In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population.
The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Verified step by step guidance

1. Step 1: Calculate the mean of the population. The population consists of the values {1, 2, 0, 3}. Use the formula for the population mean: $\mu =\frac{\sum x}{}$_iN, where $N$ is the total number of values in the population.
2. Step 2: Calculate the population standard deviation. Use the formula: $\sqrt{\frac{\sum (x}{}}$_i - μ)2N, where $\mu$ is the population mean calculated in Step 1.
3. Step 3: Identify all possible samples of size 2 from the population {1, 2, 0, 3}. Since the sample size is 2, list all combinations of two values from the population. For example, one sample could be {1, 2}, another could be {1, 0}, and so on.
4. Step 4: For each sample, calculate the sample mean using the formula: $\frac{\sum x}{}$_in, where $n$ is the sample size (in this case, 2).
5. Step 5: For each sample, calculate the sample standard deviation using the formula: $\sqrt{\frac{\sum (x}{}}$_i - x̄)2n - 1, where $\mathrm{x̄}$ is the sample mean calculated in Step 4.

Key Concepts

Population Mean

The population mean is the average of all values in a population. It is calculated by summing all the values and dividing by the total number of values. In this case, to find the mean of the goals scored by the defenders, you would add the goals (1 + 2 + 0 + 3) and divide by 4, the total number of defenders.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean. To calculate it, you find the variance (the average of the squared differences from the mean) and then take the square root of that variance. This helps in understanding the consistency of the defenders' goal-scoring performance.

Sample Size

Sample size refers to the number of observations or data points selected from a population for analysis. In this question, a sample size of 2 means that you will select 2 defenders' goal scores to analyze. The choice of sample size can affect the accuracy and reliability of statistical estimates, such as the sample mean and standard deviation.