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Homser Lake, Oregon, has an Atlantic salmon catch and release program that has been very successful. The average fisherman's catch has been \(\mu=8.8\) Atlantic salmon per day (Source: National Symposium on Catch and Release Fishing, Humboldt State University). Suppose that a new quota system restricting the number of fishermen has been put into effect this season. A random sample of fishermen gave the following catches per day: $$\begin{array}{ccccccc} 12 & 6 & 11 & 12 & 5 & 0 & 2 \\ 7 & 8 & 7 & 6 & 3 & 12 & 12 \end{array}$$ i. Use a calculator with mean and sample standard deviation keys to verify that \(\bar{x}=7.36\) and \(s \approx 4.03.\) ii. Assuming the catch per day has an approximately normal distribution, use a \(5 \%\) level of significance to test the claim that the population average catch per day is now different from \(8.8 .\)

Step by step solution

01

Prepare the Data

First, list all given data points and verify the sample mean \(\bar{x}\) and sample standard deviation \(s\). The given sample data is: \[ 12, 6, 11, 12, 5, 0, 2, 7, 8, 7, 6, 3, 12, 12 \]

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Key Concepts

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

Understanding Sample Mean

The sample mean is a statistical measure that represents the average of a set of observations or data points. It is essential when you want to estimate the central tendency of a sample. To find the sample mean, you sum up all the data points in the sample and then divide by the number of data points.
For instance, if you have the catches per day from a sample of fishermen, you would add these up first. Using the given values \(12, 6, 11, 12, 5, 0, 2, 7, 8, 7, 6, 3, 12, 12\), the sum is \(103\). Since there are \(14\) data points in total, you divide \(103\) by \(14\) to get the sample mean \(\bar{x}\). Thus, the sample mean is approximately \(7.36\).
It's crucial because it provides a simple snapshot of the data set, but remember, it doesn't capture the variability or spread of the data.

Decoding Sample Standard Deviation

The sample standard deviation is a measure that tells us how much individual data points deviate from the sample mean. In simpler terms, it quantifies the amount of variation or dispersion in the sample.
To calculate the sample standard deviation, use the following steps:

• Find the mean of your sample.
• Subtract the mean from each data point and square the result.
• Sum all these squared differences.
• Divide this sum by one less than the number of observations (i.e., \(n-1\)).
• Take the square root of the result.

For our example, the data yields a sample standard deviation (\(s\)) of approximately \(4.03\). It means the individual catches can differ by around \(4.03\) salmon from the mean catch \(7.36\). Remember, a higher standard deviation indicates more variability in the sample.

Importance of Normal Distribution

In statistics, the normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence. This distribution is known for its bell-shaped curve where most values cluster around the mean.
In the context of hypothesis testing, assuming a normal distribution helps in predicting the probability of outcomes. If a sample is normally distributed, you can use it to make inferences about population parameters.
The assumption that the fisherman's daily catches are approximately normally distributed is crucial. It justifies using certain statistical tests that require this property, allowing for more valid and reliable conclusions.

Exploring Level of Significance

The level of significance in hypothesis testing is the threshold at which you decide whether to reject the null hypothesis. It is denoted by \( \alpha \) and commonly set at \(0.05\), which is a 5% level of risk you're willing to take when making the decision.
In the given exercise, testing at a 5% level of significance means you reject the null hypothesis if the probability of observing the test statistic under the null hypothesis is less than 5%.
This concept is vital because it helps decide whether an effect truly exists or if the observed data might have occurred by random chance. A lower level of significance indicates more stringent criteria for rejecting the null hypothesis, thus lowering the risk of incorrect conclusions.