If student A scored 42 on a test of reading and the test had a mean of 52 and a standard deviation of 8, what percentage of test respondents scored below student A?
To find the percentage of test respondents who scored below student A, you need to calculate the z-score and then use the z-table to determine the corresponding percentage.
1. Calculate the z-score:
The z-score measures how many standard deviations an individual's score is from the mean. It is calculated using the formula:
z = (x - μ) / σ, where x is the individual's score, μ is the mean, and σ is the standard deviation.
In this case, student A's score (x) is 42, the mean (μ) is 52, and the standard deviation (σ) is 8. Substituting these values into the formula:
z = (42 - 52) / 8
z = -10 / 8
z = -1.25
2. Use the z-table:
The z-table provides a standardized normal distribution and helps find the proportion (percentage) of scores below a specific z-score.
To determine the percentage of test respondents who scored below student A, you need to find the area under the standard normal curve to the left of the z-score (-1.25).
Using a z-table or a statistical software, you can find that the area to the left of -1.25 is approximately 0.1056.
3. Convert to percentage:
To convert the proportion to a percentage, multiply it by 100.
0.1056 * 100 = 10.56%
Therefore, approximately 10.56% of test respondents scored below student A.