To determine the test statistic for this hypothesis test, we can follow these steps:
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State the null and alternative hypotheses:
- Null hypothesis (\(H_0\)): \(p \geq 0.33\) (33% or more of women are against the new working hours)
- Alternative hypothesis (\(H_1\)): \(p < 0.33\) (Less than 33% of women are against the new working hours)
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Determine the sample proportion (\( \hat{p} \)):
- Number of women against the new hours = 45
- Sample size \(n\) = 150
- Sample proportion \( \hat{p} = \frac{45}{150} = 0.30 \)
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Determine the hypothesized population proportion (\(p_0\)):
- \(p_0 = 0.33\)
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Calculate the test statistic using the formula for the z-test statistic: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]
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Substituting values into the formula: \[ z = \frac{0.30 - 0.33}{\sqrt{\frac{0.33(1 - 0.33)}{150}}} \] First, we calculate \(p_0(1 - p_0)\): \[ p_0(1 - p_0) = 0.33 \times 0.67 = 0.2211 \] Now calculate the standard deviation: \[ \sqrt{\frac{0.2211}{150}} \approx \sqrt{0.001474} \approx 0.0384 \] Now we can find the z-value: \[ z = \frac{0.30 - 0.33}{0.0384} = \frac{-0.03}{0.0384} \approx -0.78125 \]
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Final Result: The value of the test statistic is approximately \(-0.7814\).
Based on the choices provided, the answer is:
\(-0.7814\)