To find the probability that the sample proportion falls between 0.37 and 0.39, we'll go through the following steps:
### Step 1: Calculate the Z-scores for the Lower and Upper Bounds
First, we need to convert the sample proportions 0.37 and 0.39 into Z-scores using the formula:
[tex]\[ Z = \frac{(X - \mu)}{\sigma} \][/tex]
where \( X \) is the sample proportion, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
- For the lower bound (0.37):
[tex]\[ Z_{\text{lower}} = \frac{(0.37 - 0.38)}{0.0485} \approx -0.206 \][/tex]
- For the upper bound (0.39):
[tex]\[ Z_{\text{upper}} = \frac{(0.39 - 0.38)}{0.0485} \approx 0.206 \][/tex]
### Step 2: Find the Corresponding Probabilities from the Z-table
Next, we look up the probabilities for these Z-scores in the Z-table:
- The probability corresponding to \( Z_{\text{lower}} = -0.206 \approx 0.37 \) is approximately 0.6443.
- The probability corresponding to \( Z_{\text{upper}} = 0.206 \approx 0.39 \) is approximately 0.6517.
### Step 3: Calculate the Probability Between the Two Bounds
To find the probability that the sample proportion is between 0.37 and 0.39, we subtract the cumulative probability at the lower bound from the cumulative probability at the upper bound:
[tex]\[ P(0.37 < X < 0.39) = P(Z < 0.206) - P(Z < -0.206) \][/tex]
[tex]\[ = 0.6517 - 0.6443 \][/tex]
[tex]\[ = 0.0074 \][/tex]
### Result
The probability that the sample proportion of registered voters who vote is between 0.37 and 0.39 is approximately 0.74%.
Thus, the answer is:
[tex]\[ 0.74\% \][/tex]
