A certain standardized test's math scores have a bell-shaped distribution with a mean of 530
and a standard deviation of 105
.
Complete parts (a) through (c).
(a) What percentage of standardized test scores is between 215
and 845
?
To find the percentage of standardized test scores between 215 and 845, we will use the standard normal distribution.
First, we need to standardize the scores by subtracting the mean and dividing by the standard deviation.
Standardize 215:
Z = (215 - 530) / 105 = -3.095
Standardize 845:
Z = (845 - 530) / 105 = 3.10
Next, we need to find the area under the standard normal distribution curve between these two standardized values.
Using a standard normal distribution table or a calculator, we can find that the area to the left of Z = -3.095 is approximately 0.00117.
Similarly, the area to the left of Z = 3.10 is approximately 0.99917.
To find the area between these two values, we subtract the smaller area from the larger area:
0.99917 - 0.00117 = 0.997
This means that approximately 99.7% of standardized test scores are between 215 and 845.
(b) What percentage of standardized test scores is less than 215
or greater than 845
?
To find the percentage of standardized test scores that are less than 215 or greater than 845, we need to find the area under the standard normal distribution curve that is to the left of -3.095 and to the right of 3.10.
Using a standard normal distribution table or a calculator, we can find that the area to the left of Z = -3.095 is approximately 0.00117 and the area to the right of Z = 3.10 is approximately 0.00083.
To find the percentage of scores less than 215 or greater than 845, we add these two areas together:
0.00117 + 0.00083 = 0.002
Therefore, approximately 0.2% of standardized test scores are less than 215 or greater than 845.
wrong answer
I apologize for the incorrect answer. Let's recalculate the correct percentage.
To find the percentage of standardized test scores that are less than 215 or greater than 845, we need to find the area under the standard normal distribution curve that is to the left of -3.095 and to the right of 3.10.
Using a standard normal distribution table or a calculator, we can find that the area to the left of Z = -3.095 is approximately 0.00117 and the area to the right of Z = 3.10 is approximately 0.99917 (as mentioned before).
To find the percentage, we need to calculate the total area outside the range of 215 and 845 by summing the two areas:
0.00117 + (1 - 0.99917) = 0.00117 + 0.00083 = 0.002
Therefore, approximately 0.2% of standardized test scores are less than 215 or greater than 845.
100%-99.7%=.30 or 30%
I apologize for the mistake in my previous response.
To find the percentage of standardized test scores that are less than 215 or greater than 845, we need to subtract the percentage of scores between 215 and 845 from 100%.
We previously found that approximately 99.7% of scores are between 215 and 845.
Therefore, the percentage of standardized test scores that is less than 215 or greater than 845 is:
100% - 99.7% = 0.3% or 0.3/100 = 0.003 or 0.3%
(c) What percentage of standardized test scores is greater than 740
?
To find the percentage of standardized test scores that are greater than 740, we need to find the area under the standard normal distribution curve to the right of Z = (740 - 530) / 105 = 2.
Using a standard normal distribution table or a calculator, we can find that the area to the left of Z = 2 is approximately 0.9772.
To find the percentage, we subtract this value from 1 (or multiply by 100%):
1 - 0.9772 = 0.0228 or 2.28%.
Approximately 2.28% of standardized test scores are greater than 740.