Suppose that a random sample of size 100 is to be drawn from...

Suppose that a random sample of size 100 is to be drawn from a population with standard deviation 10 . a. What is the probability that the sample mean will be within 20 of the value of $\mu$ ? b. For this example $(n=100, \sigma=10)$, complete each of the following statements by calculating the appropriate value: i. Approximately $95 \%$ of the time, $\bar{x}$ will be within of $\mu$. ii. Approximately $0.3 \%$ of the time, $\bar{x}$ will be farther than from $\mu$.

Suppose that a random sample of size 100 is to be drawn from a population with standard deviation 10 .
a. What is the probability that the sample mean will be within 20 of the value of $\mu$ ?
b. For this example $(n=100, \sigma=10)$, complete each of the following statements by calculating the appropriate value:
i. Approximately $95 \%$ of the time, $\bar{x}$ will be within of $\mu$.
ii. Approximately $0.3 \%$ of the time, $\bar{x}$ will be farther than from $\mu$.

Key Concepts

Confidence Interval

A confidence interval provides a range of values within which the true population parameter is likely to be found with a certain level of confidence. In problems involving the sample mean, the interval is computed using the standard error and corresponding Z-scores from the standard normal distribution to determine the desired level of certainty.

Z-score

A Z-score is a standardized measure that indicates how many standard deviations an observation is from the mean. In the context of the sample mean, it is used to convert values into a standard normal distribution so that probability calculations can be made, particularly in situations involving the central limit theorem.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will tend to be approximately normal, regardless of the population's distribution, provided the sample size is large enough. This principle is foundational for making probability statements about the sample mean when the sample size is sufficiently large.

Sampling Distribution of the Mean

The sampling distribution of the mean is the probability distribution that describes how the sample mean varies from sample to sample. When drawing samples from a population, the mean of these sample means will equal the population mean, and the distribution will have a specific spread determined by the standard error.

Standard Error

The standard error measures the variability of the sample mean around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size. This concept is crucial in quantifying the precision of the sample mean as an estimator of the population mean and for calculating probabilities and confidence intervals.

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