Assume the aptitude test score are normally distributed; mean is 140 points and standard deviation is 25 points. Within what interval centered at the mean will 95% of the scores lie?

95% = mean ± 1.96 SD

To find the interval within which 95% of the scores will lie, we can use the z-score formula. The z-score measures the number of standard deviations a value is from the mean in a normal distribution.

Since we want to find the interval centered at the mean, we need to find the z-score for the upper and lower bounds of the interval.

The z-score can be calculated using the following formula:
z = (x - μ) / σ

Where:
z is the z-score
x is the raw score
μ is the mean
σ is the standard deviation

Given:
Mean (μ) = 140 points
Standard Deviation (σ) = 25 points

To find the z-score for a 95% confidence interval, we can use the Z-table (also known as standard normal distribution table) or a calculator. The z-score for a 95% confidence interval is approximately 1.96.

Using the z-score formula, we can express the upper and lower bounds of the interval:

Upper bound:
z = (x - μ) / σ
1.96 = (x - 140) / 25
1.96 * 25 = x - 140
49 + 140 = x
x = 189

Lower bound:
z = (x - μ) / σ
-1.96 = (x - 140) / 25
-1.96 * 25 = x - 140
-49 + 140 = x
x = 91

Therefore, within a 95% confidence interval centered at the mean (140 points), the scores will lie between 91 points and 189 points.

To find the interval centered at the mean within which 95% of the scores lie, we need to use the concept of the normal distribution and the empirical rule.

The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

In this case, the mean is 140 points and the standard deviation is 25 points. So, we can calculate the interval centered at the mean that contains 95% of the scores by going two standard deviations above and below the mean.

To calculate the interval, we need to find the values of x where x is two standard deviations above and below the mean:
Lower Bound: Mean - (2 * Standard Deviation)
Upper Bound: Mean + (2 * Standard Deviation)

Using the given values, we can calculate the interval as follows:
Lower Bound: 140 - (2 * 25) = 90 points
Upper Bound: 140 + (2 * 25) = 190 points

Therefore, 95% of the scores will lie within the interval 90 points to 190 points, centered at the mean (140 points).