Question

a simple random sample of ten people is taken from the set of all registered voters in a given district to determine their average age. which of the following is a reason why you can not make a statistical inference on the population?

the sample size is not appropriate.
x the population isnt given to be approximately normal.
the sample size isnt given to be normal.
the wrong sampling method was used.

Explanation:

Analyze the sampling scenario

The problem describes a simple random sample of size \(n = 10\) taken from a population of registered voters to estimate the average age. We want to determine why a statistical inference (such as constructing a t-interval or performing a t-test for the population mean) cannot be safely made.

Evaluate inference assumptions

Using the Statistical Inference Assumptions knowledge point:
For inference about a population mean \(\mu\) using a t-distribution, we require the sampling distribution of the sample mean \(\bar{x}\) to be approximately normal. This condition is met if:

1. The underlying population distribution is approximately normal, OR
2. The sample size is large enough (typically \(n \ge 30\)) to apply the Central Limit Theorem.

Check the given conditions

Using the Central Limit Theorem Conditions knowledge point:

• Here, the sample size is \(n = 10\), which is small (\(n < 30\)). Thus, we cannot rely on the Central Limit Theorem to guarantee the normality of the sampling distribution.
• Since the Central Limit Theorem does not apply, we must assume or be given that the underlying population distribution of ages is approximately normal.
• Because the population is not stated to be approximately normal, and the sample size is too small to overcome this, we cannot safely perform statistical inference.

Evaluate the given options

• "The sample size is not appropriate": While small, a sample size of 10 can be appropriate for inference if the underlying population is known to be normal.
• "The population isn't given to be approximately normal": This is the correct reason. Since \(n = 10 < 30\), we must have a normal population to proceed.
• "The sample size isn't given to be normal": A sample size is a single number, not a distribution, so it cannot "be normal."
• "The wrong sampling method was used": A simple random sample (SRS) is the ideal sampling method.

Answer:

• The sample size is not appropriate.
• The population isn't given to be approximately normal. (Correct answer)
• The sample size isn't given to be normal.
• The wrong sampling method was used.