AP Statistics 7.4 Setting Up a Test for a Population Mean Study Notes
AP Statistics 7.4 Setting Up a Test for a Population Mean Study Notes- New syllabus
AP Statistics 7.4 Setting Up a Test for a Population Mean Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- The t-distribution may be used to model variation.
Key Concepts:
- Testing Methods for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
- Null and Alternative Hypotheses for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
- Verifying Conditions for a t-Test for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
Testing Methods for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
Testing Methods for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
When the population standard deviation \(\sigma\) is unknown, we cannot use the z-test. Instead, we use a one-sample t-test based on the sample standard deviation \(s\). This applies to:
- Single sample mean (\(\mu\)): Use a one-sample t-test.
- Matched pairs / paired differences (\(\mu_d\)): Compute the differences \(d = X_{\text{after}} – X_{\text{before}}\) and perform a one-sample t-test on the differences.
Test statistic formula (single sample mean):
\( t = \dfrac{\bar{x} – \mu_0}{s / \sqrt{n}} \)
Test statistic formula (matched pairs / mean difference):
\( t = \dfrac{\bar{d} – \mu_{d,0}}{s_d / \sqrt{n}} \)
- \(\bar{x}\) = sample mean, \(\bar{d}\) = mean of paired differences
- \(\mu_0\) = hypothesized population mean, \(\mu_{d,0}\) = hypothesized mean difference
- \(s\) = sample standard deviation, \(s_d\) = standard deviation of differences
- \(n\) = sample size (or number of pairs)
Null distribution:
- Under \(H_0\) and all conditions satisfied, the t-test statistic follows a t-distribution with \(df = n – 1\).
- This distribution is used to calculate p-values and make decisions about the null hypothesis.
Example
A researcher claims a meditation program reduces stress scores by 5 points on average. A sample of 15 participants has mean reduction \(\bar{d} = 6.2\) with standard deviation \(s_d = 2.1\). Test at \(\alpha = 0.05\) whether the program reduces stress by more than 5 points.
▶️ Answer / Explanation
Step 1: Null and alternative hypotheses:
\( H_0: \mu_d = 5 \), \( H_a: \mu_d > 5 \) (one-sided test)
Step 2: Test statistic:
\( t = \dfrac{\bar{d} – \mu_{d,0}}{s_d / \sqrt{n}} = \dfrac{6.2 – 5}{2.1 / \sqrt{15}} = \dfrac{1.2}{0.542} \approx 2.21 \)
Step 3: Degrees of freedom: \( df = 15 – 1 = 14 \)
Step 4: Find p-value (t-distribution, one-sided): \( p \approx 0.023 \)
Step 5: Decision: \( p < 0.05 \) → reject \( H_0 \).
Conclusion: The data provide evidence that the meditation program reduces stress by more than 5 points on average.
Null and Alternative Hypotheses for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
Null and Alternative Hypotheses for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
When performing a t-test for a population mean (\(\mu\)) or the mean difference in matched pairs (\(\mu_d\)):
Single sample mean (\(\mu\)):
- Null hypothesis: \(H_0: \mu = \mu_0\), where \(\mu_0\) is the hypothesized population mean.
- Alternative hypothesis: Depends on the research question:
- Two-sided: \(H_a: \mu \ne \mu_0\)
- One-sided (greater): \(H_a: \mu > \mu_0\)
- One-sided (less): \(H_a: \mu < \mu_0\)
Matched pairs / mean difference (\(\mu_d\)):
- Compute differences: \(d = X_{\text{after}} – X_{\text{before}}\)
- Null hypothesis: \(H_0: \mu_d = 0\) (or another claimed value)
- Alternative hypothesis:
- Two-sided: \(H_a: \mu_d \ne 0\)
- One-sided (increase): \(H_a: \mu_d > 0\)
- One-sided (decrease): \(H_a: \mu_d < 0\)
Example
A diet program claims to reduce average weight by 4 kg. A sample of 10 participants is measured before and after the program. Let \(d = \text{weight before} – \text{weight after}\). Identify the null and alternative hypotheses for testing the claim.
▶️ Answer / Explanation
Step 1: Define population parameter: \(\mu_d\) = true mean weight loss.
Step 2: Null hypothesis (claim to test): \(H_0: \mu_d = 4\)
Step 3: Alternative hypothesis (research question): If testing whether the program reduces more than claimed: \(H_a: \mu_d > 4\) If testing any difference: \(H_a: \mu_d \ne 4\) If testing if the reduction is less than claimed: \(H_a: \mu_d < 4\)
Step 4: State conclusion in context after performing the t-test.
Verifying Conditions for a t-Test for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
Verifying Conditions for a t-Test for a Population Mean (Unknown \(\sigma\)) and Matched Pairs
Before performing a t-test for a single mean or for matched pairs, the following conditions must be checked to ensure valid statistical inference:
Randomness / Independence:
- Data should come from a random sample or a randomized experiment.
- If sampling without replacement, check the 10% condition: \( n \le 0.1N \), where \(N\) is the population size.
- For matched pairs, each pair should be independent of other pairs.
Normality / Shape:
- The sampling distribution of the mean or mean difference should be approximately normal.
- Check the distribution of the data (or differences for paired data):
- For small sample sizes (\(n < 30\)), the population or differences should be approximately normally distributed.
- For larger sample sizes (\(n \ge 30\)), the Central Limit Theorem ensures approximate normality even if the population is not perfectly normal.
Example
A researcher wants to test whether a relaxation program reduces stress scores. A sample of 12 participants is measured before and after the program. What conditions must be verified before performing a t-test on the paired differences?
▶️ Answer / Explanation
Step 1: Randomness / Independence:
– Ensure the 12 participants were randomly selected or randomly assigned.
– Each participant’s response is independent of others.
Step 2: 10% condition:
– If participants were sampled without replacement, check that 12 ≤ 0.1 × population size. Likely satisfied if population is large.
Step 3: Normality / Shape:
– Sample size is small (\(n = 12 < 30\)), so we need to check the distribution of differences for approximate normality (no strong skew or outliers).
Conclusion: If all conditions are reasonably met, a one-sample t-test on the paired differences can be performed.