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AP Statistics 6.6 Concluding a Test for a Population Proportion Study Notes

AP Statistics 6.6 Concluding a Test for a Population Proportion Study Notes- New syllabus

AP Statistics 6.6 Concluding a Test for a Population Proportion Study Notes -As per latest AP Statistics Syllabus.

LEARNING OBJECTIVE

  • Significance testing allows us to make decisions about hypotheses within a particular context.

Key Concepts:

  • Significance Testing for a Population Proportion

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Significance Testing for a Population Proportion

Significance Testing for a Population Proportion

 Significance Level (\( \alpha \)):

  • The significance level is the probability of making a Type I error : rejecting \( H_0 \) when it is actually true.
  • Common choices: 0.05, 0.01, 0.10

 Hypotheses:

  • Null hypothesis (\( H_0 \)): specifies a value for the population proportion, usually representing no effect or a baseline claim: \( H_0: p = p_0 \)
  • Alternative hypothesis (\( H_a \)) can be:
    • One-sided: \( H_a: p > p_0 \) or \( H_a: p < p_0 \)
    • Two-sided: \( H_a: p \neq p_0 \)

 Conditions for a Valid Test:

  • Independence: Random sample and 10% condition (n ≤ 0.1N if sampling without replacement)
  • Normal approximation: \( n p_0 \ge 10 \) and \( n(1 – p_0) \ge 10 \)

Test Statistic:

\( z = \dfrac{\hat{p} – p_0}{\sqrt{\dfrac{p_0 (1 – p_0)}{n}}} \)

  • \( \hat{p} \) = sample proportion
  • \( p_0 \) = hypothesized population proportion
  • \( n \) = sample size

Null Distribution:

  • Assumes \( H_0 \) is true.
  • Can be based on a randomization distribution (simulation) or a theoretical distribution (z-distribution).

P-value:

  • The probability of observing a test statistic as extreme or more extreme than the one from the sample, assuming \( H_0 \) is true.
  • Right-tailed: proportion ≥ observed z; Left-tailed: proportion ≤ observed z; Two-tailed: sum of probabilities at ≤ -|z| and ≥ |z|.
  • Small p-values → evidence for \( H_a \). Large p-values → insufficient evidence for \( H_a \).

Decision and Interpretation:

  • Compare p-value to \( \alpha \):
    • P-value ≤ \( \alpha \) → reject \( H_0 \); sufficient evidence to support \( H_a \).
    • P-value > \( \alpha \) → fail to reject \( H_0 \); insufficient evidence to support \( H_a \).
  • Always state conclusions in context.
  • Failing to reject \( H_0 \) is NOT the same as proving \( H_0 \) is true.

Conceptual Notes:

  • Rejecting \( H_0 \) → data provide sufficient statistical evidence to support \( H_a \).
  • Failing to reject \( H_0 \) → data provide insufficient statistical evidence to support \( H_a \).
  • Small p-values indicate the observed result is unusual under \( H_0 \), giving strong evidence for \( H_a \).
  • Large p-values indicate the observed result is not unusual under \( H_0 \), providing no convincing evidence for \( H_a \) and not confirming \( H_0 \).

Example:

A company claims 80% of customers are satisfied. In a random sample of 100 customers, 74 say they are satisfied. Test at \( \alpha = 0.05 \) whether the satisfaction rate is less than 80%.

▶️ Answer / Explanation

Step 1: Hypotheses

\( H_0: p = 0.80 \)
\( H_a: p < 0.80 \) (left-tailed)

Step 2: Verify conditions

  • Random sample 
  • 10% condition: 100 ≤ 0.1N 
  • Normal approximation: \( n p_0 = 80 \), \( n(1-p_0) = 20 \) 

Step 3: Test Statistic

\( \hat{p} = 74/100 = 0.74 \)

\( z = \dfrac{0.74 – 0.80}{\sqrt{0.8*0.2/100}} = -1.5 \)

Step 4: P-value

Left-tailed: \( P(Z < -1.5) \approx 0.067 \)

Step 5: Decision & Conclusion

P-value 0.067 > 0.05 → fail to reject \( H_0 \)

Interpretation in context: There is insufficient statistical evidence to conclude that fewer than 80% of customers are satisfied. This does not prove the 80% claim is true; it only indicates the data are consistent with it.

Key Concept: Small p-values (< \( \alpha \)) provide evidence for \( H_a \); large p-values do not provide evidence for \( H_a \) nor prove \( H_0 \).

Example:

A coffee shop claims that at least 75% of its customers are satisfied with their service. A random sample of 80 customers finds that 54 are satisfied. Using a significance level of 0.05, which of the following is the correct conclusion?

  1. Reject \( H_0 \); there is sufficient evidence that fewer than 75% of customers are satisfied.
  2. Fail to reject \( H_0 \); there is insufficient evidence that fewer than 75% of customers are satisfied.
  3. Reject \( H_0 \); there is sufficient evidence that more than 75% of customers are satisfied.
  4. Fail to reject \( H_0 \); there is sufficient evidence that 75% of customers are satisfied.
▶️ Answer / Explanation

Step 1: Hypotheses

\( H_0: p = 0.75 \)
\( H_a: p < 0.75 \) (claim is “at least”, so test for fewer)

Step 2: Check conditions

  • Random sample 
  • 10% condition: 80 ≤ 0.1N 
  • Normal approximation: \( n p_0 = 80*0.75 = 60 \ge 10 \), \( n(1-p_0) = 20 \ge 10 \) 

Step 3: Test statistic

\( \hat{p} = 54/80 = 0.675 \)

\( z = \dfrac{0.675 – 0.75}{\sqrt{0.75*0.25/80}} = \dfrac{-0.075}{0.0484} \approx -1.55 \)

Step 4: P-value

Left-tailed: \( P(Z < -1.55) \approx 0.060 \)

Step 5: Decision

P-value 0.060 > 0.05 → fail to reject \( H_0 \)

Step 6: Conclusion in context

There is insufficient evidence to conclude that fewer than 75% of customers are satisfied. The coffee shop’s claim that at least 75% are satisfied is consistent with the sample data.

Correct Answer: B

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