A researcher wants to estimate the proportion of property...

A researcher wants to estimate the proportion of property owners who would pay their property taxes one month early if given a $\$ 50$ reduction in their tax bill. Would the standard error of the sample proportion $\hat{p}$ be larger if the actual population proportion were $p=0.2$ or if it were $p=0.4$ ?

Key Concepts

Standard Error of a Proportion

This concept refers to the measure of variability in the sample proportion and is calculated using the formula ?(p(1 - p)/n), where p is the true population proportion and n is the sample size. It quantifies the expected difference between the sample estimate and the true population value, playing a crucial role in statistical inference.

Population Proportion and Variability

This concept involves understanding how different values of the true proportion, p, affect the variability of the sample proportion estimate. Since the term p(1 - p) reaches its maximum when p is 0.5, proportions nearer to this value yield higher variability, while proportions further away from 0.5 produce lower variability. Therefore, the standard error is larger for values of p that yield a greater p(1-p) product.

Transcript

00:01 Following the solution to number 18, and this looks at a study of home ownership, and there's a study saying, okay, how many people own their homes with your certain age.

00:09 We're going to use this formula.

00:10 You can derive this using the margin of error formula, or you can just use your book.

00:14 It's the same thing in the book.

00:16 But it's p hat times one minus p hat times the critical value divided by the margin of error quantity squared.

00:23 And we're given most of these things in the prompt, so it says that we want a margin of error to be two percentage points.

00:30 So the margin of error is going to be .02.

00:32 And then we want 90 degree confidence.

00:34 Now, if you don't have 90 degree confidence, a critical value memorized.

00:39 First off, you should have at least a few of these memorized.

00:42 And it is 1 .645.

00:44 So you can get that from a table or you can use a calculator.

00:46 I'm going to use a ti 84.

00:48 And if you go to second distribution and you go to inverse norm, and then you just change that area to 0 .9 for the 90%.

00:57 The mean is zero.

00:58 The standard deviation is 1.

01:00 And then for confidence intervals, it's always a center tail.

01:02 Now this is a newer calculator.

01:03 If you have an older calculator, you actually have to put in, it's a little bit different, but it's alpha over two.

01:09 So you have to put in .05 and change it to left or right.

01:12 And then we paste, and then this is going to give us the critical value.

01:15 Now this one gives you two critical values, but it's just the opposite value of it.

01:18 So it's 1 .645.

01:20 So that's where i get that...

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