Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.