SAT scores are normally distributed with a mean of 1518 and a standard deviation of 325. What scores correspond to P20 and P65?
What do you mean by "P20 and P65"?
If I assume that those are percentiles, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z scores corresponding to the proportions indicated. Insert the values into the following equation.
Z = (score-mean)/SD
32.5
To find the scores that correspond to P20 and P65, we need to use the z-score formula and the standard normal distribution table.
The z-score formula is given by:
z = (x - μ) / σ
Where:
- x is the value we want to find the corresponding score for
- μ is the mean of the distribution (1518 in this case)
- σ is the standard deviation of the distribution (325 in this case)
Step 1: Calculate the z-score for P20:
To find the score corresponding to P20, we need to find the z-score that corresponds to the cumulative area of 0.20 from the left of the distribution.
Using the standard normal distribution table, we look for the z-score that corresponds to a cumulative area of 0.20. The closest value we find is 0.5793. This means that 20% of the data lies to the left of the z-score 0.5793.
Now we can use the z-score formula to find the corresponding score:
0.5793 = (x - 1518) / 325
Solving for x:
0.5793 * 325 = x - 1518
188.1025 = x - 1518
x = 1706.1025
So the score corresponding to P20 is approximately 1706.1.
Step 2: Calculate the z-score for P65:
To find the score corresponding to P65, we need to find the z-score that corresponds to the cumulative area of 0.65 from the left of the distribution.
Using the standard normal distribution table, we look for the z-score that corresponds to a cumulative area of 0.65. The closest value we find is 0.3859. This means that 65% of the data lies to the left of the z-score 0.3859.
Now we can use the z-score formula to find the corresponding score:
0.3859 = (x - 1518) / 325
Solving for x:
0.3859 * 325 = x - 1518
125.7175 = x - 1518
x = 1643.7175
So the score corresponding to P65 is approximately 1643.7.
Therefore, the scores that correspond to P20 and P65 are approximately 1706.1 and 1643.7, respectively.
To find the scores that correspond to specific percentiles, you need to use the standard normal distribution, also known as the Z-distribution. The Z-distribution has a mean of 0 and a standard deviation of 1.
In order to find the scores that correspond to P20 and P65, you will need to convert these percentiles to Z-scores, and then convert the Z-scores back to actual scores using the given mean and standard deviation for the SAT scores.
1. Finding the Z-score for P20:
To find the Z-score corresponding to the 20th percentile (P20), you can use a Z-table or a calculator.
P20 corresponds to a Z-score of -0.84 (approximately) when using a Z-table.
2. Finding the Z-score for P65:
Similarly, to find the Z-score corresponding to the 65th percentile (P65), you can use a Z-table or a calculator.
P65 corresponds to a Z-score of 0.39 (approximately) when using a Z-table.
Now let's calculate the actual scores corresponding to these Z-scores.
3. Calculating the score for P20:
The Z-score formula is Z = (x - mean) / standard deviation. Rearranging the formula, we have x = Z * standard deviation + mean.
Plugging in the values, x = (-0.84) * 325 + 1518 = 1200.15.
Therefore, the score corresponding to P20 is approximately 1200.
4. Calculating the score for P65:
Using the same formula, x = (0.39) * 325 + 1518 = 1646.75.
Therefore, the score corresponding to P65 is approximately 1647.
To summarize,
- The score corresponding to P20 is approximately 1200.
- The score corresponding to P65 is approximately 1647.
Note that these values are approximate since Z-tables provide rounded values.