2- Introduction to Hypothesis Testing Going bowling with your Mum

Source: Udemy

In the bowling example, what is the null hypothesis (H₀)?

A) Sam's true long-term bowling average is 150

B) Sam is lying about his average

C) Sam's true long-term bowling average is below 150

D) Sam's three-game sample average is 40

Hint: The null hypothesis is the claim being tested, not the counter-claim.

Answer: A) Sam's true long-term bowling average is 150

Explanation: Right on the pins! H₀ is always the claim under test — here, Sam's assertion that his true long-run average is 150. We assume it's true until the sample evidence is strong enough to reject it.

What is the alternative hypothesis (H₁) in this scenario, and what kind of test does it imply?

A) His average is below 150 — a one-tailed (left-tailed) test

B) His average is not 150 (could be higher or lower) — a two-tailed test

C) His average is above 150 — a one-tailed (right-tailed) test

D) His average equals exactly 40

Hint: Think about which direction the rejection region is shaded in the lecture's diagram.

Answer: His average is below 150 — a one-tailed (left-tailed) test

Explanation: H₁ is that his real average is actually lower than the claimed 150 (making him a 'liar'). Since the rejection region only sits on the low end of the distribution, this is a one-tailed (left-tailed) test — a detail the lecture implies visually but never states explicitly.

What does the sample statistic (x̄) represent in this example?

A) The critical cutoff value, e.g. 120

B) The claimed population average of 150

C) The observed average score over the three games actually played

D) The probability of a Type I error

Answer: The observed average score over the three games actually played

Explanation: Nice pickup! x̄ is the evidence collected from the sample — here, the average of the three games actually bowled. It's compared against the critical value to decide whether to reject H₀.

Why does the narrator choose a lower critical value (50) for Mom versus a higher one (120) for Sam?

A) Mom is statistically proven to be a better bowler

B) Lower critical values always apply to older people

C) The cost of wrongly calling Mom a liar is higher, so more evidence is required before rejecting her claim

D) It doesn't matter — critical values are fixed by a formula regardless of context

Answer: C) The cost of wrongly calling Mom a liar is higher, so more evidence is required before rejecting her claim

Explanation: Correct — that's the heart of the analogy. The critical value (and thus significance level) is a subjective choice the researcher makes based on how costly a false accusation (Type I error) would be. Since wrongly doubting Mom is worse than wrongly doubting Sam, a much lower critical value — and smaller rejection region — is used for Mom, requiring stronger evidence before rejecting her claim.

Correction check: the lecture says 'the significance level measures how sure you want to be when rejecting the claim.' What's the more precise statistical definition?

A) N/A

B) The significance level (α) is the probability of committing a Type I error — wrongly rejecting a true null hypothesis

C) The significance level measures how good a bowler someone is

D) The significance level is always fixed at 0.05 in every study

E) The significance level is the same thing as the sample average

Answer: B) The significance level (α) is the probability of committing a Type I error — wrongly rejecting a true null hypothesis

Explanation: That's the precise version! Correction: 'how sure you want to be' is intuitive but informal. Precisely, α is the probability threshold you're willing to accept for a Type I error — rejecting H₀ when it's actually true. A smaller α means a smaller rejection region and a higher bar of evidence needed to reject the claim.

What is a Type I error in this context?

A) Rejecting Sam's or Mom's claim (calling them a liar) when the claim is actually true

B) Failing to bowl three games

C) Scoring an average below the critical value

D) Believing Sam's claim when it's actually false

Answer: A) Rejecting Sam's or Mom's claim (calling them a liar) when the claim is actually true

Explanation: Exactly right. A Type I error is rejecting a true null hypothesis — here, calling Sam or Mom a liar when they were actually telling the truth (just having an off day). The lecture correctly ties a lower significance level to a lower probability of this error.

Correction check: what key piece of statistical machinery does this lecture skip over when describing the 'probability density function' of bowling scores?

A) It never explains why the sampling distribution is roughly normal (Central Limit Theorem), and skips standard error, the test statistic (z/t), and p-values

C) It skips the concept of a null hypothesis entirely

D) It forgot to mention Sam's name

E) Nothing — the explanation given is fully rigorous and complete

Answer: A) It never explains why the sampling distribution is roughly normal (Central Limit Theorem), and skips standard error, the test statistic (z/t), and p-values

Explanation: Correct — good eye for what's left out. Correction: this is a deliberately simplified, intuition-first walkthrough. It asserts the bell-shaped curve without deriving it (that's the Central Limit Theorem's job) and doesn't yet introduce standard error, z/t test statistics, or p-values — those come later in a formal treatment. That's a reasonable pedagogical simplification, not an error, but worth knowing what's missing.

In the second scenario, Sam scores an average of 140 instead of 40. Why does this make the narrator more likely to believe him?

A) A higher score always disproves the null hypothesis

B) 140 exceeds the critical value threshold used for Mom

C) 140 is closer to the claimed average of 150, and some variation around a true average is expected across just three games

D) 140 is a Type I error

Answer: C) 140 is closer to the claimed average of 150, and some variation around a true average is expected across just three games

Explanation: Spot on. A true long-term average doesn't mean every sample (or every few games) matches it exactly — natural variability is expected. Since 140 is close to 150 and well within a plausible range of variation, it doesn't give strong evidence against H₀, so the claim isn't rejected.

How does the significance level (α) relate to the size of the rejection region?

A) Significance level has no relationship to the size of the rejection region

B) A smaller significance level produces a larger rejection region

C) The rejection region is always exactly half the distribution, regardless of α

D) A smaller significance level produces a smaller rejection region, requiring more evidence to reject H₀

Answer: A smaller significance level produces a smaller rejection region, requiring more evidence to reject H₀

Explanation: Exactly — smaller α means a tougher bar to clear. The lecture states this directly: a lower significance level means you require more evidence before rejecting the claim, which shows up visually as a smaller shaded rejection region (like Mom's, cut off at 50 rather than Sam's 120). The two move together — smaller α, smaller rejection region, higher bar for rejecting H₀.

Correction check: when Sam scores 140 and the narrator doesn't reject his claim, what does that technically tell us?

A) It means the significance level was set to zero

B) It means Sam definitely scored exactly 150 in every game

C) It only means the sample evidence wasn't strong enough to reject his claim — it doesn't prove the claim is true

E) It proves with certainty that Sam's long-term average really is 150

Answer: C) It only means the sample evidence wasn't strong enough to reject his claim — it doesn't prove the claim is true

Explanation: Exactly — that's a crucial nuance the lecture doesn't spell out. Correction: the lecture's framing ('you're more likely to believe him') can be misread as proof that H₀ is true. In formal hypothesis testing, failing to reject H₀ only means the sample didn't provide strong enough evidence against it — it's not equivalent to proving Sam's average really is 150. He could still be exaggerating a bit; three games just weren't enough to catch it.