In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Standard Deviation For the following, round results to three decimal places.

a. Find the value of the population standard deviation σ.

검증된 단계별 안내

Step 1: Recall the formula for the population standard deviation (σ). The formula is: σ = sqrt((Σ(xᵢ - μ)²) / N), where xᵢ represents each data point, μ is the population mean, and N is the population size.

Step 2: Calculate the population mean (μ). Use the formula μ = Σxᵢ / N, where Σxᵢ is the sum of all data points in the population and N is the number of data points in the population.

Step 3: Subtract the population mean (μ) from each data point in the population to find the deviations (xᵢ - μ). Then, square each deviation to get (xᵢ - μ)².

Step 4: Sum all the squared deviations (Σ(xᵢ - μ)²). This gives the total squared deviation for the population.

Step 5: Divide the total squared deviation by the population size (N) and take the square root of the result to find the population standard deviation (σ). Round the final result to three decimal places as instructed.

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주요 개념

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Population Standard Deviation

The population standard deviation (σ) is a measure of the dispersion or spread of a set of values in a population. It quantifies how much the individual data points deviate from the population mean. To calculate it, you take the square root of the variance, which is the average of the squared differences from the mean. Understanding this concept is crucial for analyzing the variability within a population.

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic (like the sample mean or sample standard deviation) obtained from a large number of samples drawn from the same population. It helps in understanding how sample statistics vary from sample to sample. In this context, knowing the sampling distribution of the sample standard deviation is essential for making inferences about the population based on sample data.

Random Sampling with Replacement

Random sampling with replacement means that each time a sample is drawn from the population, the selected element is returned to the population before the next draw. This method ensures that each selection is independent and that the probability of selecting any particular element remains constant across samples. This concept is important for accurately assessing the variability and distribution of sample statistics.