Respuesta :
Answer:
Confidence interval
[tex]8.98-2.2\frac{1.29}{\sqrt{12}}=8.16[/tex]
[tex]8.98+2.2\frac{1.29}{\sqrt{12}}=9.80[/tex]
And the margin of error would be:
[tex] ME=2.2\frac{1.29}{\sqrt{12}}=0.819[/tex]
Step-by-step explanation:
For this case we have the followig dataset:
DATA: 8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10.5
We can calculate the mean and the deviation with the following formulas:
[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
[tex]s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]
And replacing we got:
[tex] \bar X= 8.98[/tex
[tex]s = 1.29[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
The degrees of freedom are given by:
[tex]df=n-1=12-1=11[/tex]
The Confidence level is 0.95 or 95%, the value of significance is [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and the critical value would be [tex]t_{\alpha/2}=2.20[/tex]
Repplacing the info we got:
[tex]8.98-2.2\frac{1.29}{\sqrt{12}}=8.16[/tex]
[tex]8.98+2.2\frac{1.29}{\sqrt{12}}=9.80[/tex] And the margin of error would be:
[tex] ME=2.2\frac{1.29}{\sqrt{12}}=0.819[/tex]
The 95% confidence interval is [tex]\mathbf{ (8.17,9.79)}[/tex], and the margin of error is 0.81
The dataset is given as:
[tex]\mathbf{x= 8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10.5}[/tex]
Start by calculating the sample mean
This is calculated as:
[tex]\mathbf{\bar x = \frac{\sum x}{n}}[/tex]
[tex]\mathbf{\bar x= \frac{8.2+ 9.1+ 7.7+ 8.6+ 6.9+11.2+ 10.1+ 9.9+ 8.9+ 9.2+ 7.5+ 10.5}{12}}[/tex]
[tex]\mathbf{\bar x= \frac{107.8}{12}}[/tex]
[tex]\mathbf{\bar x= 8.98}[/tex]
Next, calculate the sample standard deviation
This is calculated as:
[tex]\mathbf{\sigma_x = \sqrt{\frac{\sum (x - \bar x)^2}{n-1}}}[/tex]
[tex]\mathbf{\sigma_x= \sqrt{\frac{(8.2 - 8.89)^2 +.......... + (10.5- 8.89)^2}{12 - 1}}}[/tex]
[tex]\mathbf{\sigma_x= 1.29}[/tex]
Calculate the degrees of freedom
[tex]\mathbf{df = n - 1}[/tex]
[tex]\mathbf{df = 12 - 1}[/tex]
[tex]\mathbf{df = 11}[/tex]
Calculate the significance level
[tex]\mathbf{\alpha /2=\frac{1 - 95\%}2 = 0.025}[/tex]
The critical value at [tex]\mathbf{\alpha/2 = 0.025}[/tex] and df = 11 is:
[tex]\mathbf{t_{\alpha/2} = 2.20}[/tex]
Calculate the margin of error
[tex]\mathbf{E = t_{\alpha/2} \frac{\sigma}{\sqrt n}}[/tex]
So, we have:
[tex]\mathbf{E = 2.20 \times \frac{1.29}{\sqrt{12}}}[/tex]
[tex]\mathbf{E = 0.81}[/tex]
The confidence interval is then calculated as:
[tex]\mathbf{CI =\bar x \pm E}[/tex]
So, we have:
[tex]\mathbf{CI = 8.98 \pm 0.81}}[/tex]
Split
[tex]\mathbf{CI = (8.98 - 0.81,8.98 + 0.81)}[/tex]
[tex]\mathbf{CI = (8.17,9.79)}[/tex]
So, the 95% confidence interval is [tex]\mathbf{ (8.17,9.79)}[/tex], and the margin of error is 0.81
Read more about confidence intervals and margin of errors at:
https://brainly.com/question/19426829
