Final answer:
A sample of size 15 will be drawn from a population with mean 139 and standard deviation 21 is a) Yes, it is appropriate to use the normal distribution to find probabilities for xˉ. b) The probability that xˉ will be less than 122 is approximately 0.25%. c) The 85th percentile of xˉ is approximately 142.29.
Step-by-step explanation:
(a) Is it appropriate to use the normal distribution to find probabilities for xˇ?
Yes, it is appropriate to use the normal distribution to find probabilities for xˇ.
When the sample size is sufficiently large (greater than or equal to 30), the sampling distribution of the sample mean becomes approximately normally distributed, regardless of the shape of the population distribution.
This is known as the Central Limit Theorem.
(b) If appropriate, find the probability that xˇ will be less than 122.
To find the probability that xˇ will be less than 122, we can use the z-score formula: z = (xˇ - μ) / (σ / sqrt(n))
Where
xˇ is the sample mean
μ is the population mean
σ is the population standard deviation
n is the sample size.
Plugging in the values given in the question, we have z = (122 - 139) / (21 / sqrt(15)) ≈ -2.81.
We can then use a standard normal distribution table or a calculator to find the probability associated with the z-score -2.81, which is approximately 0.0025.
Therefore, the probability that xˇ will be less than 122 is approximately 0.0025, or 0.25%.
(c) If appropriate, find the 85th percentile of xˇ.
To find the 85th percentile of xˇ, we need to find the z-score associated with the 85th percentile using a standard normal distribution table or a calculator. The z-score corresponding to the 85th percentile is approximately 1.0364.
We can then use the z-score formula to find the value of xˇ associated with the z-score 1.0364: xˇ = μ + (z * (σ / sqrt(n))).
Plugging in the values given in the question, we have xˇ = 139 + (1.0364 * (21 / sqrt(15))) ≈ 142.29.
Therefore, the 85th percentile of xˇ is approximately 142.29.