Questions: Some experts believe that 24% of all freshwater fish in a country have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 250 fish tested, and 54 of them have dangerous levels of mercury. Test the hypothesis that this sample is not from a population with 24% dangerous fish, assuming that this is a random sample. Use a significance level of 0.05. Comment on your conclusion. State the null and alternative hypotheses. A. H0: P<0.24 Ha: P>0.24 B. H0: p=0.24 Ha: p>0.24 C. H0: p=0.24 Ha p ≠ 0.24 D. H0: p>0.24 Ha P<0.24 E. H0: p ≠ 0.24 H3: P=0.24

Some experts believe that 24% of all freshwater fish in a country have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 250 fish tested, and 54 of them have dangerous levels of mercury. Test the hypothesis that this sample is not from a population with 24% dangerous fish, assuming that this is a random sample. Use a significance level of 0.05. Comment on your conclusion. State the null and alternative hypotheses. A. H0: P<0.24 Ha: P>0.24 B. H0: p=0.24 Ha: p>0.24 C. H0: p=0.24 Ha p ≠ 0.24 D. H0: p>0.24 Ha P<0.24 E. H0: p ≠ 0.24 H3: P=0.24
Transcript text: Some experts believe that $24 \%$ of all freshwater fish in a country have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 250 fish tested, and 54 of them have dangerous levels of mercury. Test the hypothesis that this sample is not from a population with $24 \%$ dangerous fish, assuming that this is a random sample. Use a significance level of $0.05$. Comment on your conclusion. State the null and alternative hypotheses. A. $\mathrm{H}_{0} \cdot \mathrm{P}<0.24$ $\mathrm{H}_{\mathrm{a}}: \mathrm{P}>0.24$ B. $\mathrm{H}_{0}: \mathrm{p}=0.24$ $H_{\mathrm{a}}: p>0.24$ C. $H_{0} \cdot p=0.24$ $\mathrm{H}_{\mathrm{a}} \mathrm{p} \neq 0.24$ D. $H_{0}: p>0.24$ $\mathrm{H}_{\mathrm{a}} \mathbf{P}<0.24$ E. $H_{0}: p \neq 0.24$ $H_{3}: P=0.24$
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Solution

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Solution Steps

Step 1: State the Hypotheses

We are testing the following hypotheses:

  • Null Hypothesis: \( H_0: p = 0.24 \)
  • Alternative Hypothesis: \( H_a: p \neq 0.24 \)
Step 2: Calculate the Sample Proportion

The sample proportion of fish with dangerous levels of mercury is calculated as: \[ \hat{p} = \frac{54}{250} = 0.2160 \]

Step 3: Calculate the Test Statistic

The test statistic \( Z \) is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values: \[ Z = \frac{0.2160 - 0.24}{\sqrt{\frac{0.24(1 - 0.24)}{250}}} = -0.8885 \]

Step 4: Calculate the P-Value

The P-value associated with the test statistic \( Z = -0.8885 \) is: \[ \text{P-value} = 0.3743 \]

Step 5: Determine the Critical Region

For a significance level of \( \alpha = 0.05 \) in a two-tailed test, the critical region is defined as: \[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Step 6: Make a Decision

Since the calculated P-value \( 0.3743 \) is greater than \( \alpha = 0.05 \), we fail to reject the null hypothesis.

Final Answer

There is not enough evidence to say the sample is not from a population with \( 24\% \) dangerous fish. Thus, the answer is: \[ \boxed{H_0 \text{ is not rejected}} \]

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