Question

In a normal distribution with a mean of 30 and a standard deviation of 10, what raw scares mark the middle 34% of this distribution?

a. 25.4 and 34.6
b. 25.6 and 34.4
c. 20.0 and 40.0
d. none of the above

Answer 1

To find the raw scores that mark the middle 34% of a normal distribution with a mean of 30 and a standard deviation of 10, we can use the z-score formula and the z-table.

Step-by-step explanation:

The middle 34% of a normal distribution with a mean of 30 and a standard deviation of 10 can be found using the z-score formula.

1. First, we find the z-score for the lower boundary of the middle 34%. To find this, we use the formula: z = (x - mean) / standard deviation. Plugging in the values, we get z = (x - 30) / 10.
2. Next, we look up the z-score in the z-table to find the corresponding percentile. Since we want the middle 34%, we divide it by 2 to get 0.17.
3. Using the z-table, we find that the z-score corresponding to a percentile of 0.17 is -0.4461. We can then solve for x: -0.4461 = (x - 30) / 10.
4. After solving the equation, we find that x is approximately 25.398.
5. Finally, we find the upper boundary of the middle 34% by adding the difference between x and the mean: 30 + (30 - 25.398) = 34.602.

Therefore, the raw scores that mark the middle 34% of this distribution are approximately 25.4 and 34.6.