Answer :
Certainly! To solve this problem, we will take the following steps:
1. Understanding the Problem:
We need to find out how many students out of 48,992 scored less than 96 on a test that has a mean score of 156 and a standard deviation of 27.
2. Calculating the Z-Score:
The Z-score helps us determine how many standard deviations a particular score is from the mean. The Z-score formula is:
[tex]\[ Z = \frac{(X - \mu)}{\sigma} \][/tex]
where [tex]\(X\)[/tex] is the score we are interested in (96 in this case), [tex]\(\mu\)[/tex] is the mean (156), and [tex]\(\sigma\)[/tex] is the standard deviation (27).
Plugging in the values, we get:
[tex]\[ Z = \frac{(96 - 156)}{27} = \frac{-60}{27} \approx -2.22 \][/tex]
3. Finding the Cumulative Probability:
We use the Z-score to find the cumulative probability, which is the area to the left of the Z-score in a standard normal distribution. This probability tells us the proportion of students who scored less than 96.
For [tex]\( Z = -2.22 \)[/tex], the cumulative probability (from standard normal distribution tables or using statistical software) is approximately 0.0131.
4. Calculating the Number of Students:
To find out how many students scored less than 96, we multiply the cumulative probability by the total number of students who took the test.
[tex]\[ \text{Number of students} = \text{Cumulative Probability} \times \text{Total Students} \][/tex]
[tex]\[ = 0.0131 \times 48992 \approx 643.47 \][/tex]
So, approximately 643 students scored less than 96. The given multiple-choice answers do not include the exact number we found, but based on our calculations and the closeness of the actual number:
The closest answer among the choices given is:
[tex]\[ \boxed{643} \][/tex]
1. Understanding the Problem:
We need to find out how many students out of 48,992 scored less than 96 on a test that has a mean score of 156 and a standard deviation of 27.
2. Calculating the Z-Score:
The Z-score helps us determine how many standard deviations a particular score is from the mean. The Z-score formula is:
[tex]\[ Z = \frac{(X - \mu)}{\sigma} \][/tex]
where [tex]\(X\)[/tex] is the score we are interested in (96 in this case), [tex]\(\mu\)[/tex] is the mean (156), and [tex]\(\sigma\)[/tex] is the standard deviation (27).
Plugging in the values, we get:
[tex]\[ Z = \frac{(96 - 156)}{27} = \frac{-60}{27} \approx -2.22 \][/tex]
3. Finding the Cumulative Probability:
We use the Z-score to find the cumulative probability, which is the area to the left of the Z-score in a standard normal distribution. This probability tells us the proportion of students who scored less than 96.
For [tex]\( Z = -2.22 \)[/tex], the cumulative probability (from standard normal distribution tables or using statistical software) is approximately 0.0131.
4. Calculating the Number of Students:
To find out how many students scored less than 96, we multiply the cumulative probability by the total number of students who took the test.
[tex]\[ \text{Number of students} = \text{Cumulative Probability} \times \text{Total Students} \][/tex]
[tex]\[ = 0.0131 \times 48992 \approx 643.47 \][/tex]
So, approximately 643 students scored less than 96. The given multiple-choice answers do not include the exact number we found, but based on our calculations and the closeness of the actual number:
The closest answer among the choices given is:
[tex]\[ \boxed{643} \][/tex]
