Question: What is the chief difference between a bootstrap distribution and a sampling distribution?
Answer:
A sampling distribution is created by repeatedly taking many samples from the population and calculating a statistic (like the mean) for each sample.
A bootstrap distribution is created by repeatedly resampling from the original sample with replacement, instead of sampling from the population.
The key difference is that a sampling distribution requires many samples from the population, while a bootstrap distribution uses only one sample and resamples from it to estimate variability.
Question: Looking at the bootstrap distribution for the sample mean in Figure 8.14, between what two values would you say most values lie?
Answer:
Most of the bootstrap sample means appear to lie roughly between 1992 and 2000.
This range represents where the majority of the resampled mean values occur in the bootstrap distribution and gives a plausible range for the true population mean year of all pennies.
Question: What condition about the bootstrap distribution must be met for us to construct confidence intervals using the standard error method?
Answer:
The bootstrap distribution must be approximately normal (bell-shaped).
When the distribution looks symmetric and bell-shaped, we can apply the standard error method, which assumes that about 95% of values lie within ±1.96 standard deviations of the mean.
Question: Say we wanted to construct a 68% confidence interval instead of a 95% confidence interval. What changes would be needed?
Answer:
To construct a 68% confidence interval, we would use ±1 standard error instead of ±1.96 standard errors.
This would create a narrower interval because a lower confidence level requires a smaller range around the sample mean.