In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The test scores for the Law School Admission Test (LSAT) in a recent year are normally distributed, with a mean of 151.88 and a standard deviation of 9.95. Random samples of size 40 are drawn from this population, and the mean of each sample is determined.

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Step 1: Understand the problem. We are tasked with finding the mean and standard deviation of the sampling distribution of sample means for LSAT scores. The population mean (μ) is given as 151.88, the population standard deviation (σ) is 9.95, and the sample size (n) is 40.

Step 2: Recall the properties of the sampling distribution of sample means. The mean of the sampling distribution (μ_x̄) is equal to the population mean (μ). Therefore, μ_x̄ = μ = 151.88.

Step 3: Calculate the standard deviation of the sampling distribution of sample means, also known as the standard error (SE). The formula for the standard error is:

σ

_x̄=σn. Substituting the values,

σ

_x̄=9.9540.

Step 4: Sketch the graph of the sampling distribution. Since the population distribution is normal, the sampling distribution of sample means will also be normal. The graph will be a bell-shaped curve centered at the mean (151.88) with a spread determined by the standard error calculated in Step 3.

Step 5: Summarize the results. The mean of the sampling distribution is 151.88, and the standard deviation (standard error) is calculated using the formula in Step 3. The graph is a normal distribution centered at 151.88 with the calculated standard error as its spread.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sampling Distribution of Sample Means

The sampling distribution of sample means refers to the distribution of the means of all possible random samples of a specific size drawn from a population. According to the Central Limit Theorem, this distribution will tend to be normally distributed, regardless of the population's distribution, as the sample size increases. It is characterized by its mean, which equals the population mean, and its standard deviation, known as the standard error.

Mean

The mean is a measure of central tendency that represents the average of a set of values. In the context of sampling distributions, the mean of the sampling distribution of sample means is equal to the population mean. It provides a summary measure that indicates where the center of the data lies, making it essential for understanding the overall trend of the sample data.

Standard Deviation and Standard Error

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. In sampling distributions, the standard error is the standard deviation of the sampling distribution of sample means, calculated as the population standard deviation divided by the square root of the sample size. It indicates how much the sample means are expected to vary from the population mean, providing insight into the reliability of the sample estimates.

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