8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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Multiple Choice

A factory manager wants to estimate the proportion of defective items produced. In a batch of 20 items, the factory has produced 6 with defects. Find the margin of error for a 98% confidence interval for the true proportion of defective items.

$E=0.3$

$E=0.238$

$E=0.169$

$E=0.062$

Verified step by step guidance

First, calculate the sample proportion (p̂) of defective items. This is done by dividing the number of defective items by the total number of items in the sample. In this case, p̂ = 6/20.

Next, determine the z-score associated with a 98% confidence level. You can find this value in a standard normal distribution table or use a calculator. For a 98% confidence interval, the z-score is approximately 2.33.

Now, calculate the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n), where n is the sample size.

Use the z-score and the standard error to calculate the margin of error (E) for the confidence interval. The formula is: E = z * SE.

Finally, interpret the margin of error in the context of the problem. The margin of error provides a range within which the true proportion of defective items is likely to fall, given the sample data and confidence level.

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