Question

In a normal curve, 99% of the observations are included in the range

• A. $\bar{x} \pm 0.67$
• B. $\bar{x} \pm 1.96$
• C. $\bar{x} \pm 2.0$
• D. $\bar{x} \pm 2.58$

Hint:

Keep these key standard normal values memorized for quick recall:
- $90\% \rightarrow z = \pm 1.645$
- $95\% \rightarrow z = \pm 1.96$
- $99\% \rightarrow z = \pm 2.58$

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In a normal distribution, specific percentages of the total area under the curve fall within certain standard deviation limits from the mean.
These limits are derived from the standard normal cumulative distribution function.
Detailed Explanation:
Let us review the standard confidence intervals and area coverage for a normal distribution:
- Approximately $50\%$ of the observations lie within $\bar{x} \pm 0.6745\sigma$.
- Exactly $95\%$ of the observations lie within the range $\bar{x} \pm 1.96\sigma$.
- Approximately $95.44\%$ of the observations lie within the range $\bar{x} \pm 2\sigma$.
- Exactly $99\%$ of the observations lie within the range $\bar{x} \pm 2.576\sigma$. Rounded to two decimal places, this is $\bar{x} \pm 2.58\sigma$.
- Approximately $99.73\%$ of the observations lie within the range $\bar{x} \pm 3\sigma$.
Therefore, the range representing $99\%$ of the observations is $\bar{x} \pm 2.58\sigma$.

Step 2: Final Answer:
The correct range is $\bar{x} \pm 2.58$, matching Option (D).