Questions: A population of values has a normal distribution with μ=131.6 and σ=76.2. You intend to draw a random sample of size n=212. Find the probability that a sample of size n=212 is randomly selected with a mean greater than 124.8 . P(M>124.8)= Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z scores rounded to 3 decimal places are accepted.

A population of values has a normal distribution with μ=131.6 and σ=76.2. You intend to draw a random sample of size n=212.

Find the probability that a sample of size n=212 is randomly selected with a mean greater than 124.8 .
P(M>124.8)=

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z scores rounded to 3 decimal places are accepted.

Transcript text: A population of values has a normal distribution with $\mu=131.6$ and $\sigma=76.2$. You intend to draw a random sample of size $n=212$. Find the probability that a sample of size $n=212$ is randomly selected with a mean greater than 124.8 . \[ P(M>124.8)= \] $\square$ Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact $z$-scores or $z$ scores rounded to 3 decimal places are accepted. Question Help: Message instructor

Solution

Solution Steps

Step 1: Given Parameters

We have a population with a normal distribution characterized by the following parameters:

Mean ($ \mu $) = 131.6
Standard deviation ($ \sigma $) = 76.2
Sample size ($ n $) = 212
We want to find the probability that the sample mean is greater than 124.8.

Step 2: Calculate the Standard Error

The standard error of the mean ($ \sigma_M $) is calculated using the formula: \[ \sigma_M = \frac{\sigma}{\sqrt{n}} = \frac{76.2}{\sqrt{212}} \approx 5.227 \]

Step 3: Calculate the Z-Score

To find the Z-score for the sample mean of 124.8, we use the formula: \[ Z = \frac{M - \mu}{\sigma_M} = \frac{124.8 - 131.6}{5.227} \approx -1.2993 \]

Step 4: Find the Probability

The probability that the sample mean is greater than 124.8 can be expressed as: \[ P(M > 124.8) = 1 - P(M \leq 124.8) = 1 - \Phi(Z_{start}) \] Where $ Z_{start} = -1.2993 $. Using the cumulative distribution function $ \Phi $: \[ P(M > 124.8) = 1 - \Phi(-1.2993) \approx 0.9031 \]