To address your questions, we will analyze each part step by step:
(a) Is it appropriate to use the normal distribution to find probabilities for x?
In this case, we have a population mean of \( \mu = 131 \) and a population standard deviation of \( \sigma = 21 \). We are considering a sample of size \( n = 19 \).
For the Central Limit Theorem to apply, we typically require that the sample size is large (usually \( n \geq 30 \)) or that the population from which the sample is drawn is normally distributed. Since we do not have any information about the distribution of the population, and because the sample size is less than 30, we cannot definitively conclude that the sampling distribution of the sample mean \( \bar{x} \) is normally distributed.
However, if the population is normal or if the sample size was larger (usually \( n \geq 30 \)), we could use the normal distribution. Therefore, without additional information about the population distribution, it is not appropriate to use the normal distribution for the sample mean in this case.
(b) If appropriate find the probability that \( \bar{x} \) will be less than 118.
Since we determined that the use of the normal distribution is not appropriate in this instance due to the small sample size and lack of information about the population distribution, we cannot correctly find this probability using the normal distribution.
(c) If appropriate find the 85th percentile of \( \bar{x} \).
Once again, because we cannot assume a normal distribution for the sample mean due to the small sample size and the unknown distribution of the population, we cannot correctly determine the 85th percentile using the normal distribution.
Conclusion:
- (a) It is not appropriate to use the normal distribution to find probabilities for \( \bar{x} \).
- (b) We cannot find the probability that \( \bar{x} < 118 \) without knowing more about the population distribution.
- (c) We cannot find the 85th percentile of \( \bar{x} \) under the current conditions.
If you have more information about the population distribution (e.g., if it is known to be normal), you could proceed with further analysis.