Respuesta :
Given Information:
confidence level = 95%
standard deviation = σ = 2
Margin of error = 0.4
Required Information:
Number of students = n = ?
Answer:
Number of students = n ≤ 96 (for z-score = 1.96)
Number of students = n ≤ 100 (for z-score = 2)
Step-by-step explanation:
We know that margin of error is given by
margin of error = z*(σ/√n)
since we want to have a margin of error at most 0.4 hours
margin of error ≤ z*(σ/√n)
√n ≤ z*σ/margin of error
n ≤ (z*σ/Margin of error)²
Where z is the corresponding z-score from the z-table, n is the number of students required and σ is the standard deviation.
For 95% confidence level the exact z-score is 1.96
n ≤ (1.96*2/0.4)²
n ≤ 96.04
n ≤ 96
Therefore, a sample size of less or equal to 96 students is required to ensure a margin of error at most 0.4 hours for 95% confidence interval.
If we use the approximate z-score of 2 as given in the question than the required sample size will be
n ≤ (2*2/0.4)²
n ≤ 100
Therefore, a sample size of less or equal to 100 students is required to ensure a margin of error at most 0.4 hours for 95% confidence interval.
Using the z-distribution, it is found that the minimum sample size is of 97 students.
The first step is finding the critical value, which for a confidence level of [tex]\alpha[/tex], is given by Z with a p-value of [tex]\frac{1 + \alpha}{2}[/tex].
In this problem, 95% confidence interval, thus [tex]\alpha = 0.95[/tex], and Z has a p-value of [tex]\frac{1 + 0.95}{2} = 0.975[/tex], thus z = 1.96.
The margin of error is:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
- [tex]\sigma[/tex] is the population standard deviation.
- n is the sample size.
In this problem:
- Population standard deviation of 2 hours, thus [tex]\sigma = 2[/tex]
- Margin of error of 0.4 hours, thus [tex]M = 0.4[/tex].
- The minimum sample size is n, thus:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]0.4 = 1.96\frac{2}{\sqrt{n}}[/tex]
[tex]0.4\sqrt{n} = 3.92[/tex]
[tex]\sqrt{n} = \frac{3.92}{0.4}[/tex]
[tex](\sqrt{n})^2 = (\frac{3.92}{0.4})^2[/tex]
[tex]n = 96.04[/tex]
Rounding up, the minimum sample size is of 97 students.
A similar problem is given at https://brainly.com/question/17039768
