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Chapter 8: Problem 14

Athabasca Fishing Lodge is located on Lake Athabasca in northern Canada. In one of its recent brochures, the lodge advertises that \(75 \%\) of its guests catch northern pike over 20 pounds. Suppose that last summer 64 out of a random sample of 83 guests did, in fact, catch northern pike weighing over 20 pounds. Does this indicate that the population proportion of guests who catch pike over 20 pounds is different from \(75 \%\) (either higher or lower)? Use \(\alpha=0.05\).

Short Answer

Expert verified
The sample does not indicate that the proportion differs from 75%.

Step by step solution

01

Hypotheses Formulation

To start, we need to formulate the null hypothesis \(H_0\) and alternative hypothesis \(H_a\). The null hypothesis should represent the claim of the brochure: \(H_0: p = 0.75\), where \(p\) is the population proportion of guests catching northern pike over 20 pounds. The alternative hypothesis will test if the proportion is different: \(H_a: p eq 0.75\).
02

Determine the Test Statistic

Next, we calculate the test statistic using the formula for a proportion: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \), where \(\hat{p}\) is the sample proportion. Here, \(\hat{p} = \frac{64}{83} \approx 0.7711\), \(p_0 = 0.75\), and \(n = 83\). Substitute these values into the formula to get the test statistic.
03

Calculate the Test Statistic

Substituting the values into the formula: \( z = \frac{0.7711 - 0.75}{\sqrt{\frac{0.75 \times 0.25}{83}}} \). First, calculate the denominator \(\sqrt{\frac{0.75 \times 0.25}{83}} \approx 0.0473\). The numerator is \(0.7711 - 0.75 = 0.0211\). So, \( z \approx \frac{0.0211}{0.0473} \approx 0.446\).
04

Determine the Critical Value

Since we are performing a two-tailed test (because \( H_a: p eq 0.75 \)), we need to find the critical z-values for \(\alpha = 0.05\). The critical values for a two-tailed test with \(\alpha = 0.05\) are approximately \(z = -1.96\) and \(z = 1.96\).
05

Decision Making

Compare the calculated test statistic \(z = 0.446\) with the critical values \(-1.96\) and \(1.96\). Since \(0.446\) falls within the range defined by the critical values \(-1.96 < 0.446 < 1.96\), we do not reject the null hypothesis.
06

Conclusion

Based on the decision in Step 5, there is not enough evidence to conclude that the proportion of guests catching northern pike over 20 pounds is different from 75%. Thus, the sample result is consistent with the lodge's claim in their brochure, within the significance level of 0.05.

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Key Concepts

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

Null Hypothesis
In the context of hypothesis testing, the null hypothesis is a statement that is assumed to be true until there is evidence to suggest otherwise. It serves as a starting point for statistical analysis and helps in making decisions based on sample data. In this exercise, the null hypothesis,
  • is represented by \( H_0: p = 0.75 \)
  • claims that 75% of guests catch pike over 20 pounds.
This value of 0.75 is derived from the lodge's brochure. When we test this hypothesis, we assume initially that it is true, only rejecting it if there is strong evidence that the population proportion is different. This forms the basis of statistical testing in practice.
Test Statistic
A test statistic is a standardized value that helps in determining whether to reject the null hypothesis. For comparisons involving population proportions, like in this exercise, we use the formula for the z-test statistic:
  • \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \)
Here,
  • \( \hat{p} \) is the sample proportion, calculated as \( \hat{p} = \frac{64}{83} \approx 0.7711 \).
  • \( p_0 \) is the population proportion under the null hypothesis, \(0.75\).
  • \( n \) is the sample size, which is 83 in this case.
Substituting these values, the test statistic becomes \( 0.446 \), which tells us how far the sample proportion is from 0.75, measured in standard deviation units. The test statistic helps determine whether this deviation is statistically significant.
Critical Value
Critical values are specific points on the distribution of the test statistic that play a key role in hypothesis testing. They define the threshold beyond which we will reject the null hypothesis. In a two-tailed test, like the one in this exercise, the critical values at a significance level of \( \alpha = 0.05 \) are typically found using a standard normal (z) distribution.
  • For \( \alpha = 0.05 \), the critical z-values are approximately \( -1.96 \) and \( 1.96 \).
This means that if the calculated test statistic falls beyond these critical values,
  • we would reject the null hypothesis.
  • However, in our exercise, the test statistic of \( 0.446 \) lies within the range \(-1.96 < 0.446 < 1.96\),
  • leading us to not reject the null hypothesis.
The critical values help safeguard against making decisions that are not supported by the data.
Population Proportion
The population proportion is a parameter that represents the fraction of individuals in a population that have a particular attribute. In the context of this problem, it relates to the proportion of guests who catch northern pike over 20 pounds. We often use sample data to estimate this proportion.
  • The hypothesized population proportion, based on the lodge's claim, is \( 0.75 \).
To test this claim, we calculate the sample proportion (\( \hat{p} \)) as 0.7711 from the guest data: 64 catches out of 83 total guests.
  • This number represents our best estimate of the true population proportion based on the sample.
  • We then use this observed sample proportion to perform hypothesis testing and see if it significantly departs from the hypothesized 0.75.
Population proportions are foundational in statistics, providing insights into the characteristics of large groups based on sample observations.

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Most popular questions from this chapter

If we fail to reject (i.e., "accept \({ }^{n}\) ) the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.

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