Let's work through the given problem step by step.

Given information:

• Sample size $n = 700$
• Proportion of successes $\hat{p} = 0.40$ (40%)
• Confidence level $90\%$
• The sample is large enough that we can use normal approximation.

a) Number of successes

To find the number of successes, we multiply the sample size by the proportion of successes: $\text{Number of successes} = n \times \hat{p} = 700 \times 0.40 = 280$ So, the number of successes is 280.

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b) Find the standard error (SE) of the sample proportion

The formula for the standard error of the sample proportion is: $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Substituting the given values: $SE = \sqrt{\frac{0.40(1 - 0.40)}{700}} = \sqrt{\frac{0.40 \times 0.60}{700}} = \sqrt{\frac{0.24}{700}} = \sqrt{0.000342857} \approx 0.0185$

So, the standard error $SE$ is approximately 0.0185.

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c) Find the z-score for a 90% confidence level

For a 90% confidence level, the critical value is based on the normal distribution. Since the confidence level is 90%, we leave 5% in each tail (because $100\% - 90\% = 10\%$, and dividing this by 2 gives 5%).

Using the z-table or calculator for a 90% confidence level, the z-score corresponding to the middle 90% is: $z = 1.645$

So, the z-score is 1.645.

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d) Margin of Error (MoE)

The margin of error is calculated using the formula: $MoE = z \times SE$

Substituting the values we have: $MoE = 1.645 \times 0.0185 = 0.0304$

So, the margin of error is approximately 0.0304.

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Final answers:

• a) Number of successes = 280
• b) Standard error (SE) = 0.0185
• c) z-score = 1.645
• d) Margin of error = 0.0304

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Would you like any further details or explanations? Also, here are some related questions to explore:

1. How would the margin of error change if the sample size were smaller?
2. What would happen to the margin of error if the confidence level were increased to 95%?
3. How does the formula for margin of error change if the sample size increases?
4. What assumptions must be met for using the normal approximation in this problem?
5. How do you calculate the confidence interval for the population proportion using the margin of error?

Tip: For a larger sample size, the margin of error tends to decrease because the standard error becomes smaller, which makes the confidence interval narrower.