High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ A population of values has a normal distribution with μ=229.6 and σ=97.4. If a random sample of size n=15 is selected, find the probability that a single randomly selected value is less than 297.5. Round your answer to four decimals.

Answer :

To find the probability that a single randomly selected value is less than 297.5 from a normally distributed population, we'll use the properties of the normal distribution.

Given data:

  • Mean ([tex]\mu[/tex]) = 229.6
  • Standard deviation ([tex]\sigma[/tex]) = 97.4

We want to find [tex]P(X < 297.5)[/tex].

Step-by-step Solution

  1. Calculate the Z-score:

    The Z-score helps us understand how many standard deviations a specific value (in this case, 297.5) is away from the mean. The formula for calculating the Z-score is:

    [tex]Z = \frac{X - \mu}{\sigma}[/tex]

    Plug in the values:

    [tex]Z = \frac{297.5 - 229.6}{97.4} = \frac{67.9}{97.4} \approx 0.6971[/tex]

  2. Use the Z-score to find the probability:

    To find the probability [tex]P(X < 297.5)[/tex], we'll use the standard normal distribution table (Z-table) or a calculator with normal distribution functions.

    Find the cumulative probability corresponding to the Z-score of 0.6971.

    According to the Z-table or calculator, [tex]P(Z < 0.6971) \approx 0.7570.[/tex]

  3. Interpret the result:

    The probability that a randomly selected value from this population is less than 297.5 is approximately 0.7570.

    Therefore, there is a 75.70% chance that a value randomly chosen from this population will be less than 297.5.