For a stat class the mean of the midterm is 75 with a standard deviation of 7. What percentage of students scored between 67 and 83
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To find the percentage of students who scored between 67 and 83 on the midterm, you can use the properties of the normal distribution.
First, we need to calculate the z-scores for both scores using the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the individual score
- μ is the mean of the distribution
- σ is the standard deviation
For 67:
z1 = (67 - 75) / 7 = -1.14
For 83:
z2 = (83 - 75) / 7 = 1.14
The next step is to look up the corresponding area under the standard normal distribution curve for each z-score.
Using a standard normal distribution table or a calculator, you can find the area to the left of -1.14 is approximately 0.1271. This represents the proportion of students who scored below 67.
Similarly, the area to the left of 1.14 is also approximately 0.1271, which represents the proportion of students who scored below 83.
To find the percentage of students who scored between 67 and 83, we can subtract the two proportions:
P(67 ≤ x ≤ 83) = P(x ≤ 83) - P(x ≤ 67) = 0.1271 - 0.1271 = 0.2542
However, this gives the proportion, not the percentage. To convert to a percentage, we simply multiply by 100:
Percentage of students between 67 and 83 = 0.2542 * 100 = 25.42%
Therefore, approximately 25.42% of students scored between 67 and 83 on the midterm.