We appreciate your visit to Suppose the ages of the residents of Georgetown are normally distributed with a mean of 38 2 years and a standard deviation of 8 6. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem, we need to determine the probability that the mean age of a sample of 30 residents is less than 38 years when the ages of the residents follow a normal distribution with a mean of 38.2 years and a standard deviation of 8.6 years.
Here's how we can figure this out step by step:
1. Identify the Given Values:
- Population mean ([tex]\(\mu\)[/tex]) = 38.2 years
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 8.6 years
- Sample size ([tex]\(n\)[/tex]) = 30
- Target sample mean = 38 years
2. Calculate the Standard Error of the Mean:
- The standard error of the mean is calculated using the formula:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Plugging in the values:
[tex]\[
\text{Standard Error} = \frac{8.6}{\sqrt{30}} \approx 1.5701
\][/tex]
3. Calculate the Z-Score:
- The z-score measures how many standard errors the target sample mean is from the population mean. It is calculated using the formula:
[tex]\[
z = \frac{\text{Target Sample Mean} - \mu}{\text{Standard Error}}
\][/tex]
- Using the given values:
[tex]\[
z = \frac{38 - 38.2}{1.5701} \approx -0.1274
\][/tex]
4. Find the Probability:
- The probability that the sample mean is less than 38 is found using a standard normal distribution table or a calculator that provides cumulative distribution function values.
- A z-score of approximately [tex]\(-0.1274\)[/tex] translates to a probability of about 0.4493.
Therefore, the probability that the mean age of the sample is less than 38 years is approximately 0.4493, or 44.93%.
Here's how we can figure this out step by step:
1. Identify the Given Values:
- Population mean ([tex]\(\mu\)[/tex]) = 38.2 years
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 8.6 years
- Sample size ([tex]\(n\)[/tex]) = 30
- Target sample mean = 38 years
2. Calculate the Standard Error of the Mean:
- The standard error of the mean is calculated using the formula:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Plugging in the values:
[tex]\[
\text{Standard Error} = \frac{8.6}{\sqrt{30}} \approx 1.5701
\][/tex]
3. Calculate the Z-Score:
- The z-score measures how many standard errors the target sample mean is from the population mean. It is calculated using the formula:
[tex]\[
z = \frac{\text{Target Sample Mean} - \mu}{\text{Standard Error}}
\][/tex]
- Using the given values:
[tex]\[
z = \frac{38 - 38.2}{1.5701} \approx -0.1274
\][/tex]
4. Find the Probability:
- The probability that the sample mean is less than 38 is found using a standard normal distribution table or a calculator that provides cumulative distribution function values.
- A z-score of approximately [tex]\(-0.1274\)[/tex] translates to a probability of about 0.4493.
Therefore, the probability that the mean age of the sample is less than 38 years is approximately 0.4493, or 44.93%.
Thanks for taking the time to read Suppose the ages of the residents of Georgetown are normally distributed with a mean of 38 2 years and a standard deviation of 8 6. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
