The percent of fat calories that a person in America consumes each day is normally distributed with a mean of 36 and a standard deviation of about 10. Suppose that 16 individuals are randomly chosen for an experiment.

A) Is it appropriate to use the normal approximation for the sampling distribution in this case? Why or Why Not?

B)Assuming its appropriate to use the normal approximation, find the first quartile for the average percent of fat calories and write a sentence to interpret it in the context of this problem?

I know the 1st quartile is 34.31 but not sure how to write sentence to interpret it.

Thanks

A) small sample. What experiment?

B) If that is the first quartile, then 25% would consume 34.31 fat calories or less.

Consumption ≤ 34.51

A) Yes, it is appropriate to use the normal approximation for the sampling distribution in this case. This is because the sample size is relatively large (16 individuals), which satisfies the central limit theorem, stating that for a large sample size, the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution.

B) To find the first quartile, we need to find the value below which 25% of the data lies. Using a standard normal distribution table or a calculator, the z-score corresponding to the first quartile is approximately -0.674.

To find the first quartile for the average percent of fat calories:
1. Calculate the standard error of the mean (SE):

SE = standard deviation / sqrt(sample size)
= 10 / sqrt(16)
= 10 / 4
= 2.5

2. Find the actual value for the first quartile:

First quartile = mean + (z-score * SE)
= 36 + (-0.674 * 2.5)
= 36 - 1.685
≈ 34.315

Sentence to interpret the first quartile:
The first quartile for the average percent of fat calories consumed by individuals in America is approximately 34.315. This means that 25% of the individuals in the sample have an average percent of fat calories below 34.315, indicating a relatively lower intake of fat calories compared to the overall mean.

A) Yes, it is appropriate to use the normal approximation for the sampling distribution in this case. The sample size of 16 individuals satisfies the condition for the Central Limit Theorem (CLT) which states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. Therefore, we can approximate the sampling distribution as a normal distribution.

B) To find the first quartile of the average percent of fat calories, we need to find the z-score corresponding to the first quartile. Since the population mean is 36 and the standard deviation is given as 10, we can calculate the z-score using the formula:

z = (x - μ) / σ

where x is the value corresponding to the first quartile, μ is the mean, and σ is the standard deviation. Rearranging the formula, we can solve for x:

x = z * σ + μ

Given that the first quartile corresponds to a z-score of approximately -0.674, we can substitute the values into the equation:

x = (-0.674) * 10 + 36 ≈ 29.26

Therefore, the first quartile for the average percent of fat calories is approximately 29.26.

To interpret this result in the context of the problem, we can say that approximately 25% of the randomly chosen individuals consume an average percent of fat calories that is less than or equal to 29.26. This suggests that a quarter of the individuals in the population have a lower average percent of fat calories than this value.