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.
the population isnt given to be approximately normal.
the sample size isnt given to be normal.
the wrong sampling method was used.

Explanation:

Identify the goal and given conditions

The problem asks why we cannot make a statistical inference about the population mean age based on a simple random sample of ten people. We are given:

• Sample size \(n = 10\).
• The sampling method is a simple random sample (SRS).
• The target parameter is the population average age.

Analyze the requirements for inference

To perform statistical inference (such as constructing a confidence interval or conducting a hypothesis test) for a population mean, certain conditions must be met:

1. Randomness: The data must come from a random sample. This condition is met because a simple random sample was used.
2. Independence: The sample size should be less than 10% of the population. This is highly likely to be met for registered voters in a district.
3. Normality: The sampling distribution of the sample mean must be approximately normal. This is satisfied if:

• The population distribution is approximately normal, OR
• The sample size is large enough (typically \(n \ge 30\)) by the Central Limit Theorem.

Evaluate the given sample size and population

Here, the sample size is \(n = 10\), which is small (\(n < 30\)). Because the sample size is small, we cannot rely on the Central Limit Theorem to guarantee that the sampling distribution of the mean is approximately normal. Therefore, for statistical inference to be valid, we must know that the underlying population distribution of ages is approximately normal. Since this is not stated or known, we cannot safely perform the inference.

Evaluate the given options

• The sample size is not appropriate: While \(n = 10\) is small, a small sample size is not inherently "inappropriate" if the underlying population is normal.
• The population isn't given to be approximately normal: This is the correct reason. Since \(n = 10\) is small, we require the population to be approximately normal to proceed with standard inference procedures.
• The sample size isn't given to be normal: A sample size is a single number (\(10\)), so it cannot be "normal."
• The wrong sampling method was used: A simple random sample is the gold standard for sampling, so the method is correct.

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.