math 146 I need help

Question

A population of values has a normal distribution with μ=155.5μ=155.5 and σ=9.8σ=9.8. You intend to draw a random sample of size n=173n=173.

Find P₆₁, which is the score separating the bottom 61% scores from the top 39% scores. 

P₆₁ (for single values) = 

Find P₆₁, which is the mean separating the bottom 61% means from the top 39% means. 

P₆₁ (for sample means) = 

Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

To find the percentile (for a single value) for a normally distributed random variable, we can use the following formula:

Percentile Value = μ + zσ

1. where: μ: Mean
2. z: z-score from z table that corresponds to percentile value
3. σ: Standard deviation

For example: to find the 15%, you need to find the value of Z (using the appropriate table) that closest to 0.15.

To find the percentile (for the sample mean), the CLT tells us that the distribution of the mean is also normal with the same mean as the population mean, and the std. dev. is = sigma/√N. Now you can follow the same steps as above(using the formula, but the std. dev. will be a different value, which will yield a different percentile.

Answer 2

Here are some hints to get you started:

1.) Part 1 is asking you for the value corresponding to a single observation drawn from a normally distributed population with a known standard deviation. Start with finding the Z score corresponding to p61. Convert that Z score to the original scale via, Do a bit of algebra on the definition of a Z score, Z = (x - mu)/sigma to solve for X. You are given mu and sigma in the problem.

2.) Part 2 is asking you about the sampling distribution of the mean. The central limit theorem tells us that this distribution is also normal and has the same mean as the population mean. The standard deviation of the sampling distribution of the mean, also known as the standard error, can be calculated as sigma/sqrt(N). So, you can take the Z score from part 1 and solve for x, but be sure to use the standard error you calculated as the "sigma" in the equation you used earlier.