Respuesta :
Answer:
The 95% confidence interval for the mean breaking weight for this type cable is (767.47 lb, 777.13 lb).
Step-by-step explanation:
Our sample size is 41
The first step to solve this problem is finding our degrees of freedom, that is, the sample size subtracted by 1. So
[tex]df = 41-1 = 40[/tex]
Then, we need to subtract one by the confidence level [tex]\alpha[/tex] and divide by 2. So:
[tex]\frac{1-0.95}{2} = \frac{0.05}{2} = 0.025[/tex]
Now, we need our answers from both steps above to find a value T in the t-distribution table. So, with 40 and 0.025 in the two-sided t-distribution table, we have [tex]T = 2.021[/tex]
Now, we find the standard deviation of the sample. This is the division of the standard deviation by the square root of the sample size. So
[tex]s = \frac{15.3}{\sqrt{41}} = 2.39[/tex]
Now, we multiply T and s
[tex]M = T*s =2.021*2.39 = 4.83[/tex]
Then
The lower end of the confidence interval is the mean subtracted by M. So:
[tex]L = 772.3 - 4.83 = 767.47[/tex]
The upper end of the confidence interval is the mean added to M. So:
[tex]L = 772.3 + 4.83 = 777.13[/tex]
The 95% confidence interval for the mean breaking weight for this type cable is (767.47 lb, 777.13 lb).
The confidence interval is a one of the estimates that can be determined from an observed data
The confidence interval is: [tex]\mathbf{ (767.47,777.13)}[/tex]
The given parameters are:
[tex]\mathbf{ n = 41}[/tex]
[tex]\mathbf{CI = 95\%}[/tex]
[tex]\mathbf{\mu = 772.3}[/tex]
[tex]\mathbf{\sigma = 15.3}[/tex]
Start by calculating the standard deviation of the sample
[tex]\mathbf{\sigma_x = \frac{\sigma}{\sqrt n}}[/tex]
[tex]\mathbf{\sigma_x = \frac{15.3}{\sqrt{41}}}[/tex]
[tex]\mathbf{\sigma_x = \frac{15.3}{6.40}}[/tex]
[tex]\mathbf{\sigma_x = 2.39}[/tex]
Next, calculate the degree of freedom (df)
[tex]\mathbf{ df = n-1}[/tex]
So, we have:
[tex]\mathbf{ df = 41-1}[/tex]
[tex]\mathbf{ df = 40}[/tex]
Calculate the significance level
[tex]\mathbf{\alpha = \frac{1 - CI}{2}}[/tex]
[tex]\mathbf{\alpha = \frac{1 - 95\%}{2}}[/tex]
[tex]\mathbf{\alpha = \frac{0.05}{2}}[/tex]
[tex]\mathbf{\alpha = 0.025}\\[/tex]
The T value, when [tex]\mathbf{\alpha = 0.025}\\[/tex] and [tex]\mathbf{ df = 40}[/tex] us:
[tex]\mathbf{T = 2.021}[/tex]
The endpoints of the confidence interval is calculated as:
[tex]\mathbf{CI = \mu \pm T \times \sigma}[/tex]
So, we have:
[tex]\mathbf{CI = 772.3 \pm 2.021 \times 2.39}[/tex]
[tex]\mathbf{CI = 772.3 \pm 4.83}[/tex]
Split
[tex]\mathbf{CI = (772.3 - 4.83,772.3 + 4.83)}[/tex]
[tex]\mathbf{CI = (767.47,777.13)}[/tex]
Hence, the confidence interval is: [tex]\mathbf{ (767.47,777.13)}[/tex]
Read more about confidence intervals at:
https://brainly.com/question/24131141
