Answer :
In this question, the z-score associated with the 97.5 percent confidence interval is option e) 1.960.
In statistics, the z-score is used to determine the number of standard deviations a particular value is away from the mean in a normal distribution. The z-score is commonly used in confidence interval calculations, where it corresponds to a certain level of confidence.
The 97.5 percent confidence interval corresponds to a two-tailed test, meaning we need to find the z-score that captures 97.5 percent of the area under the normal distribution curve, with 2.5 percent of the area in each tail.
Looking up the z-score in a standard normal distribution table or using statistical software, we find that the z-score associated with the 97.5 percent confidence interval is approximately 1.960.
Therefore, the correct answer is e) 1.960. This z-score is used when constructing a 97.5 percent confidence interval, which means there is a 97.5 percent probability that the true population parameter lies within the interval calculated using this z-score.
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Final answer:
The z-score for a 97.5 percent confidence interval is e) 1.960. This value corresponds to the central 95% of the normal distribution curve, leaving 2.5% in the tails. It is consistent with the empirical rule, which helps explain the distribution of values within standard deviations from the mean.
Explanation:
The z-score associated with the 97.5 percent confidence interval suggests that we are looking for a critical value at which the area under the standard normal distribution curve to the left of the z-score is 0.975. This is because a 97.5 percent confidence interval implies that the total area in both tails (100% - 97.5%) is 2.5%, and since the normal distribution is symmetrical, each tail would contain 1.25% of the total area. The correct z-score is e) 1.960. This value is often used in constructing a 95% confidence interval, which effectively captures the central 95% of the normal distribution, leaving 2.5% in each tail.
To further illustrate, the empirical rule, also known as the 68-95-99.7 rule, helps us understand that for normally distributed data: about 68% of values lie within one standard deviation of the mean (z-scores of -1 to +1), about 95% within two standard deviations (z-scores of -2 to +2), and about 99.7% within three standard deviations (z-scores of -3 to +3).
