Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=36, find the probability of a sample mean being less than 12,750 or greater than 12,753 when mu=12750 and 1.7.

검증된 단계별 안내

Step 1: Identify the given values. The population mean (μ) is 12,750, the population standard deviation (σ) is 1.7, and the sample size (n) is 36. We are tasked with finding the probability of the sample mean being less than 12,750 or greater than 12,753.

Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is SE = σ / √n. Substitute the given values into the formula to compute the standard error.

Step 3: Convert the sample mean values (12,750 and 12,753) into z-scores using the formula z = (X̄ - μ) / SE, where X̄ is the sample mean. Compute the z-scores for both 12,750 and 12,753.

Step 4: Use the standard normal distribution table (or a statistical software) to find the probabilities corresponding to the calculated z-scores. For the z-score of 12,750, find the cumulative probability. For the z-score of 12,753, find the cumulative probability and subtract it from 1 to get the upper tail probability.

Step 5: Add the probabilities from Step 4 to find the total probability of the sample mean being less than 12,750 or greater than 12,753. Finally, determine if this probability is unusual by comparing it to a significance threshold (e.g., 0.05).

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주요 개념

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Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n ≥ 30). This theorem is crucial for calculating probabilities related to sample means.

Standard Error

Standard Error (SE) measures the dispersion of sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size (SE = σ/√n). Understanding SE is essential for determining how much variability to expect in sample means and for calculating probabilities.

Z-scores

A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula Z = (X - μ) / SE, where X is the sample mean, μ is the population mean, and SE is the standard error. Z-scores are used to find probabilities in a standard normal distribution, helping to assess whether a sample mean is unusual.