Construct a confidence interval of the population proportion at the given level of confidence. x

Construct a confidence interval of the population proportion...

Key Concepts

Population Proportion

The population proportion is the true fraction of individuals in the entire population possessing a particular attribute. It is the parameter of interest when estimating how common a characteristic is within the population, and is typically unknown, which is why we use sample statistics to infer its value.

Sample Proportion

The sample proportion is the ratio calculated from the observed data, defined as the number of successes divided by the total sample size. This statistic serves as an estimate of the population proportion and is central to constructing confidence intervals for the population parameter.

Confidence Interval

A confidence interval provides a range of plausible values for the population parameter based on the sample data. It is constructed so that, with a specified level of confidence, the interval is expected to capture the true population parameter if the sampling process were repeated many times.

Standard Error

The standard error measures the variability of the sample proportion from sample to sample. It is calculated using the formula ?[p?(1 ? p?)/n] and quantifies the precision of the sample estimate with respect to the true population proportion.

Critical Value (z-score)

The critical value is a point from the standard normal distribution corresponding to the desired level of confidence. It determines how far, in standard error units, the interval extends from the sample proportion. For a 90% confidence level, the critical value is typically obtained from the z-distribution.

Central Limit Theorem

The Central Limit Theorem ensures that, for large enough sample sizes, the sampling distribution of the sample proportion will be approximately normally distributed, regardless of the population distribution. This justifies using the normal (z) approximation for constructing confidence intervals around the sample proportion.

Transcript

00:01 To find a 90 % confidence interval for a population proportion p, you want to recall that the form of the confidence interval is p hat the point estimate of the population proportion minus e, the margin of error for our lower bound, to p hat the point estimate of the population proportion plus e, the margin of error as our upper bound.

00:28 Now recall p hat, we find.

00:30 By taking x, the number in the sample that have the specific characteristic divided by n, the sample size.

00:40 And e, the margin of error for a confidence interval for a population proportion, we find by taking the critical value z sub alpha over 2 for the specific confidence interval level times the square root of p hat times one minus p hat, divided by n.

01:04 So here our p hat is equal to, well, x is the number in the sample that had that specific characteristics.

01:12 So that's 30 for our x.

01:15 And then divided by our n is our sample size 150.

01:26 So our p hat comes out to be 0 .20.

01:32 Now for e, the critical value z sub alpha over 2 for a 90 % confidence level is z sub alpha over 2 is equal to 1 .645.

01:56 And recall we got that by in our normal distribution looking at equally spaced over the middle, that's where your 0 .90 confidence level goes.

02:12 Subtracting that from 1, the total area gives us how much is left in the two tails combined.

02:22 That's your alpha, is 0 .10 for this example.

02:26 So alpha divided by 2 is dividing that 0 .10 by 2 and getting 0 .05.

02:35 Is the amount of area left in each of the tails.

02:40 So it's the area in there...

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