scores on the SAT critical reading test in 2013 follow a normal distribution with a mean score of 496 and a standard deviation of 115. what proportion of scores are higher thzn 530? what proportin of scores are between 400 and 500? the highest 10% of scored are higher than a SAT score of?

keep this link handy:

http://davidmlane.com/hyperstat/z_table.html

To find the proportions of scores in different ranges, we can use the properties of the normal distribution. In this case, we are given the mean score (496) and the standard deviation (115) for the SAT critical reading test in 2013.

1. Proportion of scores higher than 530:

To calculate this, we need to calculate the z-score first. The z-score represents the number of standard deviations an individual score is from the mean. We can use the formula:

z = (X - μ) / σ

Where X is the value (530), μ is the mean (496), and σ is the standard deviation (115).

So, for X = 530:

z = (530 - 496) / 115 = 0.2957

Next, we find the proportion of scores higher than 530 by calculating the area under the standard normal distribution curve to the right of z = 0.2957. We can use a table or a calculator to find this area.

Using a standard normal distribution table or calculator, we find that the proportion of scores higher than 530 is approximately 0.3830.

2. Proportion of scores between 400 and 500:

To find the proportion of scores between 400 and 500, we need to calculate the z-scores for the lower and upper bounds of the range.

For X = 400:

z1 = (400 - 496) / 115 = -0.8348

For X = 500:

z2 = (500 - 496) / 115 = 0.0348

To find the proportion of scores between these two z-scores, we calculate the area under the standard normal distribution curve between z1 and z2. Again, we can use a table or a calculator to find this area.

Using a standard normal distribution table or calculator, we find that the proportion of scores between 400 and 500 is approximately 0.2876.

3. Highest 10% of scores:

To find the SAT score that corresponds to the highest 10% of scores, we need to find the z-score that encloses the upper 10% of the standard normal distribution.

Using a standard normal distribution table or calculator, we find that the z-score corresponding to the upper 10% is approximately 1.28.

We can then use the formula to calculate the SAT score:

X = μ + (z * σ)

X = 496 + (1.28 * 115) = 641.2

Therefore, the highest 10% of the scores are higher than an SAT score of approximately 641.2.