91Ó°ÊÓ

Is fishing better from a boat or from the shore? Pyramid Lake is located on the Paiute Indian Reservation in Nevada. Presidents, movie stars, and people who just want to catch fish go to Pyramid Lake for really large cutthroat trout. Let row \(B\) represent hours per fish caught fishing from the shore, and let row \(A\) represent hours per fish caught using a boat. The following data are paired by month from October through April (Source: Pyramid Lake Fisheries, Paiute Reservation, Nevada). $$ \begin{array}{l|ccccccc} \hline & \text { Oct. } & \text { Nov. } & \text { Dec. } & \text { Jan. } & \text { Feb. } & \text { March } & \text { April } \\ \hline \text { B: Shore } & 1.6 & 1.8 & 2.0 & 3.2 & 3.9 & 3.6 & 3.3 \\ \hline \text { A: Boat } & 1.5 & 1.4 & 1.6 & 2.2 & 3.3 & 3.0 & 3.8 \\ \hline \end{array} $$ Use a \(1 \%\) level of significance to test if there is a difference in the population mean hours per fish caught using a boat compared with fishing from the shore.

Step by step solution

01

Define Hypotheses

To test if there's a significant difference in the population mean hours caught per fish between fishing from the boat and from the shore, we set up our hypotheses. \(H_0: \mu_A = \mu_B\) (no difference in means), and \(H_1: \mu_A eq \mu_B\) (there is a difference).

02

Calculate Differences

Calculate the differences between each pair of data points (Boat time - Shore time) for October through April:- Oct: 1.5 - 1.6 = -0.1- Nov: 1.4 - 1.8 = -0.4- Dec: 1.6 - 2.0 = -0.4- Jan: 2.2 - 3.2 = -1.0- Feb: 3.3 - 3.9 = -0.6- March: 3.0 - 3.6 = -0.6- April: 3.8 - 3.3 = +0.5So respective differences are: \(-0.1, -0.4, -0.4, -1.0, -0.6, -0.6, +0.5\).

03

Compute Mean and Standard Deviation of Differences

Calculate the mean of differences, \( \bar{d} \), as \( \bar{d} = \frac{-0.1 - 0.4 - 0.4 - 1.0 - 0.6 - 0.6 + 0.5}{7} \).Calculate the standard deviation of differences using:\[ s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \] where \(n=7\).

04

Compute Test Statistic

The test statistic is given by: \[ t = \frac{\bar{d}}{s_d/\sqrt{n}} \]Substitute in the mean, standard deviation of differences, and \(n=7\) to compute \(t\).

05

Determine Critical Value and Decision

Find the critical value from the t-distribution table for \(n-1=6\) degrees of freedom at a 1% significance level (two-tailed). Compare the computed \(t\) statistic with the critical value to determine if \(H_0\) can be rejected.

06

Conclusion

If \(t\) statistic exceeds the critical \(t\) value from the table, reject \(H_0\); otherwise, do not reject \(H_0\). This concludes whether there is a significant difference in fishing times between shore and boat fishing at Pyramid Lake.

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Key Concepts

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

Paired t-test

The paired t-test is a statistical method used to compare two related groups. It is especially useful when you have measurements in pairs, such as the same subjects measured under two different conditions. In the context of the example, we are comparing the fishing times needed to catch a fish from a boat versus from the shore each month. These are considered paired because each condition is aligned to a specific month.

When using the paired t-test, one key step is to calculate the differences between the two sets of results for each pair. The focus is mainly on these differences rather than the raw data itself. This is a common problem setup in paired t-tests:

• Data is in pair form, with each pair having a measurement for each condition (e.g., boat vs. shore).
• The hypothesis test revolves around the differences of these pairs.
• The goal is to determine if the average difference is significantly greater or less than zero.

Understanding the paired t-test helps when the goal is to detect differences between two sets of closely related data, thus providing insights into which condition might be more effective or efficient.

Significance Level

The significance level in hypothesis testing is a crucial concept that determines the threshold for making statistical decisions. Typically represented as alpha (\( \alpha \)), it signifies the probability of rejecting the null hypothesis when it is actually true. This is also known as the Type I error rate.

In this exercise, the significance level is set at 1% (\( \alpha = 0.01 \)). This means there is only a 1% risk of incorrectly concluding that there is a difference in mean hours per fish caught between boat and shore under this hypothesis testing. Here are some points about significance levels:

• Lower significance levels (e.g., 1%) reduce the chance of a Type I error but might increase the chance of a Type II error (failing to reject the null hypothesis when it is false).
• Common significance levels are 0.01 (1%), 0.05 (5%), and 0.10 (10%).
• The choice of significance level affects how conservative your test is.

Setting the correct significance level is vital for making accurate statistical inferences and deciding whether differences in your data are statistically significant or just a result of random chance.

Population Mean Comparison

Population mean comparison is the main focus of this statistical exercise. It involves comparing the mean of two populations to ascertain if there is a significant difference between them.

In our case, we are interested in comparing the average hours it takes to catch a fish from a boat compared to from the shore—a classic example of comparing two population means. This involves calculating parameters such as:

• The mean difference in hours needed for fishing for each method.
• The variance and standard deviation of these differences.
• Finally, using the test statistic (like the t-statistic) to see how these compare statistically.

The null hypothesis is that these population means are equal, while the alternative hypothesis states they are not. Comparing population means is crucial for any field tasked with comparing two different conditions, treatments, or behaviors to see which may yield a better or different outcome.