Statistical Hypothesis Testing and Confidence Intervals: A Comprehensive Analysis
1. Analyzing Beats Per Minute in Dance Songs
Hypotheses and Parameter
The parameter of interest is μ = mean beats per minute for all dance songs.
- H0: μ = 120.5 beats per minute
- Ha: μ > 120.5 beats per minute
T-Procedure Validity
Even with a slightly skewed distribution of beats per minute, the t-procedure remains valid due to its robustness to deviations from normality.
Test Statistic and P-value
Assuming necessary assumptions are met, the test statistic and P-value are calculated using a t-test (refer to calculator output for specific values).
P-value Interpretation
The P-value (e.g., 0.0023) represents the probability of obtaining a sample mean of 122.5 beats per minute or greater in a sample of size 35, assuming the true mean beats per minute for all dance songs is 120.5.
2. Comparing Side Effects of Two Whooping Cough Drugs
Parameters and Hypotheses
- p1 = Proportion of all whooping cough patients experiencing side effects with drug A
- p2 = Proportion of all whooping cough patients experiencing side effects with drug B
- H0: p1 = p2
- Ha: p1 ≠ p2
Test Statistic and P-value
The test statistic and P-value are calculated using a two-proportion z-test (refer to calculator output for specific values).
Conclusion
Based on the P-value, we lack evidence to conclude a difference in the proportion of whooping cough patients experiencing side effects between drug A and drug B.
Conclusion at α = 0.01
With a P-value greater than 0.01, we would fail to reject the null hypothesis at the α = 0.01 level.
Type II Error
A Type II error would mean concluding no difference in side effect rates between the two drugs when, in reality, a difference exists.
3. Analyzing Attitudes Toward Gun Control Legislation
Confidence Interval Interpretation
The 99% confidence interval of (0.21, 0.03) for p1 – p2 suggests with 99% confidence that the percentage of women favoring stricter gun control legislation is between 3 and 21 percentage points higher than that of men.
Standard Error Calculation
The standard error of p1 – p2 is calculated as follows:
(0.03 – (-0.21)) / 2 = 0.09 = Margin of Error
2.576 * SE(p̂1 – p̂2) = 0.09
SE(p̂1 – p̂2) = 0.09 / 2.576 = 0.0349
Hypotheses for Comparison
- H0: p1 = p2 (Proportion of men favoring stricter gun control is equal to that of women)
- Ha: p1 < p2 (Proportion of men favoring stricter gun control is less than that of women)
Conclusion Based on Confidence Interval
Since the entire 99% confidence interval is less than 0, strong evidence suggests that the proportion of women favoring stricter gun control legislation is higher than that of men.
4. Probability of Sample Proportion Exceeding a Threshold
Given a random sample of 1024 Americans and assuming 44% of all Americans prioritize environmental protection over energy production, the probability of the sample proportion exceeding 46% is calculated using the Central Limit Theorem.
Since the sample size is large enough (n = 1024), the sampling distribution of the sample proportion (p̂) is approximately normal with mean 0.44 and standard error √((0.44)(0.56) / 1024) = 0.0155.
P(p̂ > 0.46) ≈ 0.0985 (calculated using a normal distribution calculator)
5. Analyzing Left-Handedness Among Artists
Hypotheses
- H0: p = 0.10 (Proportion of left-handed artists is the same as the general population)
- Ha: p > 0.10 (Proportion of left-handed artists is higher than the general population)
Test Statistic and P-value
Based on a sample of 50 artists with 12 left-handed, the test statistic and P-value are calculated using a one-proportion z-test (refer to calculator output for specific values).
Conclusion
With a small P-value (e.g., 0.0005), strong evidence suggests that artists are more likely to be left-handed than the general population.
P-value for Two-Sided Alternative
For a two-sided alternative hypothesis, the P-value would be 2 * 0.0005 = 0.001.
Type I Error
A Type I error would mean concluding that artists tend to be left-handed when, in reality, their likelihood of being left-handed is no different from the general population.
6. Comparing Average Commute Times of Alice and Bob
Parameters of Interest
- μA = True mean travel time for Alice
- μB = True mean travel time for Bob
95% Confidence Interval
A 95% confidence interval for the difference between Alice and Bob’s mean travel times is calculated using a two-sample t-interval (refer to calculator output for specific values).
Conclusion
If the confidence interval contains 0, it indicates that there may be no significant difference between Alice and Bob’s average commute times.