People end up tossing 12% of what they buy at the grocery...

People end up tossing 12$\%$ of what they buy at the grocery store (Reader's Digest, March, 2009 . Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior. a. Show the sampling distribution of $\overline{p},$ the proportion of grocenies thrown out by your sample respondents. b. What is the probability that your survey will provide a sample proportion within $\pm .03$ of the population proportion? c. What is the probability that your survey will provide a sample proportion within $\pm .015$ of the population proportion?

People end up tossing 12$\%$ of what they buy at the grocery store (Reader's Digest, March, 2009 . Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.
a. Show the sampling distribution of $\overline{p},$ the proportion of grocenies thrown out by your
sample respondents.
b. What is the probability that your survey will provide a sample proportion within $\pm .03$ of the population proportion?
c. What is the probability that your survey will provide a sample proportion within $\pm .015$ of the population proportion?

Key Concepts

Sampling Distribution of a Proportion

This concept describes the probability distribution of the sample proportion (p-hat) when repeatedly drawing random samples from a population. Its mean is the true population proportion (p) and its spread is determined by the variability inherent in the sampling process. Understanding this distribution is fundamental in estimating population parameters and their uncertainty in survey sampling.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the original population distribution, the distribution of the sample proportion (or mean) will approximate a normal distribution for sufficiently large sample sizes. This is particularly useful when applying normal probability methods to compute probabilities involving the sample proportion.

Standard Error

The standard error is a measure of the dispersion or variability of the sampling distribution and is computed for proportions as the square root of (p(1-p)/n). It quantifies the expected deviation of the sample proportion from the true population proportion and is a key component in constructing confidence intervals and hypothesis tests.

Z-Score Transformation

The Z-score transformation converts a sample proportion into a standardized form by subtracting the population proportion (the mean) and dividing by the standard error. This allows for the use of the standard normal distribution to calculate probabilities and assess how far the sample statistic is from the expected value under the null hypothesis.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate, within a certain confidence level. It is derived using the standard error and the appropriate critical value from the normal (or t-) distribution, and it helps in understanding the precision of the estimate drawn from the sample.

Transcript

00:01 All right, we're giving the statistic that 12 % of all groceries bought are eventually thrown out.

00:05 And we're assuming that's the true population proportion.

00:08 And we're also given a sample size and equals 540.

00:12 So for part a, we want to sketch the sampling distribution.

00:17 For that, we're going to need our mean of p, of our sample p and our standard deviation of p.

00:27 So our mean is just going to equal p.

00:31 So that's 0 .12.

00:32 And our standard deviation, we have a formula that's p times 1 minus p all over n.

00:41 So that's going to be 0 .12 times 0 .88 all over 540.

00:51 Take the square root of that.

00:53 This equals 0 .0138.

00:57 So we sketch out our normal distribution, look something like that.

01:02 I'm going to make that look slightly more symmetrical.

01:06 There we go.

01:09 Hold on.

01:11 Oops.

01:12 I don't know how that one happened.

01:16 One of the world? okay, it's not letting me erase, so i'm just going to shift the bell curve so it's closer to the center then.

01:26 Anyway, this is 0 .12, our mean.

01:31 We're going to go out about one standard deviation on either side, so i'm just going to denote that in red.

01:43 So right here will be 0 .1068, and over here will be 0 .13398.

01:55 Wait, not 8, this should be a 2.

01:58 Hold on, wait.

02:02 Okay, yeah, it really won't let me erase.

02:04 So there, now it's a dot.

02:09 Anyway, so problem b, or part b, we want to find the probability that the sample of proportion we find is within 0 .03 of our population mean, or sorry, the sample mean we find.

02:27 I think i just said the population mean refined.

02:30 Anyway, so this is a z score problem.

02:32 We're going to find our z lower and z upper.

02:36 So our z lower is going to be negative 0 .03 divided by our standard deviation of our population distribution.

02:43 So that's going to be zero point, or sorry, standard deviation of the sampling distribution...

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