Final answer:
To find the probability that a single randomly selected value is less than $975, we use the Z-score formula. The probability is approximately 0.5144. To find the probability that a sample of size n=70 is randomly selected with a mean less than $975, we use the Central Limit Theorem. The probability is approximately 0.6255.
Step-by-step explanation:
To find the probability that a single randomly selected value is less than $975, we need to use the Z-score formula. The Z-score formula is calculated as Z = (X - μ) / σ, where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
- Calculate the Z-score: Z = (975 - 967) / 206 = 0.0388.
- Look up the Z-score in the Z-table or use a calculator to find the corresponding probability. For a Z-score of 0.0388, the probability is approximately 0.5144.
Therefore, the probability that a single randomly selected value is less than $975 is 0.5144.
To find the probability that a sample of size n=70 is randomly selected with a mean less than $975, we can use the Central Limit Theorem. The Central Limit Theorem states that the sample means of sufficiently large samples from any population will be approximately normally distributed.
- Calculate the standard error: SE = σ / sqrt(n) = 206 / sqrt(70) = 24.637.
- Calculate the Z-score: Z = (975 - 967) / 24.637 = 0.3242.
- Look up the Z-score in the Z-table or use a calculator to find the corresponding probability. For a Z-score of 0.3242, the probability is approximately 0.6255.
Therefore, the probability that a sample of size n=70 is randomly selected with a mean less than $975 is 0.6255.