Data Science Statistical Concepts and Inference 2 — Questions and Answers
Question 1: A 95% confidence interval for a population mean means which of the following?
- 95% of the data falls within the interval
- If we repeated sampling many times, about 95% of such intervals would contain the true mean (Correct answer)
- There is a 95% probability the sample mean is correct
- The interval contains 95% of all possible sample means
Correct answer: If we repeated sampling many times, about 95% of such intervals would contain the true mean
A confidence level refers to the long-run proportion of constructed intervals that capture the true parameter, not the probability for a single fixed interval.
Question 2: Which of the following increases the width of a confidence interval, all else equal?
- Increasing the sample size
- Decreasing the confidence level
- Increasing the population variance (Correct answer)
- Reducing the standard error
Correct answer: Increasing the population variance
Greater population variance increases the standard error, widening the interval.
Question 3: A Type I error in hypothesis testing is best described as:
- Failing to reject a false null hypothesis
- Rejecting a true null hypothesis (Correct answer)
- Accepting the alternative when it is false because of low power
- Using the wrong test statistic
Correct answer: Rejecting a true null hypothesis
A Type I error is a false positive: rejecting a null hypothesis that is actually true.
Question 4: What does the p-value represent?
- The probability the null hypothesis is true
- The probability of observing data at least as extreme as the sample, assuming the null is true (Correct answer)
- The probability the alternative hypothesis is true
- The size of the effect being tested
Correct answer: The probability of observing data at least as extreme as the sample, assuming the null is true
A p-value is the probability of results as or more extreme than observed under the assumption the null hypothesis holds.
Question 5: The Central Limit Theorem states that, for a large sample size, the sampling distribution of the sample mean is approximately:
- Uniform regardless of population shape
- Normal regardless of population distribution (Correct answer)
- Identical to the population distribution
- Exponential for skewed populations
Correct answer: Normal regardless of population distribution
The CLT guarantees the sample mean's distribution approaches normality as sample size grows, no matter the population shape.
Question 6: Statistical power is the probability of:
- Rejecting the null when the alternative is actually true (Correct answer)
- Failing to reject a true null
- Making a Type I error
- Obtaining a statistically significant result by chance
Correct answer: Rejecting the null when the alternative is actually true
Power is the probability of correctly rejecting a false null hypothesis, equal to 1 minus the Type II error rate.
Question 7: Which factor does NOT increase statistical power?
- Larger sample size
- Larger true effect size
- Smaller significance level (alpha) (Correct answer)
- Lower measurement variance
Correct answer: Smaller significance level (alpha)
Decreasing alpha makes it harder to reject the null, which reduces power.
Data Science Statistical Concepts and Inference 2
A 95% confidence interval for a population mean means which of the following?