Question

The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds Construct a 95% confidence interval for the population mean weight of newborn elephants. State the confidence interval (Round your answers to two decimal places.) Sketch the graph. (Round your answers to two decimal places.) CL - 0.95 X Calculate the error bound (Round your answer to two decimal places)

Answer 1

The 95% confidence interval for the population mean weight of newborn elephants is between 237.32 pounds and 250.68 pounds. The error bound for this confidence interval is 6.68 pounds.

Step-by-step explanation:

To calculate the 95% confidence interval for the population mean weight of newborn elephants, we can use the formula:

`\[ \text{Confidence interval} = \text{Sample mean} \pm \left( \text{Critical value} * \frac{\text{Sample standard deviation}}{\sqrt{\text{Sample size}}} \right) \]`

Given: Sample mean = 244 pounds, Sample standard deviation = 11 pounds, Sample size = 50, and for a 95% confidence level, the critical value is 1.96 (z-value).

Calculating the margin of error:

`\[ \text{Margin of error} = \text{Critical value} * \frac{\text{Sample standard deviation}}{\sqrt{\text{Sample size}}} = 1.96 * (11)/(√(50)) \approx 3.34 \text{ pounds} \]`

Thus, the 95% confidence interval is:

`\[ 244 \pm 3.34 = (240.66, 247.34) \text{ pounds} \]`

Rounded to two decimal places, the confidence interval is between 237.32 pounds and 250.68 pounds.

The error bound is half of the width of the confidence interval, which is \( \frac{247.34 - 240.66}{2} = 3.34 \) pounds, rounded to two decimal places.

Answer 2

The 95% confidence interval for the mean weight of newborn elephant calves is approximately (239.46, 248.54) pounds. The error bound is 4.54 pounds.

Step-by-step explanation:

To construct the confidence interval, we use the formula:

`\[ \text{Confidence Interval} = \text{Sample Mean} \pm \left( \text{Critical Value} * \frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \]`

Given that the sample mean
`(\(\bar{X}\))` is 244 pounds, the sample standard deviation (\(S\)) is 11 pounds, and the sample size (\(n\)) is 50, we find the critical value for a 95% confidence interval (CL = 0.95) using a standard normal distribution table. The critical value is approximately 1.96. Substituting these values into the formula, we get the confidence interval:

`\[ 244 \pm (1.96 * (11)/(√(50))) \approx (239.46, 248.54) \]`

The error bound is calculated as half the width of the confidence interval:

`\[ \text{Error Bound} = \frac{\text{Confidence Interval Width}}{2} \approx ((248.54 - 239.46))/(2) \approx 4.54 \]`

Thus, the final answer is a 95% confidence interval of (239.46, 248.54) pounds with an error bound of 4.54 pounds.