Question

In a study of human blood types in nonhuman primates, a sample of 71 orangutans were tested, and 14 were found to be blood type \(\mathrm{B}^{10}\) Construct a \(95 \%\) confidence interval for the relative frequency of blood type \(\mathrm{B}\) in the orangutan population.

Short answer

With \( \hat{p} = \frac{14}{71} \) and \( z^* = 1.96 \) for a 95% CI, calculate \( SE \) and CI to find that the 95% confidence interval for the relative frequency of blood type B in the orangutan population is approximately (0.118, 0.277).

Step-by-step solution

Calculate the sample proportion

First, determine the sample proportion of orangutans with blood type B by dividing the number of individuals with blood type B by the total number of individuals in the sample. In symbols, the sample proportion \( \hat{p} \) is calculated as \( \hat{p} = \frac{\text{number with blood type B}}{\text{total in sample}} = \frac{14}{71} \).

Determine the z-score for a 95% confidence level

For a 95% confidence interval, the z-score corresponds to the critical value \(z^*\) that captures the central 95% of the standard normal distribution. This z-score can be found in a z-table or by using an appropriate calculator function, typically approximately equal to 1.96.

Calculate the standard error

The standard error (SE) of the proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( n \) is the sample size. Here, \( SE = \sqrt{\frac{(14/71)(1 - 14/71)}{71}} \).

Compute the confidence interval

The confidence interval (CI) is calculated as \( CI = \hat{p} \pm z^* \times SE \). Plug the values \( \hat{p} \) and SE from the previous steps and the z-score into this formula to find the interval.

Finalize the confidence interval

After calculating the lower and upper bounds using the CI formula, round the results to an appropriate number of decimal places for the context of the problem, typically three decimal places.

Key concepts

Sample Proportion

When researchers want to estimate the characteristics of a larger population based on a smaller group, they often use a sample proportion. This proportion is simply the number of times a characteristic occurs divided by the total number of observations. For instance, in the given exercise on orangutan blood types, we find the sample proportion ( ) of those with blood type B by dividing the number orangutans with this blood type (14) by the total sample size (71).
This proportion provides us with a point estimate for what the true proportion might be in the entire population. It's the building block of constructing a confidence interval, enabling us to infer population characteristics from our sample data.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of confidence intervals, we use the z-score to determine the range around our sample proportion which contains the true population proportion with a certain degree of certainty.
For a 95% confidence interval, we typically use a z-score of about 1.96. This tells us that if we were to take multiple samples and calculate intervals using this z-score, about 95% of these intervals would contain the true population proportion.

Standard Error

The concept of standard error (SE) quantifies how much we would expect our sample proportion to vary from one sample to another, purely because of chance. It's a way of gauging the precision of our sample estimate with respect to the true population characteristic.
The formula used for calculating the SE in the context of a proportion is rooted in the principles of probability and involves the sample proportion and the sample size. Larger samples tend to have smaller standard errors, indicating more stable estimates of the population parameter.

Statistical Significance

The concept of statistical significance helps us understand whether a particular finding is likely due to an actual effect or just random chance. When we construct a 95% confidence interval, we are essentially saying that we're 95% confident the interval contains the true parameter - and if an estimate or test result falls outside this interval, it might be statistically significant.
A result is typically considered statistically significant if the probability of the observation occurring by chance is below a pre-defined threshold, often 5%. It's important to remember that statistical significance doesn't necessarily imply practical significance - it's a measure of confidence in the result rather than its magnitude.

Normal Distribution

The normal distribution is a fundamental concept in statistics, often depicted as a bell-shaped curve. It describes a distribution where most of the observations cluster around the central mean and the probabilities for values further away from the mean taper off symmetrically in both directions.
The assumption of a normal distribution of sample proportions underpins the construction of a confidence interval. This is justified by the Central Limit Theorem, which states that the sampling distribution of the sample proportion will approximate a normal distribution as the sample size grows, even if the population distribution itself is not normal.