Question

the city wishes to estimate the average commute distance for all city employees. they collect a random sample of 83 employees and find a sample mean of \\(\bar{x} = 9.6\\) miles. the city wants to estimate \\(\mu\\), the average commute distance for all city employees with 95% confidence.

check the conditions for performing a one-sample \\(t\\)-interval.

select two answers

the sample size not sufficiently large so the sampling distribution cant be assumed approximately normal
the sample size is sufficiently large so the sampling distribution should be approximately normal
the population distribution is approximately normal so the sample size does not matter
the sample is a random sample and representative of the population

Explanation:

Evaluate randomness condition

Using the t-Interval Conditions knowledge point

• The problem states: "They collect a random sample of 83 employees."
• This satisfies the randomness condition, meaning the sample is representative of the population.

Evaluate normality condition

Using the t-Interval Conditions knowledge point

• The sample size is \(n = 83\).
• Since \(n \ge 30\), the Central Limit Theorem applies.
• Thus, the sampling distribution of the sample mean \(\bar{x}\) is approximately normal.

Answer:

• The sample size not sufficiently large so the sampling distribution can't be assumed approximately normal
• The sample size is sufficiently large so the sampling distribution should be approximately normal (Correct answer)
• The population distribution is approximately normal so the sample size does not matter
• The sample is a random sample and representative of the population (Correct answer)