celesteh12

2022-10-13

A population of values has a normal distribution with μ=211.7$\mu =211.7$ and σ=23.8$\sigma =23.8$. You intend to draw a random sample of size n=232$n=232$. Please answer the following questions, and show your answers to 1 decimal place.

Find the value separating the bottom 30% values from the top 70% values.

Find the sample mean separating the bottom 30% sample means from the top 70% sample means.

Don Sumner

To solve this problem, let's first define the given values:
$\mu$ = Population mean = 211.7
$\sigma$ = Population standard deviation = 23.8
$n$ = Sample size = 232
(a) Finding the value separating the bottom 30% values from the top 70% values:
To find the value that separates the bottom 30% values from the top 70% values, we need to find the z-score corresponding to the cumulative probability of 0.30.
Using the z-score formula:
$z=\frac{x-\mu }{\sigma }$
Rearranging the formula to solve for x:
$x=z·\sigma +\mu$
To find the z-score corresponding to the cumulative probability of 0.30, we can use a standard normal distribution table or a calculator. The z-score for a cumulative probability of 0.30 is approximately -0.524.
Substituting the values into the formula, we have:
$x=-0.524·23.8+211.7$
$x\approx 198.60$
Therefore, the value separating the bottom 30% values from the top 70% values is approximately 198.60.
(b) Finding the sample mean separating the bottom 30% sample means from the top 70% sample means:
Since the sample mean follows the same distribution as the population mean, we can use the same formula as in part (a) to find the sample mean that separates the bottom 30% sample means from the top 70% sample means.
Using the given values, the sample mean separating the bottom 30% sample means from the top 70% sample means is approximately 198.60.

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