Question

Suppose you wish to estimate the mean \(\mathrm{pH}\) of rainfalls in a heavily polluted area. You know that \(\sigma\) is approximately \(.5 \mathrm{pH},\) and you wish your estimate to lie within .1 of \(\mu,\) with a probability near \(.95 .\) Approximately how many rainfalls must be included in your sample (one pH reading per rainfall)? Would it be valid to select all of your water specimens from a single rainfall? Explain.

Short answer

Answer: Approximately 97 rainfalls are needed to achieve the desired margin of error with a 95% confidence level. It is not valid to select all water specimens from a single rainfall.

Step-by-step solution

Identify the known values

We are given the following values: - Standard deviation, \(\sigma = .5\ \mathrm{pH}\) - Margin of error, \(E = .1\ \mathrm{pH}\) - Confidence level, \(C = .95\) or \(95\%\)

Calculate the z-score for the confidence level

First, we need to find the z-score associated with the given confidence level. This score represents how many standard deviations away from the mean a value is. For a \(95\%\) confidence level, we look up the z-score in a standard normal (z) table or use a calculator that has an inverse cumulative distribution function. In this case, the z-score is: \(\textit{z} \approx 1.96\)

Apply the formula for sample size

We will use the formula for estimating the sample size when the standard deviation is known: \(n = \left(\dfrac{z\cdot\sigma}{E}\right)^2\) Substituting the given values, we get: \(n = \left(\dfrac{1.96 \times .5}{.1}\right)^2\) Calculate the value: \(n \approx 96.04\)

Round up to the nearest whole number

As we cannot have a fractional number of samples, we need to round up the value of n to the nearest whole number: \(n = 97\)

Determine if selecting water specimens from a single rainfall is valid

Selecting all water specimens from a single rainfall would not be valid for this estimation. This is because the pH readings might be similar and would not give a representative sample of the mean pH for the heavily polluted area. For the estimated sample size to be applicable, the samples need to be randomly selected from different rainfalls, ensuring the independence of observations and eliminating any bias. #Conclusion# Approximately 97 rainfalls must be included in the sample to achieve the desired margin of error with a 95% confidence level. It would not be valid to select all water specimens from a single rainfall.

Key concepts

Standard Deviation

Standard deviation is a statistical measure that describes the spread or dispersion of a set of data points. In simpler terms, it tells us how much the values in a dataset tend to deviate from the mean (average) value. The formula for standard deviation is the square root of the variance, where variance is the average of the squared differences from the mean. In our exercise, it relates to the variation in pH levels of rainfalls.

When working with sample sizes, knowing the standard deviation is important as it influences the width of confidence intervals for the mean. A smaller standard deviation means that the values are clustered closely around the mean, and we can be more confident in our estimates with fewer samples. Conversely, a larger standard deviation requires a larger sample size to achieve the same level of confidence.

Confidence Level

Confidence level is the probability that the value of a parameter falls within a specified range of values, called the confidence interval. Essentially, it tells us how sure we can be about our estimations. For instance, a 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we expect about 95 of those intervals to contain the true mean.

The most common confidence levels in statistics are 90%, 95%, and 99%. Choosing a higher confidence level widens the confidence interval, requiring a larger sample size to estimate the parameter with the same margin of error. The exercise specifies a 95% confidence level, which strikes a balance between precision and assuredness.

Margin of Error

The margin of error is the range within which we expect our sample estimate to reflect the true population parameter. It quantifies the maximum expected difference between the observed sample mean and the true population mean. In practical terms, in our pH example, a margin of error of 0.1 pH means we are aiming for our sample mean to be within 0.1 units of the true mean pH level.

The margin of error is affected by the size of the sample and the standard deviation of the population. To achieve a smaller margin of error (more precision), we can either increase the sample size or decrease the standard deviation if possible. The formula incorporates the z-score and standard deviation to help calculate the necessary sample size for the desired margin of error.

Z-Score

A z-score indicates how many standard deviations an element is from the mean. It's a way of standardizing scores on different scales to a common scale, with a mean of zero and a standard deviation of one. In the context of confidence intervals, the z-score helps determine how far the estimate is expected to be from the mean if the true distribution is normal (bell-shaped).

The z-score corresponding to a 95% confidence level is approximately 1.96, meaning the sample mean is expected to be within 1.96 standard deviations of the population mean 95% of the time. This value is integral to the formula for calculating the required sample size to meet the specified margin of error and confidence level.

Random Sampling

Random sampling is a fundamental concept in statistics that ensures each member of the population has an equal chance of being selected for the sample. This method promotes the representativeness and independence of the sample, which are key to producing unbiased estimations of the population parameters.

In the exercise, selecting all water specimens from a single rainfall would not constitute random sampling and would likely introduce bias, since the pH levels from a single rainfall might not be indicative of the variety found across multiple rainfalls in a heavily polluted area. True random sampling in this context would involve selecting different rainfalls randomly to collect pH readings, ensuring a more accurate and generalizable estimate of the mean pH level for the entire area.