<a, or μ>a, or μ≠a; there is a difference, there is an effect that is not caused by random chance. AKA H₁
Test if the sample (i.e. its statistic x̄ and its size n) provides enough evidence to overthrow ("warrant rejection of") the null hypothesis. Is the sample statistic x̄ extreme enough (i.e. very improbable under the null hypothesis).
Either "reject" or "fail to reject" the null hypothesis; never "accept" it. Rejecting it ≡ "support" the alternative.
The alternative hypothesis is neither rejected nor accepted.
Nothing is ever "proven". (would need entire population to prove anything) We never have certainty.

T-Test for mean μ. Uses μ, s, x̄, and n. Test statistic is t.

Assumes population is normal or n≥30 and no outliers or heavy tails.
Sample must be SRS random.

The test statistic t is a measure of discrepancy between the sample statistic x̄ and the H₀ claimed value of the population parameter μ.
t = (observed x̄ - expected μ) / SEM

Given null hypothesis H0: μ = a
Choose one:
HA: parameter μ < a	"HA < H0"	Left-tailed
HA: parameter μ > a	"HA > H0"	Right-tailed
HA: parameter μ ≠ a	"HA ≠ H0"	Two-tailed

"μ":
x̄:    s:
n:

t:		df:	estimated SEM=s/√n:	skip: set t and df, look at p-value below

   Critical value: α=0.05: α=0.01:
If t is outside/beyond the bounds of the critical value, i.e. |t|>|crit val|, reject H₀.
   If Left-tailed and t≤CritValue then Reject H₀ at that α level.
   If Right-tailed and t≥CritValue then Reject H₀ at that α level.
   If Two-tailed and |t|≥|CritValue| then Reject H₀ at that α level.

p-value (TCDF(t,df)): if p < α, reject H₀

Chance that the test statistic t would be this much or more if H₀ were true.
                            (this little or less) (this extreme or more).
"If the p is low, the null must go." Too unlikely to have occurred by chance if H₀ were true.
Typically the critical/rejection region ("level of significance", α) is chosen to be .05 or .01, so if p is less than it reject H₀; if p is not less than the critical value don't reject H₀ ("fail to reject" = "cannot disprove").
Probability (area) in a tail (or two) of the test statistic's PDF curve.
If p is high (bigger than α), can't reject H₀.
Selecting Two-tailed case doubles the p-value over the One-tailed cases.
Mean One-tailed tests are "symmetric".
Tip: if the p-value is like .9, check that you selected the appropriate "tail" above before failing to reject.

A confidence interval and a two-sided hypothesis test are two sides of the same coin:
 Any value of μ that lies inside a 95% confidence interval would not be rejected by a two-sided test at α=0.05.
 Any value of μ that lies outside the 95% confidence interval would be rejected by that same test.
So the confidence interval is essentially the set of all hypothesized values that 
 are still considered plausible after seeing the data.

Exs.
Worksheet
μ=100 x̄=100 s n irrelevant. Can't reject claim that μ is 100.
μ=100 x̄=105 s=10 n=36 →SEM=1.66 An x̄≈3σ very unlikely → Reject claim that μ is 100.
μ=100 x̄=101.7 s=10 n=36 →SEM=1.66 An x̄≈1σ likely → Can't reject claim that μ is 100.. Can't reject claim that μ is 100.

T-test: μ=100, s=10, n=30. Try x̄= 102, 103, 104, 105. Ha>H0
T-test: μ=100, s=10, n=30. Try x̄= 102, 103, 104, 105. Ha≠H0

Effect of s:
T-test: μ=100, s=5, n=30. Try x̄= 101, 102, 103. Ha>H0
T-test: μ=100, s=5, n=30. Try x̄= 101, 102, 103. Ha≠H0

Effect of n:
T-test: μ=100, s=10, n=100. Try x̄= 101, 102, 103. Ha>H0
T-test: μ=100, s=10, n=100. Try x̄= 101, 102, 103. Ha≠H0