Calculating a confidence interval for the proportion...

Calculating a confidence interval for the proportion requires a minimum sample size. Calculate a confidence interval, using any confidence level of $90 \%$ or higher, for the population proportion for each of the following. a. $n=200 \quad$ and $\quad \hat{p}=.01$ b. $n=160 \quad$ and $\quad \hat{p}=.9875$ Explain why these confidence intervals reveal a problem when the conditions for using the normal approximation do not hold.

Calculating a confidence interval for the proportion requires a minimum sample size. Calculate a confidence interval, using any confidence level of $90 \%$ or higher, for the population proportion for each of the following.
a. $n=200 \quad$ and $\quad \hat{p}=.01$
b. $n=160 \quad$ and $\quad \hat{p}=.9875$
Explain why these confidence intervals reveal a problem when the conditions for using the normal approximation do not hold.

Key Concepts

Minimum Sample Size Requirement

For using the normal approximation in constructing confidence intervals for a proportion, there is a requirement that the sample size be large enough so that the expected numbers of successes (np) and failures (n(1-p)) both meet a minimum value. When these conditions fail, the interval may not capture the true uncertainty in the estimate, leading to unreliable confidence intervals.

Confidence Interval for a Proportion

A confidence interval for a proportion is a range of values, derived from sample data, that is likely to contain the true population proportion with a certain level of confidence. It is constructed using the point estimate (sample proportion), standard error, and a critical value from a relevant probability distribution (typically the normal distribution) assuming the underlying assumptions hold.

Normal Approximation and Conditions

The normal approximation is used when the sampling distribution of the sample proportion is assumed to be approximately normal. This assumption holds if the sample size is sufficiently large, typically when both np and n(1-p) are greater than or equal to a minimum threshold (commonly 10). If these conditions aren’t met, the distribution may be too skewed for the normal approximation to be reliable.

Issues When Conditions Are Not Met

When the conditions for the normal approximation do not hold, especially because the sample size is too small or the sample proportion is extremely low or high, the standard formula can produce confidence intervals that are either too narrow, too wide, or even nonsensical (such as limits outside the [0,1] range). This highlights the need for alternative methods or adjustments when the approximation conditions fail.

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