Question 1160891: Sharlize conducts an experiment with 200 students and finds that her control group had a mean of 24 questions right, with the lowest being 10 questions right and the highest being 30. If the standard deviation is 2.7, how many students received between and 18 and 25 questions correct. Explain your process using normalcdf.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2285)   (Show Source):
You can put this solution on YOUR website! ```python
import scipy.stats as stats
mean = 24
std = 2.7
total_students = 200
# NormalCDF for 18 to 25
prob = stats.norm.cdf(25, mean, std) - stats.norm.cdf(18, mean, std)
expected_students = prob * total_students
print(f"Prob: {prob:.6f}")
print(f"Expected students: {expected_students:.2f}")
print(f"Rounded: {round(expected_students)}")
```
```text
Prob: 0.631313
Expected students: 126.26
Rounded: 126
```
To find the number of students who scored between 18 and 25 questions correct, assuming the distribution of test scores follows a normal distribution, we calculate the proportion of the area under the normal curve between 18 and 25 and multiply it by the total number of students.
---
### Step 1: Identify the Given Parameters
* **Mean ($\mu$):** $24$
* **Standard Deviation ($\sigma$):** $2.7$
* **Lower Bound ($a$):** $18$
* **Upper Bound ($b$):** $25$
* **Total Number of Students ($n$):** $200$
---
### Step 2: Use `normalcdf` to Find the Probability
On a graphing calculator (such as a TI-83/84), the `normalcdf` function calculates the probability (area) between a lower bound and an upper bound for a given mean and standard deviation:
$$\text{normalcdf}(\text{lower bound}, \text{upper bound}, \text{mean}, \text{standard deviation})$$
Entering the values:
$$\text{normalcdf}(18, 25, 24, 2.7) \approx 0.631313$$
This means approximately **$63.13\%$** of the students scored between 18 and 25 questions correct.
---
### Step 3: Calculate the Expected Number of Students
Multiply the probability by the total sample size ($200$ students):
$$\text{Expected Students} = 0.631313 \times 200 = 126.2626$$
Rounding to the nearest whole student gives **126 students**.
Answer by ikleyn(54018)  (Show Source):
You can put this solution on YOUR website! .
Sharlize conducts an experiment with 200 students and finds that her control group
had a mean of 24 questions right, with the lowest being 10 questions right
and the highest being 30. If the standard deviation is 2.7, how many students
received between and 18 and 25 questions correct.
Explain your process using normalcdf.
~~~~~~~~~~~~~~~~~~~~~~
From this problem and from its solution given by @CPhill, it is unclear to me
for which reason the values "lowest being 10 questions right and highest being30"
are given in the problem, because they are not used in the solution.
Are they given only to confuse a reader?
Moreover, it seems to me that these given lowest and highest values are not consistent
with the hypothesis about a normal distribution.
In any case, if the solver pretends to be an Artificial Intelligence,
it cannot leave such circumstances without comment.
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