Question

In a random sample of 30 young bears, the average weight at the age of breeding is 312 pounds. Assuming the population ages are normally distributed with a population standard deviation is 30 pounds, use the Empirical Rule to construct a 68% confidence interval for the population average of young bears at the age of breeding. Do not round intermediate calculations. Round only the final answer to the nearest pound. Remember to enter the smaller value first, then the larger number.

Answer 1

Answer: (307,317)

Explanation:

The population standard deviation is known (and is assumed to be normally distributed). Using the Empirical Rule, we know that approximately 68% of the data lies within one standard deviation of the mean. So, the confidence coefficient is zα2=1, approximately.The point estimate for the population mean, μ, is the sample mean, x¯=312. The standard error for the sampling distribution is σx¯=σn√, where σ is the population standard deviation and n is the sample size. So, the standard error is σx¯=30n√. We did not list a decimal approximation because the instructions stated that only the final answer should be rounded. The error bound for the mean (EBM) is given by the formula EBM=zα2(σn√)=1(3030√). Again, we did not list a decimal approximation because the instructions stated that only the final answer should be rounded. Finally, we can calculate the confidence interval at the desired level of significance using the formula (point estimate−EBM , point estimate+EBM)(312−1(3030‾‾‾√),312+1(3030‾‾‾√))(307,317)

Answer 2

We have that the empirical rule to construct a 68% confidence interval can be expressed with the following general rule:

`(\mu-\sigma,\mu+\sigma)`

in this case, we have the following information:

`\begin{gathered} \mu=312 \\ \sigma=30 \end{gathered}`

then, the confidence interval with 68% of confidence is:

`(312-30,312+30)=(282,342)`

therefore, the confidence interval with 68% of confidence is (282,342)