To determine whether the normal sampling distribution can be used, we need to check the conditions for normal approximation in a hypothesis test about a population proportion $p$. Specifically, we use the large sample conditions:

1. Condition 1: $n \cdot p_0 \geq 5$
2. Condition 2: $n \cdot (1 - p_0) \geq 5$

Where:

• $n$ is the sample size,
• $p_0$ is the hypothesized population proportion,
• $\hat{p}$ is the sample proportion.

Step 1: Check conditions

Given:

• $p_0 = 0.23$,
• $n = 100$,
• $\alpha = 0.05$.

Calculate:

1. $n \cdot p_0 = 100 \cdot 0.23 = 23$ (meets the condition: $\geq 5$).
2. $n \cdot (1 - p_0) = 100 \cdot (1 - 0.23) = 100 \cdot 0.77 = 77$ (meets the condition: $\geq 5$).

Since both conditions are met, the normal sampling distribution can be used.

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Step 2: Hypothesis test

We are testing the claim $p \neq 0.23$ at $\alpha = 0.05$.

Hypotheses:

• Null hypothesis ($H_0$): $p = 0.23$,
• Alternative hypothesis ($H_1$): $p \neq 0.23$ (two-tailed test).

Test statistic:

The test statistic for proportions is given by: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

Substitute the values: $\hat{p} = 0.17, \quad p_0 = 0.23, \quad n = 100$

= \frac{-0.06}{\sqrt{\frac{0.23 \cdot 0.77}{100}}} = \frac{-0.06}{\sqrt{0.1771 / 100}} = \frac{-0.06}{0.04207}$$ $$z \approx -1.43$$ --- #### Critical values and $$p$$-value: For a two-tailed test at $$\alpha = 0.05$$, the critical $$z$$-values are: $$z = \pm 1.96$$ The $$p$$-value is the area under the standard normal curve outside $$z = -1.43$$ and $$z = 1.43$$. Using a standard normal table or calculator: $$P(Z < -1.43) = 0.0764$$ Since it’s two-tailed: $$p\text{-value} = 2 \cdot 0.0764 = 0.1528$$ --- ### Step 3: Conclusion - The $$p$$-value $$0.1528 > 0.05$$, so we **fail to reject the null hypothesis**. - There is insufficient evidence to support the claim that $$p \neq 0.23$$. --- ### Summary - The normal sampling distribution can be used. - The hypothesis test results in failing to reject $$H_0$$ at $$\alpha = 0.05$$. - **Interpretation**: The sample data does not provide enough evidence to suggest that the population proportion $$p$$ is different from 0.23. --- Would you like a breakdown of calculations or a graph to visualize this result? ### Relative Questions: 1. Why are $$n \cdot p_0$$ and $$n \cdot (1 - p_0)$$ conditions necessary for using the normal approximation? 2. How is the critical value determined in a two-tailed test for proportions? 3. What factors can influence the $$p$$-value in this type of hypothesis test? 4. How does the sample size $$n$$ impact the test's reliability? 5. Could a smaller significance level $$\alpha$$ (e.g., 0.01) change the result of this test? ### Tip: Always double-check the two-tailed nature of the test when interpreting $$p$$-values and critical regions!