Answer :
To determine the probability that a sample proportion lies between 0.37 and 0.39, we'll follow these steps:
### Step-by-Step Solution:
1. Identify the Given Values:
- Mean proportion ([tex]\(\mu\)[/tex]): 0.38
- Standard deviation ([tex]\(\sigma\)[/tex]): 0.0485
- Lower proportion bound ([tex]\( p_{lower} \)[/tex]): 0.37
- Upper proportion bound ([tex]\( p_{upper} \)[/tex]): 0.39
2. Calculate the Z-Score for Lower Bound:
The Z-score formula is:
[tex]\[ z = \frac{(x - \mu)}{\sigma} \][/tex]
For [tex]\( p_{lower} = 0.37 \)[/tex]:
[tex]\[ z_{lower} = \frac{(0.37 - 0.38)}{0.0485} \approx -0.206 \][/tex]
3. Calculate the Z-Score for Upper Bound:
For [tex]\( p_{upper} = 0.39 \)[/tex]:
[tex]\[ z_{upper} = \frac{(0.39 - 0.38)}{0.0485} \approx 0.206 \][/tex]
4. Find the Probabilities Corresponding to the Z-Scores:
Using the standard normal table provided:
- For [tex]\( z_{lower} \approx -0.206 \)[/tex]: The probability is the corresponding value for [tex]\( z \approx -0.21 \)[/tex], which is roughly 0.6443.
- For [tex]\( z_{upper} \approx 0.206 \)[/tex]: The probability is the corresponding value for [tex]\( z \approx 0.21 \)[/tex], which is roughly 0.6517.
5. Calculate the Probability Between the Two Z-Scores:
The probability that a proportion is between these two bounds is the difference in the corresponding probabilities:
[tex]\[ P(0.37 < p < 0.39) = P(z_{upper}) - P(z_{lower}) \][/tex]
[tex]\[ P(0.37 < p < 0.39) = 0.6517 - 0.6443 = 0.0074 \][/tex]
6. Convert the Result to a Percentage:
[tex]\[ 0.0074 \times 100 = 0.74\% \][/tex]
Final Result:
The probability that a sample chosen at random has a proportion of registered voters who vote between 0.37 and 0.39 is approximately 0.74%.
### Step-by-Step Solution:
1. Identify the Given Values:
- Mean proportion ([tex]\(\mu\)[/tex]): 0.38
- Standard deviation ([tex]\(\sigma\)[/tex]): 0.0485
- Lower proportion bound ([tex]\( p_{lower} \)[/tex]): 0.37
- Upper proportion bound ([tex]\( p_{upper} \)[/tex]): 0.39
2. Calculate the Z-Score for Lower Bound:
The Z-score formula is:
[tex]\[ z = \frac{(x - \mu)}{\sigma} \][/tex]
For [tex]\( p_{lower} = 0.37 \)[/tex]:
[tex]\[ z_{lower} = \frac{(0.37 - 0.38)}{0.0485} \approx -0.206 \][/tex]
3. Calculate the Z-Score for Upper Bound:
For [tex]\( p_{upper} = 0.39 \)[/tex]:
[tex]\[ z_{upper} = \frac{(0.39 - 0.38)}{0.0485} \approx 0.206 \][/tex]
4. Find the Probabilities Corresponding to the Z-Scores:
Using the standard normal table provided:
- For [tex]\( z_{lower} \approx -0.206 \)[/tex]: The probability is the corresponding value for [tex]\( z \approx -0.21 \)[/tex], which is roughly 0.6443.
- For [tex]\( z_{upper} \approx 0.206 \)[/tex]: The probability is the corresponding value for [tex]\( z \approx 0.21 \)[/tex], which is roughly 0.6517.
5. Calculate the Probability Between the Two Z-Scores:
The probability that a proportion is between these two bounds is the difference in the corresponding probabilities:
[tex]\[ P(0.37 < p < 0.39) = P(z_{upper}) - P(z_{lower}) \][/tex]
[tex]\[ P(0.37 < p < 0.39) = 0.6517 - 0.6443 = 0.0074 \][/tex]
6. Convert the Result to a Percentage:
[tex]\[ 0.0074 \times 100 = 0.74\% \][/tex]
Final Result:
The probability that a sample chosen at random has a proportion of registered voters who vote between 0.37 and 0.39 is approximately 0.74%.
