Question

Appropriate use of the interval $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. $$ \begin{array}{l} \text { a. } n=100 \text { and } \hat{p}=0.70 \\ \text { b. } n=40 \text { and } \hat{p}=0.25 \\ \text { c. } n=60 \text { and } \hat{p}=0.25 \end{array} $$ d. \(n=80\) and \(\hat{p}=0.10\)

Short answer

The sample sizes in cases 'a', 'b', and 'c' are large enough for the interval to be appropriate. However, the sample size in case 'd' is not large enough.

Step-by-step solution

Calculate \(n\hat{p}\)

For each case, calculate \(n\hat{p}\). In case 'a', this would be \(100 \times 0.70 = 70\), for case 'b' it would be \(40 \times 0.25 = 10\), for 'c' it would be \(60 \times 0.25 = 15\) and for 'd' it would be \(80 \times 0.10 = 8\).

Calculate \(n(1-\hat{p})\)

Again, calculate this for each case. In case 'a', this would be \(100 \times (1 - 0.70) = 30\), for case 'b' it would be \(40 \times (1 - 0.25) = 30\), for 'c' it would be \(60 \times (1 - 0.25) = 45\) and for 'd' it would be \(80 \times (1 - 0.10) = 72\).

Check the condition

Now, if both these values are greater than or equal to 10 for each case, then the sample size is considered large enough for the interval to be appropriate. In case 'a' and 'b', both values are greater than or equal to 10. However, in cases 'c' and 'd', while \(n(1-\hat{p})\) is greater than or equal to 10, \(n\hat{p}\) in case 'd' is less than 10. Therefore, the sample size is not large enough in case 'd'.

Key concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, which is likely to contain the value of an unknown population parameter. To put it simply, it gives an estimated range of values which is assumed to cover the true value of the parameter with a certain degree of confidence, typically expressed as a percentage. This is crucial because we often cannot measure an entire population, but we try to infer about it from a sample.

When constructing a confidence interval for a population proportion, the formula typically used is:
\[\begin{equation}\hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\end{equation}\]
Here, \(\hat{p}\) is the sample proportion, \(z\) is the z-score corresponding to the desired level of confidence, and \(n\) is the sample size. The term \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) represents the margin of error for the confidence interval. The larger the sample size or the lower the confidence level, the narrower the confidence interval will be, providing a more precise estimate of the population parameter.

Population Proportion

Population proportion, denoted as \( P \), represents the fraction of individuals in a population that possess a certain characteristic, feature, or attribute. In contrast, a sample proportion, \(\hat{p}\), refers to the fraction of individuals with the characteristic in a sample drawn from the population.

Estimating the population proportion can be challenging because it is not feasible or practical to investigate every individual in a large population. Hence, statisticians use the sample proportion as an estimator of the population proportion, which involves collecting data from a subset of the whole population. The accuracy of the estimation improves with larger samples and random sampling methods, as it reduces the chance of bias and increases representation of the varied population characteristics.

Normal Approximation

The normal approximation method is used in statistics to estimate the probability distribution of a binomial variable using a normal distribution when certain conditions are met. This becomes particularly useful when the sample size is large, as it is easier to work with a normal distribution than a binomial one.

The binomial distribution can be approximated with a normal distribution when both \( n\hat{p} \) and \( n(1-\hat{p}) \) are greater than or equal to 10—the larger these products, the better the approximation. When these conditions are met, the sample proportion, \(\hat{p}\), has an approximately normal distribution, allowing us to make inferences about the population proportion with confidence intervals or hypothesis tests.

Sample Proportion

The sample proportion, \(\hat{p}\), is a statistic that estimates the proportion of elements in a population with a certain characteristic based on a sample. It's calculated by dividing the number of individuals in the sample with the characteristic by the total number of individuals in the sample.

For example, if you have a sample size of \(n\) individuals and \(x\) of them have the characteristic you're analyzing, the sample proportion is:
\[\begin{equation}\hat{p} = \frac{x}{n}\end{equation}\]
This sample proportion is key when conducting statistical tests or constructing confidence intervals. It's a snapshot of what we might expect in the wider population, and it plays a critical role in determining the representativeness of the sample. A larger sample size generally leads to a sample proportion that is a better estimator of the actual population proportion, primarily due to the reduction in the margin of error and the effects of random variability.