Question

Refer to Exercise 13. Suppose you wish to estimate the difference between the mean acidity for rainfalls at two different locations, one in a relatively unpolluted area and the other in an area subject to heavy air pollution. If you wish your estimate to be correct to the nearest. \(1 \mathrm{pH}\), with probability near 90 , approximately how many rainfalls (pH values) would have to be included in each sample? (Assume that the variance of the pH measurements is approximately .25 at both locations and that the samples will be of equal size.)

Short answer

Answer: Approximately 27 rainfalls.

Step-by-step solution

Identify variables

In this problem, we are given: - Variance (σ²) = 0.25 - Desired margin of error (ME) = 1 pH - Confidence level (CL) = 90%

Find the standard deviation (σ)

The standard deviation (σ) is the square root of the variance (σ²). σ = sqrt(σ²) = sqrt(0.25) = 0.5

Calculate the critical value (z)

Using a standard normal distribution table or calculator, find the critical value (z) for a 90% confidence level. The critical value is the number of standard deviations away from the mean that corresponds to the desired confidence level. z = 1.645 (for a 90% confidence level)

Calculate the standard error (SE) for the difference between two means

The formula for the standard error for the difference between two means is: SE = sqrt(SE₁² + SE₂²) Since we are considering equal sample sizes (n) for both locations, we can rewrite the equation as: SE = sqrt(2 * (σ²/n))

Set up and solve the equation involving margin of error and standard error

The margin of error (ME) is the product of the critical value (z) and the standard error (SE). ME = z * SE Plug in the values we have (ME, z, σ) to find the sample size (n). 1 = 1.645 * sqrt(2 * (0.25/n)) After rearranging and solving for n, we get: n ≈ 27 Therefore, approximately 27 rainfalls (pH values) would have to be included in each sample.

Key concepts

Confidence Interval

When scientists conduct experiments or researchers analyze data, they often need to make inferences about population parameters based on sample statistics. The concept of a confidence interval provides a range of values, estimated from the data, that is likely to contain the true population parameter with a certain probability, known as the confidence level.

In our exercise, the intention is to conclude about the mean acidity differences between two locations. We utilize the concept of a confidence interval to estimate this difference with a specified level of certainty. The 90% confidence level implies that if we were to take many samples and build intervals in the same way, we would expect about 90% of these intervals to contain the true mean difference. This statistical technique allows researchers to quantify the uncertainty of their estimate, which is crucial when decisions or conclusions are to be drawn from the data.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.

In the context of our exercise, the standard deviation of the pH measurements helps us understand how much individual measurements can be expected to vary around the mean. Knowing that the variance is 0.25, we calculated the standard deviation to be 0.5. This calculation is essential as it feeds into further computations, such as calculating the standard error and subsequently the sample size needed to achieve the desired margin of error for our confidence interval. The smaller the standard deviation, the closer our sample mean is likely to be to the true population mean, which suggests that fewer samples may be needed to achieve a certain level of accuracy.

Sample Size Calculation

Determining the appropriate sample size for a study is crucial to ensure reliable and valid results. The sample size affects the precision of the estimates and the power of the study to detect a true effect. The larger the sample size, the smaller the margin of error and the more confident we can be that our sample statistically reflects the population.

In our exercise, we calculated that approximately 27 rainfalls would need to be included in each sample to achieve a margin of error of 1 pH unit with 90% confidence. The formula used incorporates the variance, desired margin of error, and confidence level. Calculating an adequate sample size is a balancing act between the desired precision and the practicalities and costs of data collection. This step is indispensable in planning statistical research, as it underpins the credibility and generalizability of the results.

Margin of Error

The margin of error quantifies the uncertainty in an estimate and is typically associated with the confidence interval. It reflects the range within which we can expect the true population parameter to fall, with a certain level of confidence. The margin of error is influenced by two main factors: the standard deviation of the population and the size of the sample.

As we saw in the exercise, to keep the margin of error at a desired level (1 pH unit), we had to manipulate the sample size based on the standard deviation and the confidence level. The margin of error also helps gauge the precision of an estimate—the smaller the margin of error, the closer the sample statistic is likely to be to the true population value. Understanding and controlling the margin of error is vital for researchers to ensure their findings are both accurate and reliable.