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Chapter 8: Problem 14

Fishing: Northern Pike 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
No, there is not enough evidence to suggest that the catch proportion differs from 75%.

Step by step solution

01

Define the Hypotheses

To determine if the proportion of guests catching pike over 20 pounds is different from 75%, we set up the null and alternative hypotheses. The null hypothesis (\(H_0\)) is that the true proportion \(p\) is 75% (\(p = 0.75\)). The alternative hypothesis (\(H_a\)) is that \(p\) is different from 75%, so \(H_a: p eq 0.75\).
02

Determine the Test Statistic

We will use a z-test for a proportion. The test statistic \(z\) is calculated using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion under the null hypothesis, and \(n\) is the sample size. Here, \(\hat{p} = \frac{64}{83}\), \(p_0 = 0.75\), and \(n = 83\).
03

Calculate the Sample Proportion

Compute the sample proportion \(\hat{p}\): \[ \hat{p} = \frac{64}{83} \approx 0.7711 \]
04

Compute the Test Statistic

Using the values calculated: \[ z = \frac{0.7711 - 0.75}{\sqrt{\frac{0.75 \times 0.25}{83}}} \]\[ z \approx \frac{0.0211}{0.0484} \approx 0.4357 \]
05

Determine the Critical Value and Make Decision

For a two-tailed test with \(\alpha = 0.05\), the critical z-values are approximately \(\pm 1.96\). Since the calculated z-value (0.4357) is within the range (-1.96, 1.96), we fail to reject the null hypothesis.
06

Conclusion

Since we failed to reject the null hypothesis, we conclude that there is not enough evidence to suggest that the true proportion of guests catching northern pike over 20 pounds is different from 75%.

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Key Concepts

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

z-test for proportion
When conducting a hypothesis test about a population proportion, particularly to determine if it differs from a specific value, the z-test for proportion becomes an essential tool. The z-test evaluates if the observed sample proportion significantly deviates from the stated population proportion under the null hypothesis.

Here鈥檚 a quick rundown of how this test works:

  • Sample Proportion: This represents the proportion observed in your sample, often denoted as \( \hat{p} \).
  • Null Hypothesis Proportion: This is the proportion you specify in your null hypothesis, \( p_0 \).
  • Test Statistic: Calculated using the formula \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \). This formula helps gauge the number of standard deviations the sample proportion is from the null hypothesis proportion.

The z-test is advantageous because it considers the variance of the population proportion and standardizes the observed difference, allowing you to conclude whether any deviation is due to random chance or a real discrepancy.
population proportion
In statistics, the population proportion is a parameter that defines the fraction of the entire population that exhibits a particular attribute. For instance, if we say that 75% of the guests at Athabasca Fishing Lodge catch pike over 20 pounds, 75% is our population proportion.

To effectively understand population proportion:

  • Defined Group: This proportion applies to the entire set, like all guests in the Athabasca example.
  • Used in Hypothesis Testing: It helps in determining what we should expect under the null hypothesis.
  • Comparison with Sample Proportion: The population proportion often acts as a benchmark against which the sample proportion is compared.

By knowing the population proportion, researchers can decide if a certain outcome in a sample is typical or indicates a significant deviation, thus reflecting potential changes in underlying conditions.
null and alternative hypotheses
Before any hypothesis testing, clearly defining the null and alternative hypotheses is paramount. These hypotheses set the scene for testing and allow a structured approach to decision making regarding the data.

Here鈥檚 how they work:

  • Null Hypothesis \( (H_0) \): This is your default or starting assumption. It's usually a statement of no effect or a status quo. In the pike fishing example, the null claims that the proportion remains 75%.
  • Alternative Hypothesis \( (H_a) \): This counters the null. It indicates that there is a significant effect or change from the null assumption. Here, the alternative posits that the proportion differs from 75%, either higher or lower.
  • Establishing Hypotheses is Critical: Precisely defining these two hypotheses before conducting a test helps you maintain objectivity and rigor in your analysis.
  • Decision Making: Comparing your sample data outcome against these hypotheses lets us decide which is more likely to be true.

Adopting a rigorous approach to establishing these hypotheses lays the foundation for reliable and interpretable statistical testing.

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Most popular questions from this chapter

Consider a hypothesis test of difference of means for two independent populations \(x_{1}\) and \(x_{2}\) (a) What does the null hypothesis say about the relationship between the two population means? (b) If the sample test statistic has a \(z\) distribution, give the formula for \(z\) (c) If the sample test statistic has a \(t\) distribution, give the formula for \(t\).

If sample data is such that for a one-tailed test of \(\mu\) you can reject \(H_{0}\) at the \(1 \%\) level of significance, can you always reject \(H_{0}\) for a two-tailed test at the same level of significance? Explain.

What terminology do we use for the probability of rejecting the null hypothesis when it is, in fact, false?

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding \(z\) value. (c) Find the \(P\)-value of the test statistic. Sketch the sampling distribution and show the area corresponding to the \(P\)-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Focus Problem: Benford's Law Again, suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 7). You draw a random sample of \(n=228\) numbers from this file and \(r=92\) have a first nonzero digit of \(1 .\) Let \(p\) represent the population proportion of all numbers in the computer file that have a leading digit of \(1 .\) i. Test the claim that \(p\) is more than \(0.301 .\) Use \(\alpha=0.01.\) ii. If \(p\) is in fact larger than \(0.301,\) it would seem there are too many numbers in the file with leading Is. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. iii. Comment on the following statement: "If we reject the null hypothesis at level of significance \(\alpha,\) we have not proved \(H_{0}\) to be false. We can say that the probability is \(\alpha\) that we made a mistake in rejecting \(H_{0}\) " Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Myers-Briggs: Extroverts Are most student government leaders extroverts? According to Myers-Briggs estimates, about \(82 \%\) of college student government leaders are extroverts (Source: Myers -Briggs Type Indicator Atlas of Type Tables). Suppose that a Myers-Briggs personality preference test was given to a random sample of 73 student government leaders attending a large national leadership conference and that 56 were found to be extroverts. Does this indicate that the population proportion of extroverts among college student government leaders is different (either way) from \(82 \% ?\) Use \(\alpha=0.01.\)
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