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Bass Fishing The pH factor is a measure of the acidity or alkalinity of water. A reading of 7.0 is neutral; values in excess of 7.0 indicate alkalinity; those below 7.0 imply acidity. Loren Hill states that the best chance of catching bass occurs when the \(\mathrm{pH}\) of the water is in the range 7.5 to \(7.9 .{ }^{18}\) Suppose you suspect that acid rain is lowering the \(\mathrm{pH}\) of your favorite fishing spot and you wish to determine whether the \(\mathrm{pH}\) is less than 7.5 . a. State the alternative and null hypotheses that you would choose for a statistical test. b. Does the alternative hypothesis in part a imply a one- or a two-tailed test? Explain. c. Suppose that a random sample of 30 water specimens gave \(\mathrm{pH}\) readings with \(\bar{x}=7.3\) and \(s=.2\). Just glancing at the data, do you think that the difference \(\bar{x}-7.5=-.2\) is large enough to indicate that the mean \(\mathrm{pH}\) of the water samples is less than \(7.5 ?\) (Do not conduct the test.) d. Now conduct a statistical test of the hypotheses in part a and state your conclusions. Test using \(\alpha=.05 .\) Compare your statistically based decision with your intuitive decision in part \(\mathrm{c} .\)

Step by step solution

01

State the null and alternative hypotheses

The null hypothesis (H鈧�) is that there is no difference in the pH value, and it is equal to 7.5. The alternative hypothesis (H鈧�) is that the pH of the water is less than 7.5. H鈧�: 碌 = 7.5 H鈧�: 碌 < 7.5

02

Determine if it's a one-tailed or two-tailed test

The alternative hypothesis in this situation implies that the pH is less than 7.5. Therefore, it is a one-tailed test, as we are looking for a specific direction (lower than 7.5) rather than a difference in either direction.

03

Examine given data

Without conducting a test, we can take a quick look at the given data from the random sample of 30 water specimens with a mean pH value of 7.3 and a standard deviation of 0.2. Taking note of the difference between the mean and 7.5 (-0.2), we can say that there might be a chance that the pH is lower than 7.5.

04

Conduct a statistical test

To conduct a statistical test, we will use a t-test, since the sample size is less than 30 and the population standard deviation is unknown. The t-value can be calculated as follows: t = (x - 碌) / (s / 鈭歯) Where x is the sample mean, 碌 is the population mean, s is the sample standard deviation, and n is the sample size. Substituting the values, we get: t = (7.3 - 7.5) / (0.2 / 鈭�30) = -0.2 / (0.2 / 鈭�30) = -2.1909 Now, we need to determine the critical value for this one-tailed test with a significance level (伪) of 0.05. Since we have 29 degrees of freedom (n-1), after consulting the t-table, we find that the critical value is -1.699 for a one-tailed test at 伪 = 0.05.

05

Compare the t-value with the critical value and draw a conclusion

Since the calculated t-value (-2.1909) is less than the critical value (-1.699), we can reject the null hypothesis. Therefore, there is sufficient evidence to claim that the pH of the water in the fishing spot is less than 7.5. Our statistically based decision matches our intuitive decision in part c, as both suggest that there is sufficient evidence to support the claim that the pH of the water in the fishing spot is less than 7.5.

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Key Concepts

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

pH measurement

Understanding pH measurement is important in various fields such as chemistry, biology, and environmental science. The pH scale ranges from 0 to 14, where a pH of 7 is considered neutral. Values less than 7 indicate acidity, while values greater than 7 indicate alkalinity. The measurement is crucial, for example, in determining the quality of water in aquatic environments.
For bass fishing enthusiasts, as noted in the original exercise, the optimal pH range is between 7.5 and 7.9. Outside this range, the acidity or alkalinity can affect fish behavior and health, including their likelihood to bite. Therefore, accurately measuring the pH is essential for maintaining a suitable fishing environment.
To measure pH, tools such as pH meters or pH paper are often used in field surveys. These tools help in gauging the hydrogen ion concentration of a sample, yielding the pH measurement. Consistently monitoring these values can help in understanding long-term changes or impacts, such as that caused by acid rain.

statistical test

Statistical tests are fundamental in data analysis and research. They help determine if there is enough evidence to support a specific hypothesis. In the context of the exercise, a statistical test is used to determine whether the average pH of a body of water is significantly different from an expected value (in this case, lower than 7.5).
These tests are based on probability theory and allow researchers to make inferences about a population based on sample data. Setting up a statistical test involves:

• Choosing a significance level, often denoted as \( \alpha \), e.g., 0.05 which represents a 5% chance of making a Type I error or rejecting a true null hypothesis.
• Selecting an appropriate test statistic (like a t-statistic for small sample sizes without a known population standard deviation).

Once the statistical test is conducted, researchers can compare the calculated test statistic to a theoretical distribution (e.g., using a t-table) to see if the null hypothesis can be rejected.

one-tailed test

A one-tailed test is a statistical hypothesis test where the probability of a distribution is tested against a single outcome. It examines whether a parameter is either greater than or less than a certain value, not both.
In the example from the original exercise, the alternative hypothesis suggests that the pH is less than 7.5. This directly points to a one-tailed test where the interest lies only in the values that are lower, not higher.
One-tailed tests can be powerful because they put all significance into one end of the distribution. This can make it easier to detect significant differences in the direction specified by the alternative hypothesis. However, it's crucial to predetermine which direction is relevant, as after-the-fact changes can lead to biased results.

t-test

The t-test is a popular statistical method used to determine if there are significant differences between two groups or if a sample mean differs significantly from a known population mean. It's particularly useful when dealing with small sample sizes (often n < 30) and when the population standard deviation is unknown.
In using a t-test, as seen in the exercise, the following steps are usually followed:

• Calculate the t-statistic using the formula: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \] Where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.
• Determine the degrees of freedom, typically \(n - 1\), to find the critical value from a t-distribution table.

The t-test helps determine if the sample mean is significantly different from the hypothesized population mean. In the exercise, the t-test showed that the mean pH was significantly lower than 7.5, leading to the rejection of the null hypothesis.