In a random sample of 95 manufacturing firms, 67 indicated...

In a random sample of 95 manufacturing firms, 67 indicated that their company attained ISO certification within the last two years. Find a $99 \%$ confidence interval for the population proportion of companies that have been certified within the last 2 years.

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample proportion approximates a normal distribution. This principle underpins the use of z-scores and normal probability methods in constructing confidence intervals for proportions.

Margin of Error

The margin of error represents the maximum expected difference between the true population proportion and the sample proportion, computed as the product of the critical value and the standard error. It is crucial in forming the limits of the confidence interval.

Critical Value

The critical value is a point on the standard normal distribution that corresponds to the desired level of confidence (such as a z-score for a 99% confidence level). It determines how many standard errors to extend from the sample proportion to form the confidence interval.

Confidence Interval for a Population Proportion

This concept refers to creating an estimated range of values that is likely to include the true proportion of a population. It involves using the sample proportion, its standard error, and a critical value from the normal distribution corresponding to the chosen confidence level to generate a lower and upper bound.

Standard Error of a Proportion

The standard error of a proportion quantifies the variability or spread in the sampling distribution of the proportion. It is calculated using the formula ?[p?(1-p?)/n], where p? is the sample proportion and n is the sample size, and it is essential for determining the margin of error in the interval estimation.

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