Confidence Interval Estimate for Population Proportion - CFA Level 1 Exam

Confidence Interval Estimate for Population Proportion

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Question

Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence, what is the interval estimate for the population proportion (to the nearest tenth of a percent)?

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A. B. C. D. E.

B

Interval estimate can be found from p +/- z[p(1-p)/n]^0.5. Here we have n = 2000, p = 1600/2000 = 0.8 and z = 1.96 (for 95%).

Therefore 0.8 +/- 1.96*0.008944 and we get 0.7825 and 0.8175.

To calculate the interval estimate for the population proportion, we can use the formula for the confidence interval for a proportion. The formula is:

Confidence Interval=Sample Proportion±Margin of Error\text{Confidence Interval} = \text{Sample Proportion} \pm \text{Margin of Error}

Given information:

  • Sample size (n) = 2,000 (the number of registered voters sampled)
  • Number of voters planning to vote for the Republican candidate (x) = 1,600

First, we need to calculate the sample proportion (p̂), which represents the proportion of voters planning to vote for the Republican candidate in the sample. We can calculate it by dividing the number of voters planning to vote for the Republican candidate (x) by the sample size (n):

Sample Proportion (p)=xn=1,6002,000=0.8\text{Sample Proportion (p̂)} = \frac{x}{n} = \frac{1,600}{2,000} = 0.8

Next, we need to calculate the margin of error (E), which represents the range around the sample proportion that we are confident the true population proportion lies within. The margin of error can be calculated using the formula:

Margin of Error (E)=Critical Value×Standard Error\text{Margin of Error (E)} = \text{Critical Value} \times \text{Standard Error}

The critical value is determined based on the desired confidence level. In this case, the confidence level is 0.95, which corresponds to a 95% confidence level. For a proportion, the critical value can be obtained from a standard normal distribution table or by using a statistical calculator. For a 95% confidence level, the critical value is approximately 1.96.

The standard error (SE) is a measure of the variability in the sample proportion and is calculated using the formula:

Standard Error (SE)=p^⋅(1−p^)n\text{Standard Error (SE)} = \sqrt{\frac{p̂ \cdot (1 - p̂)}{n}}

Substituting the values we have:

Standard Error (SE)=0.8⋅(1−0.8)2,000=0.0112\text{Standard Error (SE)} = \sqrt{\frac{0.8 \cdot (1 - 0.8)}{2,000}} = 0.0112

Now, we can calculate the margin of error:

Margin of Error (E)=1.96×0.0112=0.0219\text{Margin of Error (E)} = 1.96 \times 0.0112 = 0.0219

Finally, we can calculate the confidence interval by adding and subtracting the margin of error from the sample proportion:

Confidence Interval=0.8±0.0219\text{Confidence Interval} = 0.8 \pm 0.0219

Converting this to a percentage range, we get:

Confidence Interval=(0.8−0.0219)×100%\text{Confidence Interval} = (0.8 - 0.0219) \times 100\% to (0.8+0.0219)×100%(0.8 + 0.0219) \times 100\%

Simplifying:

Confidence Interval=77.81%\text{Confidence Interval} = 77.81\% to 82.19%82.19\%

Rounded to the nearest tenth of a percent, the interval estimate for the population proportion is:

A. 77.7% to 82.3%

Therefore, the correct answer is A. 77.7% to 82.3%.

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