Question

is the nutritional information listed on food items accurate? researchers randomly sampled 12 frozen dinners of a certain type from production during a particular period. the calorie content was determined. the stated calorie content on the package was 240 calories. researchers wanted to determine if there was evidence to indicate that the mean calorie count was not equal to 240 calories using the one - sample t - methods. which of the following statements is true?
choose the correct answer below
○ a. the distribution of sample means will be normal only if the population data (that is, calorie content of all frozen dinners of this type produced during this particular period) follow a normal distribution.
○ b. the distribution of sample means will be normal regardless of the shape of the data in the population (that is, calorie content of all frozen dinners of this type produced during this particular period).
○ c. since the sample size is less than 30, the population data cannot be normally distributed. therefore, the distribution of sample means will not be normally distributed.
○ d. the distribution of sample means will not be normal regardless of the shape of the data in the population because the sample size is less than 30.

Explanation:

• The Central Limit Theorem (CLT) states that for a sample of size \(n\), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal if \(n\) is large (\(n\geq30\)). When \(n < 30\), the sampling distribution of the sample mean is exactly normal if the population from which the sample is drawn is normal.
• Option B is incorrect because the CLT does not apply for \(n=12\) (since \(n < 30\)) to make the sampling distribution of the sample mean normal regardless of the population distribution.
• Option C is wrong. A sample size less than 30 does not imply that the population is not normal. The population can be normal even if \(n<30\).
• Option D is incorrect. If the population is normal, the sampling distribution of the sample mean is normal even for \(n < 30\).
• Option A is correct. When \(n = 12<30\), the distribution of sample means (sampling distribution of \(\bar{X}\)) is normal only if the population is normal.

Answer:

A. The distribution of sample means will be normal only if the population data (that is, calorie content of all frozen dinners of this type produced during this particular period) follow a normal distribution.