Question

Given a normal distribution with mean μ=100 and standard deviation σ=10, and given you select a random sample of n=25 determine the following probabilities. round all answers to 4 decimal places.

Answer 1

To find the probability that the sample mean is between 85 and 92, standardize the values using the z-score formula and find the corresponding probabilities using a standard normal distribution table or calculator. The probability is approximately 0.6997.

Step-by-step explanation:

To find the probability that the sample mean is between 85 and 92, we need to standardize the values using the formula z = (x - μ) / (σ / √n), where z is the standard score, x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Using this formula, we can find the z-scores for 85 and 92 and then use a standard normal distribution table or a calculator to find the corresponding probabilities.

1. 
   
3. Calculate the z-score for 85: z = (85 - 90) / (15 / √25) = -5 / 3 = -1.6667.
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6. Calculate the z-score for 92: z = (92 - 90) / (15 / √25) = 2 / 3 = 0.6667.
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9. Look up the probability associated with each z-score in a standard normal distribution table or use a calculator. The probability for z = -1.6667 is approximately 0.0478, and the probability for z = 0.6667 is approximately 0.7475.
10. 
    
12. Subtract the probability associated with the lower z-score from the probability associated with the higher z-score to find the probability that the sample mean is between 85 and 92: 0.7475 - 0.0478 = 0.6997.