Sample Size Suppose x has a normal distribution with σ=1.2 ....

Sample Size Suppose $x$ has a normal distribution with $\sigma=1.2 .$ (a) Find the minimal sample size required so that for a $90 \%$ confidence interval, the maximal margin of error is $E=0.5$. (b) Check Requirements Based on this sample size and the $x$ distribution, can we assume that the $\bar{x}$ distribution is approximately normal? Explain.

Sample Size Suppose $x$ has a normal distribution with $\sigma=1.2 .$
(a) Find the minimal sample size required so that for a $90 \%$ confidence interval, the maximal margin of error is $E=0.5$.
(b) Check Requirements Based on this sample size and the $x$ distribution, can we assume that the $\bar{x}$ distribution is approximately normal? Explain.

Key Concepts

Margin of Error and Sample Size Determination

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is directly influenced by the sample size, the variability in the data, and the critical value from the z-distribution. Determining the appropriate sample size to achieve a specific margin of error involves rearranging the confidence interval formula, so that as the sample size increases the standard error decreases, leading to a smaller margin of error.

Z-Distribution and Critical Values

When the population standard deviation is known and the data are normally distributed, the z-distribution is used to calculate critical values corresponding to the desired confidence level. These z-scores are crucial for determining both the width of a confidence interval and the sample size required for a given margin of error.

Central Limit Theorem

The central limit theorem states that, regardless of the shape of the original population distribution, the distribution of the sample mean will tend to be approximately normal if the sample size is sufficiently large. This principle justifies using the normal distribution to evaluate the behavior of the sample mean, especially in the context of constructing confidence intervals, even when the sample is drawn from a non-normally distributed population.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. When the population standard deviation is known and the underlying data is normally distributed, the confidence interval for the mean is calculated using the z-distribution, ensuring that the interval captures the desired level of confidence (e.g., 90%).

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