To determine the probability that the sample proportion is less than 0.27, we can use the normal approximation of the binomial distribution.
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Define the known parameters:
- Population proportion (\( p \)): 0.34
- Sample size (\( n \)): 137
- Sample proportion to evaluate (\( \hat{p} \)): 0.27
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Calculate the mean and standard deviation of the sampling distribution:
- The mean of the sampling distribution \( \mu_{\hat{p}} \) is given by: \[ \mu_{\hat{p}} = p = 0.34 \]
- The standard deviation of the sampling distribution \( \sigma_{\hat{p}} \) is given by: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.34(1-0.34)}{137}} \]
First, calculate \( 1 - p \): \[ 1 - p = 1 - 0.34 = 0.66 \]
Now compute \( \sigma_{\hat{p}} \): \[ \sigma_{\hat{p}} = \sqrt{\frac{0.34 \times 0.66}{137}} = \sqrt{\frac{0.2244}{137}} \approx \sqrt{0.00164} \approx 0.0405 \]
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Calculate the z-score: The z-score for the sample proportion \( \hat{p} = 0.27 \) is calculated as: \[ z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.27 - 0.34}{0.0405} \] \[ z = \frac{-0.07}{0.0405} \approx -1.7284 \]
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Find the probability: We will look up the z-score in the standard normal distribution table (or use a calculator). The z-score of -1.7284 corresponds to a probability.
Using standard normal distribution tables or calculators gives us: \[ P(Z < -1.7284) \approx 0.04211 \]
So, the probability that the sample proportion is less than 0.27 is approximately:
\[ \text{Answer} = 0.0421 \]
Thus, your answer is:
\[ \boxed{0.0421} \]