A population of values has a normal distribution with $$ \mu=143.3 $$ and $$ \sigma=74.1 $$. You intend to draw a random sample of size $$ n=147 $$. Find the prodability that a single randomly selected value is between 134.7 and 162.9 . $$ P(134.7

We start with a normally distributed variable $$ X $$ with mean $$\mu=143.3$$ and standard deviation $$\sigma=74.1$$. A random sample of size $$ n=147 $$ is drawn. ### 1. Single Randomly Selected Value We want to find \[ P(134.7 < X < 162.9). \] We first standardize to convert $$ X $$ to a standard normal variable $$ Z $$ using \[ Z=\frac{X-\mu}{\sigma}. \] **For $$ X=134.7 $$:** \[ Z_{\text{low}}=\frac{134.7-143.3}{74.1}=\frac{-8.6}{74.1}\approx -0.1161. \] **For $$ X=162.9 $$:** \[ Z_{\text{high}}=\frac{162.9-143.3}{74.1}=\frac{19.6}{74.1}\approx 0.2645. \] The probability is then \[ P(134.7\[ P(134.7 < X < 162.9) = 0.1520, \] \[ P(134.7 < M < 162.9) = 0.9200. \]

[Solved]: A population of values has a normal distribution with mu=143.3 and sigma=74.1. You intend to draw a ...

The Deep Dive

To solve this problem, we will use the properties of the normal distribution. Let's calculate the two probabilities one by one. ### Probability for a Single Randomly Selected Value 1. First, we need to calculate the Z-scores for the values 134.7 and 162.9. \[ Z = \frac{X - \mu}{\sigma} \] For $ X = 134.7 $: \[ Z_{134.7} = \frac{134.7 - 143.3}{74.1} \approx -0.115 \] For $ X = 162.9 $: \[ Z_{162.9} = \frac{162.9 - 143.3}{74.1} \approx 0.263 \] 2. Now, we will look up these Z-scores in the standard normal distribution table or use a calculator. - The probability for $ Z = -0.115 $ is approximately 0.4543. - The probability for $ Z = 0.263 $ is approximately 0.6032. 3. To find the probability that a single randomly selected value is between 134.7 and 162.9: \[ P(134.7 < X < 162.9) = P(Z < 0.263) - P(Z < -0.115) \approx 0.6032 - 0.4543 = 0.1489 \] ### Probability for the Sample Mean 1. We need to find the Z-scores for the sample mean. We use the standard error of the mean (SEM): \[ SEM = \frac{\sigma}{\sqrt{n}} = \frac{74.1}{\sqrt{147}} \approx 6.094 \] 2. We will calculate the Z-scores for the sample mean. For $ M = 134.7 $: \[ Z_{134.7} = \frac{134.7 - 143.3}{6.094} \approx -1.42 \] For $ M = 162.9 $: \[ Z_{162.9} = \frac{162.9 - 143.3}{6.094} \approx 3.25 \] 3. Again, we look up these Z-scores: - The probability for $ Z = -1.42 $ is approximately 0.0772. - The probability for $ Z = 3.25 $ is approximately 0.9994. 4. Now, we find the probability that a sample mean is between 134.7 and 162.9: \[ P(134.7 < M < 162.9) = P(Z < 3.25) - P(Z < -1.42) \approx 0.9994 - 0.0772 = 0.9222 \] ### Final Answers So, the probabilities are: - $ P(134.7 < X < 162.9) \approx 0.1489 $ - $ P(134.7 < M < 162.9) \approx 0.9222 $

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Statistics Questions