This problem involves using a normal approximation to the binomial distribution because we are dealing with a large sample size of 1,000 voters.

Step 1: Define the variables

• The sample size $n = 1,000$
• The proportion of white male voters who voted for Obama $p = 0.40$
• We are looking for the probability that between 37% and 43% of the voters (i.e., between 0.37 and 0.43) voted for Obama.

Step 2: Calculate the standard error (SE)

The standard error of the proportion is calculated using the formula:

$SE = \sqrt{\frac{p(1-p)}{n}}$

Substitute $p = 0.40$, $1 - p = 0.60$, and $n = 1,000$:

$SE = \sqrt{\frac{0.40 \times 0.60}{1,000}} = \sqrt{\frac{0.24}{1,000}} = \sqrt{0.00024} \approx 0.0155$

Thus, the standard error is $SE \approx 0.016$ (rounded to 3 decimal places).

Step 3: Convert the percentages to z-scores

Now, we convert the given percentages (0.37 and 0.43) to z-scores using the formula:

$z = \frac{\hat{p} - p}{SE}$

• For $\hat{p} = 0.37$:

$z_1 = \frac{0.37 - 0.40}{0.016} = \frac{-0.03}{0.016} \approx -1.875$

• For $\hat{p} = 0.43$:

$z_2 = \frac{0.43 - 0.40}{0.016} = \frac{0.03}{0.016} \approx 1.875$

Step 4: Find the probabilities

Using the z-table (or a standard normal distribution calculator), we find the probabilities corresponding to these z-scores:

• For $z_1 = -1.875$, the cumulative probability is approximately 0.0304.
• For $z_2 = 1.875$, the cumulative probability is approximately 0.9696.

Step 5: Calculate the probability between 0.37 and 0.43

The probability that the proportion of voters is between 0.37 and 0.43 is:

$P(0.37 \leq \hat{p} \leq 0.43) = P(z_2) - P(z_1) = 0.9696 - 0.0304 = 0.9392$

Thus, the probability is approximately $0.9392$.

Final Answer:

The probability that between 37% and 43% of white male voters voted for Obama in a random sample of 1,000 voters is 0.9392.

Would you like further details or clarification?

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Here are 5 related questions to explore:

1. How does increasing the sample size affect the standard error in this case?
2. What if the percentage of white male voters for Obama was 45% instead of 40%?
3. How would the probability change if we were interested in a narrower range, say between 38% and 42%?
4. What are the limitations of using a normal approximation for a binomial distribution?
5. How would you calculate the probability if the sample size was smaller, say 100 instead of 1,000?

Tip: When dealing with large sample sizes, the binomial distribution can often be approximated using the normal distribution, simplifying the calculations.