Question

An entomologist wishes to estimate the average development time of the citrus red mite, a small spider-like insect that causes damage to leaves and fruit, correct to within . 5 day. From previous experiments it is known that \(\sigma\) is approximately 4 days. How large a sample should the entomologist take to be \(95 \%\) confident of her estimate?

Short answer

Answer: The entomologist needs a sample size of 246 citrus red mites to achieve the desired margin of error and confidence level.

Step-by-step solution

Find the Z score corresponding to the desired confidence level.

The problem states that we want a 95% confidence level. For a 95% confidence interval, the Z score corresponding to the \(\frac{\alpha}{2}\) is approximately 1.96 (which can be found in a standard normal Z-table or using a calculator).

Insert known values into the sample size formula.

We have all the necessary values to plug into the sample size formula: $$n = \left(\frac{1.96 \times 4}{0.5}\right)^2$$

Calculate the sample size.

Now, we can calculate the sample size: $$n = \left(\frac{7.84}{0.5}\right)^2 = (15.68)^2 = 245.7024$$ Since we cannot have a fraction of a sample, we round up the result to the nearest whole number to ensure the desired level of confidence and precision: $$n = 246$$ The entomologist needs to take a sample of 246 citrus red mites to be 95% confident that the average development time estimation is correct to within 0.5 days.

Key concepts

Confidence Intervals

Understanding confidence intervals is vital when interpreting statistical data. Confidence intervals provide a range of values, which is estimated to contain the true population parameter (like a mean or proportion) with a specified level of confidence. Think of confidence intervals as a net that tries to catch the true value – the wider the net, the more likely it is to catch the value, but the less precise the estimate becomes.

In our entomology example, the researcher aims to estimate the average development time of the citrus red mite. To state this estimate not just as a single number but within a range that captures the true mean with a given certainty, a confidence interval is used. When we say we are 95% confident, it doesn't mean that there's a 95% chance the true parameter is in our interval. What it does mean is that if we took many samples and built confidence intervals for each one, we expect approximately 95% of those intervals to contain the true population mean.

Standard Deviation

Standard deviation is a measure of how spread out numbers are in a data set, or in simpler terms, how much the numbers differ from the mean (average). It's what statisticians use to gauge the variability or diversity within a set of numbers. A low standard deviation means that most of the numbers are close to the average, while a high standard deviation indicates that the numbers are more spread out.

In the citrus red mite study, a standard deviation of 4 days indicates how much variation there is in the development time among individual mites. It’s important for determining sample size because the more variable the data (the larger the standard deviation), the larger the sample size needed to achieve a certain level of confidence and precision in the results.

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. If a Z-score is 0, it represents the score is identical to the mean score.

Z-scores are crucial when creating confidence intervals or performing hypothesis testing. They are the cornerstone of how we calculate the probability of a score occurring within a normal distribution and a key part in determining how many standard deviations an element is from the mean. In the context of our confidence interval for the mite's development time, the Z-score of 1.96 corresponds to the 95% confidence level. This tells us that if we take repeated samples from the population, the average development time will fall within 1.96 standard deviations of the sample mean 95% of the time.

Statistics in Entomology

Statistics in entomology and other biological sciences is an essential tool for understanding and interpreting variability within natural systems. Researchers use statistics to analyze data from experiments and observations, which can include things like the development time of insects, population size, or the effect of a treatment on pest control.

In our textbook problem, statistics are being applied to determine how large a sample should be to estimate the development time of citrus red mites with a certain level of confidence. This process, which uses knowledge about standard deviations and Z-scores to create confidence intervals, illustrates the role statistics plays in designing experiments and making decisions based on data in the complex field of entomology.