Questions: Homework 8 Question 12 of 23 (1 point) Question Attempt: 1 of Unlimited Number of giraffes Sample mean Sample standard deviation 14 5.898 1.164 Take Sample Enter the values of the sample size, the point estimate of the mean, the sample standard deviation, and the critical value for the confidence interval. (Choose the correct critical value from the table of critical values provided.) When you are done, select Compute. Sample size: Point estimate: Sample standard deviation: Critical value: Standard error: Margin of error: 99% confidence interval: Critical values t0.005 = 3.012 t0.010 = 2.650 t0.025 = 2.160 t0.050 = 1.771 t0.100 = 1.350 Compute Check

Solution Steps

The standard error (SE) is calculated using the formula:

\[ SE = \frac{s}{\sqrt{n}} \]

where \( s \) is the sample standard deviation and \( n \) is the sample size. Given \( s = 1.164 \) and \( n = 14 \):

\[ SE = \frac{1.164}{\sqrt{14}} \approx 0.3111 \]

The margin of error (ME) is calculated using the formula:

\[ ME = t \cdot SE \]

where \( t \) is the critical value for the t-distribution at the desired confidence level. For a 99% confidence level, \( t \approx 3.012 \). Thus, the margin of error is:

\[ ME = 3.012 \cdot 0.3111 \approx 0.8013 \]

The confidence interval (CI) for the mean is given by:

\[ \bar{x} \pm ME \]

where \( \bar{x} \) is the sample mean. Given \( \bar{x} = 5.898 \):

\[ CI = 5.898 \pm 0.8013 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 5.898 - 0.8013 \approx 5.0967 \] \[ \text{Upper Bound} = 5.898 + 0.8013 \approx 6.6993 \]

Thus, the 99% confidence interval is:

\[ (4.9609, 6.8351) \]

Final Answer