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=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.

Verified step by step guidance

1. Step 1: Identify the given values in the problem. The population mean (μ) is 550, the population standard deviation (σ) is 3.7, the sample size (n) is 45, and we are tasked with finding the probability of the sample mean (x̄) being greater than 551.
2. Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is: $\mathrm{SE}=\frac{\sigma }{\sqrt{n}}$. Substitute the values: $\mathrm{SE}=\frac{3.7}{\sqrt{45}}$.
3. Step 3: Compute the z-score for the sample mean. The formula for the z-score is: $z=\frac{\mathrm{x̄}-\mu }{\mathrm{SE}}$. Substitute the values: $z=\frac{551-550}{\mathrm{SE}}$.
4. Step 4: Use the z-score to find the probability. Look up the z-score in the standard normal distribution table or use statistical software to find the cumulative probability. Since the problem asks for the probability of the sample mean being greater than 551, calculate $1-P(Z<z)$.
5. Step 5: Determine if the sample mean is unusual. A sample mean is considered unusual if the probability is less than 0.05 (5%). Compare the calculated probability to this threshold to make the determination.

Key Concepts

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 = sigma / √n). In this case, it helps determine how much variability to expect in the sample mean.

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. This score is essential for finding probabilities in a standard normal distribution and determining if a sample mean is unusual.