Answer:
[tex]CI =6.5 \± 1.09[/tex]
Step-by-step explanation:
Given
[tex]Data: 4, 5, 7, 5, 8, 7, 5, 9, 10, 5, 6, 7, 2, 11, 6, 3, 2, 11, 10, 7[/tex]
Required
Determine the 99% t-confidence interval
First, calculate the mean
[tex]\bar x =\frac{\sum x}{n}[/tex]
[tex]\bar x =\frac{4+ 5+ 7+ 5+ 8+ 7+ 5+ 9+10+ 5+ 6+ 7+ 2+ 11+ 6+ 3+ 2+ 11+ 10+ 7}{20}[/tex]
[tex]\bar x =\frac{130}{20}[/tex]
[tex]\bar x =6.5[/tex]
Calculate the standard deviation
[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)}{n}[/tex]
[tex]\sigma =\sqrt{\frac{(4-6.5)^2+ (5-6.5)^2+ ........+ (2-6.5)^2+ (11-6.5)^2+ (10-6.5)^2+ (7-6.5)^2}{20}}[/tex]
[tex]\sigma =\sqrt{\frac{143}{20}}[/tex]
[tex]\sigma =\sqrt{7.15}[/tex]
[tex]\sigma =2.67[/tex]
The z score for 99% confidence interval is: 2.576
The confidence interval is calculated as:
[tex]CI =\bar x \± z * \frac{\sigma}{\sqrt n}[/tex]
[tex]CI =6.5 \± 2.576 * \frac{2.67}{\sqrt {40}}[/tex]
[tex]CI =6.5 \± 2.576 * \frac{2.67}{6.32}}[/tex]
[tex]CI =6.5 \± 1.09[/tex]
