Final answer:
The question discusses constructing 95% confidence intervals for the slope of the population regression line. Each tail of the distribution for the 95% confidence interval holds 2.5% of the excluded probability, making the interval wider than the 90% interval. Approximately 95% of these intervals from repeated samples would contain the true population mean.
Step-by-step explanation:
The question refers to constructing a 95% confidence interval for the slope of the population regression line. In statistics, specifically in regression analysis, a confidence interval gives a range of values for the slope parameter that is likely to contain the true slope of the population regression line with a certain level of confidence, in this case, 95%.
To divide the probability for the two-sided 95% confidence interval, we allocate the remaining 5% probability to the tails of the distribution, which means each tail holds 2.5% of the probability. Consequently, if the 90% confidence interval is given by (67.18, 68.82), it's noted that the 95% confidence interval is wider at (67.02, 68.98), because we seek more certainty that the interval contains the true value.
This idea is further supported when comparing the widths of confidence intervals constructed at different confidence levels (e.g., 90% versus 95% versus 99%).
A key concept here is that for any given data set, it is expected that a higher confidence level leads to a wider confidence interval. This is due to the fact that including a larger percentage of the data increases the likelihood that the true parameter (slope in our context) lies within the interval.
Lastly, from repeated sampling, approximately 95% of the confidence intervals computed would contain the true population mean, provided the confidence level is set at 95%.