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=64, find the probability of a sample mean being less than 24.3 when Mu=24 and sigma=1.25.

검증된 단계별 안내

Step 1: Identify the given values in the problem. The population mean (μ) is 24, the population standard deviation (σ) is 1.25, the sample size (n) is 64, and we are tasked with finding the probability that the sample mean (x̄) is less than 24.3.

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

Step 3: Convert the sample mean (x̄ = 24.3) to a z-score using the formula z = (x̄ - μ) / SE. Substitute the values for x̄, μ, and SE into the formula to calculate the z-score.

Step 4: Use the z-score to find the corresponding probability. Refer to the standard normal distribution table (or use statistical software) to find the cumulative probability associated with the calculated z-score.

Step 5: Interpret the result. Compare the probability to the threshold for unusual events (commonly 0.05 or 5%). If the probability is less than this threshold, the sample mean would be considered unusual; otherwise, it would not be considered unusual.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. This theorem is crucial for understanding how to calculate probabilities related to sample means, especially when the sample size is large, such as n=64 in this case.

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 = sigma/√n). In this scenario, the SE will help determine how likely it is for the sample mean to be less than 24.3 given the population parameters.

Z-Score

A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula Z = (X - Mu) / SE, where X is the sample mean. By calculating the Z-score for the sample mean of 24.3, we can assess whether this value is unusual compared to the expected distribution of sample means.