Math Problem Statement

A normally distributed population has a mean of 475 and a standard deviation of 30.

a. Determine the probability that a random sample of size 25 selected from this population will have a sample mean less than 457.

b. Determine the probability that a random sample of size 9 selected from the population will have a sample mean greater than or equal to 500.

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Part 1 a.

Upper P left parenthesis x overbar less than 457 right parenthesisPx<457equals=(answer) ​(Round to four decimal places as​ needed.)

Part 2 b.

Upper P left parenthesis x overbar greater than or equals 500 right parenthesisPx≥500equals=enter your response here

Solution

To solve these problems, we will use the sampling distribution of the sample mean, which is normally distributed with a mean μxˉ\mu_{\bar{x}} equal to the population mean μ\mu and a standard deviation σxˉ\sigma_{\bar{x}} given by:

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

where σ\sigma is the population standard deviation, and nn is the sample size.

Given dataTo solve these problems, we will use the sampling distribution of the sample mean, which is normally distributed with a mean μxˉ\mu_{\bar{x}} equal to the population mean μ\mu and a standard deviation σxˉ\sigma_{\bar{x}} given by:

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

where σ\sigma is the population standard deviation, and nn is the sample size.

Given

  • Population mean μ=475\mu = 475
  • Population standard deviation σ=30\sigma = 30

Part (a): Probability that the sample mean is less than 457 for n=25n = 25

  1. Calculate the standard deviation of the sample mean: σxˉ=3025=305=6\sigma_{\bar{x}} = \frac{30}{\sqrt{25}} = \frac{30}{5} = 6

  2. Calculate the z-score for xˉ=457\bar{x} = 457: z=xˉμσxˉ=4574756=186=3z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{457 - 475}{6} = \frac{-18}{6} = -3

  3. Find P(xˉ<457)P(\bar{x} < 457) using the z-score**: Using a z-table, the probability corresponding to z=3z = -3 is approximately 0.00130.0013.

So, the probability that the sample mean is less than 457 is: P(xˉ<457)0.0013P(\bar{x} < 457) \approx 0.0013


Part (b): Probability that the sample mean is greater than or equal to 500 for n=9n = 9

  1. Calculate the standard deviation of the sample mean: σxˉ=309=303=10\sigma_{\bar{x}} = \frac{30}{\sqrt{9}} = \frac{30}{3} = 10

  2. Calculate the z-score for xˉ=500\bar{x} = 500: z=xˉμσxˉ=50047510=2510=2.5z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} = \frac{500 - 475}{10} = \frac{25}{10} = 2.5

  3. Find P(xˉ500)P(\bar{x} \geq 500) using the z-score**: For z=2.5z = 2.5, the cumulative probability is approximately 0.99380.9938. Therefore, P(xˉ500)=10.9938=0.0062P(\bar{x} \geq 500) = 1 - 0.9938 = 0.0062.

So, the probability that the sample mean is greater than or equal to 500 is: P(xˉ500)0.0062P(\bar{x} \geq 500) \approx 0.0062


Summary of Answers

  • (a) P(xˉ<457)0.0013P(\bar{x} < 457) \approx 0.0013
  • (b) P(xˉ500)0.0062P(\bar{x} \geq 500) \approx 0.0062

Would you like more details, or have any questions?


  1. How does the sample size affect the standard deviation of the sample mean?
  2. What is the significance of a z-score in finding probabilities in a normal distribution?
  3. How would the answers change if the population standard deviation were different?
  4. Why is the cumulative probability used to calculate probabilities for z-scores?
  5. How can we interpret the results in terms of likelihood or rarity of the events?

Tip: When working with z-scores, remember that a positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Sampling Distribution
Z-score

Formulas

Standard deviation of the sample mean: σ_x̄ = σ / √n
Z-score formula: z = (x̄ - μ) / σ_x̄

Theorems

Central Limit Theorem

Suitable Grade Level

College-level Statistics