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6.128 Are Florida Lakes Acidic or Alkaline? The \(\mathrm{pH}\) of a liquid is a measure of its acidity or alkalinity. Pure water has a pH of \(7,\) which is neutral. Solutions with a pH less than 7 are acidic while solutions with a \(\mathrm{pH}\) greater than 7 are basic or alkaline. The dataset ForidaLakes givesinformation, including pH values, for a sample of lakes in Florida. Computer output of descriptive statistics for the \(\mathrm{pH}\) variable is shown:(a) How many lakes are included in the dataset? What is the mean pH value? What is the standard deviation? (b) Use the descriptive statistics above to conduct a hypothesis test to determine whether there is evidence that average pH in Florida lakes is different from the neutral value of 7 . Show all details of the test and use a \(5 \%\) significance level. If there is evidence that it is not neutral, does the mean appear to be more acidic or more alkaline? (c) Compare the test statistic and p-value you found in part (b) to the computer output below for the same data: One-Sample T: pH Test of \(m u=7\) vs not \(=7\) Variable \(\begin{array}{rrrr}\mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ 53 & 6.591 & 1.288 & 0.177\end{array}\) \(\mathrm{pH}\) \(\begin{array}{crr}95 \% \mathrm{Cl} & \mathrm{T} & \mathrm{P} \\\ {[6.235,6.946)} & -2.31 & 0.025\end{array}\) 16

Step by step solution

01

Understand the given data

The given dataset represents the pH levels of some florida lakes. We know from the data that there are 53 lakes (N=53), the mean pH level is 6.591 or roughly 6.59, and the standard deviation is 1.288 or approximately 1.29.

02

Conduct a hypothesis test

We need to set up the null and alternative hypothesis for the test. The null hypothesis assumes the mean pH of Florida lakes, \( \mu \), is 7 (neutral). So, \( H_0 : \mu = 7 \). The alternative hypothesis assumes the mean pH is different from 7. So, \( H_a : \mu \neq 7 \). This is a two-tailed test. Since we know the mean and the standard deviation, we can calculate the test statistic (t-score) using the formula, \( t = (\bar{x} - \mu) / (\sigma / \sqrt N) \), where \( \bar{x} \) is the sample mean, \( \mu \) is the claimed population mean in the null hypothesis, \( \sigma \) is the standard deviation and N is the number of values in the sample. Inserting the given values into the formula, we can calculate \( t \). Next, we compare the calculated t-value with the table value to decide whether to reject or not to reject the null hypothesis.

03

Compare the test statistic and p-value

After calculating, we then compare the test statistic and p-value with the computer output. We look at the t-value and the P-value to draw up conclusions about the hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

pH Levels

Understanding pH levels is essential in assessing the acidity or alkalinity of a solution. The pH scale ranges from 0 to 14, where:

• A pH less than 7 indicates acidity.
• A pH of exactly 7 is neutral.
• A pH greater than 7 indicates alkalinity.

In this exercise, we're focusing on the pH of water in Florida lakes. Pure water at neutral pH 7 serves as a reference point. To determine if Florida's lakes are more acidic or basic, we use statistical analysis on pH data from a sample of these lakes. The average pH, standard deviation, and other statistical metrics provide insights into whether these lakes maintain a neutral pH or differ significantly from it.

Descriptive Statistics

Descriptive statistics provide a summary of the data, giving us key insights before performing more complex analyses. In this case, the dataset contains pH values for 53 lakes. Some important metrics include:

• N: The number of observations, which is 53.
• Mean pH: The average pH value, which is 6.591.
• Standard Deviation (StDev): A measure of the variability in pH levels, which is 1.288.

These statistics offer a crucial overview of the central tendency and dispersion of the data. By interpreting these values, we can gain an initial understanding of how much the pH levels vary across the sampled lakes.

Significance Level

The significance level in hypothesis testing is a threshold that determines whether a result is statistically meaningful. Commonly set at 0.05 (or 5%), the significance level represents the probability of rejecting the null hypothesis when it is actually true.
In this exercise, we use a 5% significance level to assess the pH levels of Florida lakes. A calculated p-value less than 0.05 indicates significant evidence against the null hypothesis, suggesting the mean pH differs from neutral. Conversely, a p-value above 0.05 implies insufficient evidence to discard the null hypothesis. Choosing an appropriate significance level is crucial as it affects the sensitivity and specificity of the hypothesis test.

T-test

The T-test is a statistical tool used to compare sample means to a known value or another sample's mean to determine if there's a significant difference. A T-test helps us decide whether the observed data deviate from the null hypothesis enough to conclude there's a real difference.

• Null Hypothesis (H_0): Assumes no real effect or difference exists, which, in this case, means the mean pH is 7.
• Alternative Hypothesis (H_a): Suggests there is a difference, meaning the mean pH is not equal to 7.
• T-score formula: \[ t = \frac{\bar{x} - \mu}{\sigma / \sqrt{N}} \]where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation, and \(N\) is the sample size.

The output of the T-test provides us with a t-value and a p-value, allowing us to assess the significance of our results. Comparing computed values against critical values helps determine whether to reject the null hypothesis in favor of the alternative.