Question

Acid rain, caused by the reaction of certain air pollutants with rainwater, appears to be a growing problem in the northeastern United States. (Acid rain affects the soil and causes corrosion on exposed metal surfaces.) Pure rain falling through clean air registers a pH value of 5.7 (pH is a measure of acidity: 0 is acid; 14 is alkaline). Suppose water samples from 40 rainfalls are analyzed for \(\mathrm{pH}\) and \(\bar{x}\) and \(s\) are equal to 3.7 and \(.5,\) respectively Find a \(99 \%\) confidence interval for the mean \(\mathrm{pH}\) in rainfall and interpret the interval. What assumption must be made for the confidence interval to be valid?

Short answer

Answer: The 99% confidence interval for the mean pH in rainfall is (3.486, 3.914).

Step-by-step solution

Calculate degrees of freedom

Degrees of freedom (df) is calculated as the sample size (\(n\)) minus 1. In this case, the sample size is 40. $$df = n - 1 = 40 - 1 = 39$$

Find the t-distribution critical value

We need to find the t-distribution critical value with \(\frac{\alpha}{2}\) and 39 degrees of freedom. The confidence level is 99%, so \(\alpha = 1 - 0.99 = 0.01\). Therefore, we need to look for the value of \(t_{0.005, 39}\). Using a t-distribution table or calculator, we find \(t_{0.005, 39} \approx 2.707\).

Calculate the margin of error

Now that we have the critical value, we can calculate the margin of error for the confidence interval: $$E = t_{0.005, 39} \cdot \frac{s}{\sqrt{n}} = 2.707 \cdot \frac{0.5}{\sqrt{40}} \approx 0.214$$

Calculate the confidence interval

Now, we can calculate the 99% confidence interval for the population mean \(\mu\): $$\bar{x} \pm E = 3.7 \pm 0.214$$ The 99% confidence interval for the mean pH in rainfall is (3.486, 3.914).

Interpret the confidence interval

We can interpret the 99% confidence interval by saying that we are 99% confident that the true mean pH of rainfall in the northeastern United States is between 3.486 and 3.914.

Identify the underlying assumption

The underlying assumption needed for the confidence interval to be valid is that the pH measurements are taken from a random sample of rainfalls that follows a normal distribution. This is important for the t-distribution method to provide accurate results.

Key concepts

t-distribution

The t-distribution is an essential statistical tool used when working with small sample sizes or when the population standard deviation is unknown. It is similar in shape to the normal distribution but has thicker tails, meaning it accommodates for more variability in data. The t-distribution becomes quite useful in creating confidence intervals when you can't assume a normal distribution due to smaller sample sizes.

For the problem of finding a confidence interval for the pH level of rainfall, the t-distribution was employed because only a sample— rather than the entire population— was analyzed. With a sample size of 40, the degrees of freedom, which are calculated as the sample size minus one ( -1 = 39), were used to find a critical t-value. This t-value helps in determining the margin of error in your confidence interval.

• Key feature: Greater variability
• Used with smaller sample sizes
• Important for accuracy in non-population standard deviation data sets

sample size

Sample size, typically denoted as 'n', is the number of observations in a sample. It is a critical factor when constructing a confidence interval, as it affects the degrees of freedom in the t-distribution and thus impacts the margin of error in the interval calculation.

In this scenario, a sample of 40 rainfall events was considered. With more data points (i.e., a larger sample size), the approximation of the sample mean to the true population mean becomes more accurate, and the margin of error reduces. However, because we aim for a 99% confidence level, the larger t-value makes the confidence interval broader to reflect the higher certainty.

• Sample size affects precision
• Higher sample sizes lead to smaller margins of error
• Sample size drives the degrees of freedom in t-distribution

pH scale

The pH scale is a measure of hydrogen ion concentration, representing the acidity or alkalinity of a solution. It ranges from 0 to 14, with 7 being neutral. Values less than 7 indicate acidity, whereas values greater than 7 indicate alkalinity.

For pure rain, which falls through clean air, the typical pH is around 5.7, slightly acidic due to naturally dissolved carbon dioxide forming carbonic acid. However, in areas affected by acid rain, pH values can drop significantly lower, as observed in the northeastern United States scenario. Lower pH values can result in more acidic rain, which damages the environment and corrodes infrastructure.

• pH levels: 0-14 scale, with 7 being neutral
• Lower values signal higher acidity
• Environmental implications, such as acid rain

normal distribution assumption

The assumption of normal distribution is crucial for creating accurate confidence intervals using the t-distribution. This assumption suggests that the dataset follows a bell curve shape where most of the observations cluster around the mean and fewer lie in the tails.

In the analysis of rainfall pH levels, this assumption underlines that the pH values collected from the 40 samples are normally distributed. This assumption is necessary because it ensures the validity of using the t-distribution to infer the population's mean pH. While small deviations from normality can be tolerated, significant skewness would require adjustments or different analyses.

• Normal distribution = bell curve
• Essential for proper statistical inference
• Validates t-distribution for confidence intervals