Final answer:
The statement that a 90% confidence interval not including 0 suggests potential rejection of the null hypothesis is correct. A 90% confidence interval signifies that approximately 90 out of 100 intervals from repeated samples will include the true population mean. As the confidence level decreases, the interval becomes narrower, indicating a lower level of certainty about the estimated parameter.
Step-by-step explanation:
Yes, the statement is correct that if we set the significance level at 10% (i.e., α = 0.10), and then the 90% confidence interval for the parameter does not include 0, it suggests there is some effect or difference present, and that the null hypothesis may potentially be rejected. This is because in hypothesis testing, if zero is not contained within the confidence interval, it typically indicates that the estimated parameter (such as a mean difference, proportion, or regression coefficient) is statistically significantly different from zero given the chosen level of confidence.
When we construct a 90% confidence interval, we're essentially saying that if we were to take repeated random samples from the population and build a confidence interval from each sample, approximately 90 out of 100 of those intervals would contain the true population mean. This means that there is a 90% chance that any given confidence interval we construct will include the true population mean. Confidence intervals provide a range of values which are estimated to contain the population parameter with a certain level of confidence.
As confidence level decreases from 99% to 90%, the confidence interval becomes narrower since it will be excluding more of the distribution (from excluding 1% in each tail to excluding 5% in each). This narrower range reflects less certainty about the estimated parameter—in other words, we are less confident that our interval contains the true population mean. Consequently, the confidence interval is a specific range of values, not a distribution of individual data points.