Calculating a Confidence Interval for the Population Proportion | Step-by-Step Guide and Examples

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion PN=110, x=55, 88% confidence

To construct a confidence interval for the population proportion, we can use the formula:

Confidence Interval = sample proportion ± margin of error

Where the sample proportion is calculated by dividing the number of successes (x) by the total sample size (N) and the margin of error represents the range within which the true population proportion lies

To construct a confidence interval for the population proportion, we can use the formula:

Confidence Interval = sample proportion ± margin of error

Where the sample proportion is calculated by dividing the number of successes (x) by the total sample size (N) and the margin of error represents the range within which the true population proportion lies.

First, let’s calculate the sample proportion (p-hat):

p-hat = x/N
p-hat = 55/110
p-hat = 0.5

For a 88% confidence level, we need to calculate the margin of error by using the appropriate critical value from the standard normal distribution table.

To find the critical value, we subtract the confidence level from 100% and divide by 2 (since it is a two-tailed interval).

(100% – 88%) / 2 = 6%

To find the critical z-score for the 6% probability, we look up 0.03 in the standard normal distribution table, which corresponds to approximately 1.55.

Now, let’s calculate the margin of error:

Margin of Error = critical value * standard deviation
Margin of Error = 1.55 * √(p-hat * (1 – p-hat) / N)
Margin of Error = 1.55 * √(0.5 * (1 – 0.5) / 110)
Margin of Error ≈ 0.107

Finally, we can construct the confidence interval:

Confidence Interval = p-hat ± margin of error
Confidence Interval = 0.5 ± 0.107
Confidence Interval ≈ [0.393, 0.607]

Therefore, with 88% confidence, the true population proportion is estimated to be between 0.393 and 0.607.

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