How To Construct A Confidence Interval For Population Proportion Using One-Sample Z-Interval

One Sample Z-Interval for Population Proportion INTERVAL

1. Random sample or assignment2. 10% Condition n ≤ 1/10(N)3. a) Population distribution is stated as NORMAL b) n ≥ 30 c) if n < 30, check graph of sample data to verify no strong skew or outliers

A confidence interval is a range of values that is likely to contain the true value of a population parameter. We can use the z-distribution to construct a confidence interval for a population proportion.

The formula for a one sample z-interval for population proportion is:

z = (p̂ – p) / √(p * (1 – p) / n)

Where:
p̂ is the sample proportion
p is the hypothesized proportion (if there is no hypothesized proportion, use p = 0.5 for the midpoint)
n is the sample size

To calculate the confidence interval, we then use the formula:

CI = p̂ ± z*√(p̂ * (1 – p̂) / n)

Where CI is the confidence interval and z is the critical value obtained from the standard normal distribution table, based on the desired level of confidence.

For example, suppose we want to estimate the proportion of people in a city who support a new policy. We randomly sample 300 people and find that 210 of them support the policy. We want to construct a 95% confidence interval for the true proportion of people in the city who support the policy.

First, we calculate the sample proportion:

p̂ = 210/300 = 0.7

Next, we determine the critical value from the standard normal distribution table for a 95% confidence level, which is 1.96.

Then, we calculate the standard error of the proportion:

SE(p̂) = √(p̂ * (1 – p̂) / n) = √(0.7 * 0.3 / 300) = 0.0346

Using the formula for the confidence interval, we have:

CI = 0.7 ± 1.96*0.0346
= (0.6336, 0.7664)

Therefore, we are 95% confident that the true proportion of people in the city who support the policy is between 63.36% and 76.64%.

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