A politician claims that she will receive at least 61% of the vote in an upcoming election. The results of a properly designed random sample of 100 voters showed that 54 of those sampled will vote for her. Is it likely that her assertion is correct at the 0.05 level of significance? (a) State the appropriate null and alternative hypotheses. H0: p

To evaluate the politician's claim, we can use a one-sample proportion z-test. Let's calculate step by step:

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(a) State the hypotheses:

• Null hypothesis ($H_0$): $p = 0.61$ (the politician's claim is correct).
• Alternative hypothesis ($H_a$): $p < 0.61$ (the politician's claim is incorrect, and she will receive less than 61% of the vote).

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(ii) Find the z-score:

The test statistic $z$ for a proportion is calculated using the formula:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Where:

• $\hat{p} = \frac{\text{Number of successes}}{\text{Sample size}} = \frac{54}{100} = 0.54$
• $p_0 = 0.61$ (the claimed population proportion)
• $n = 100$ (sample size)

Substitute the values:

$z = \frac{0.54 - 0.61}{\sqrt{\frac{0.61 (1 - 0.61)}{100}}}$

First, calculate the standard error ($SE$):

$SE = \sqrt{\frac{0.61 \times (1 - 0.61)}{100}} = \sqrt{\frac{0.61 \times 0.39}{100}} = \sqrt{0.002379} \approx 0.0488$

Now calculate $z$:

$z = \frac{0.54 - 0.61}{0.0488} = \frac{-0.07}{0.0488} \approx -1.43$

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(iii) Find the p-value:

The p-value corresponds to the probability of observing a z-score as extreme as the one calculated, assuming the null hypothesis is true. For a one-tailed test, the p-value is given by:

$p = P(Z < -1.43)$

Using standard z-tables or a calculator:

$p \approx 0.0764$

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Final Results:

• (ii) z = -1.43 (rounded to two decimal places).
• (iii) p-value = 0.0764 (rounded to four decimal places).

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Decision:

At the 0.05 significance level, the p-value ($0.0764$) is greater than $0.05$. Therefore, we fail to reject the null hypothesis. This suggests that there is insufficient evidence to claim that the politician will receive less than 61% of the vote, but the sample does not strongly support her claim either.

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Do you need further clarification or more details? Here are 5 follow-up questions to expand on this topic:

1. What does the p-value represent in the context of this hypothesis test?
2. How would the conclusion change if the significance level was 0.10 instead of 0.05?
3. What is the interpretation of a Type I error in this scenario?
4. How would the test change if the sample size was increased to 500 voters?
5. What role does the standard error play in hypothesis testing for proportions?

Tip: Always compare the p-value to the significance level to make a clear conclusion about the hypothesis test.