The Cincinnati Enquirer reported that, in the United States,...

The Cincinnati Enquirer reported that, in the United States, 66$\%$ of adults and 87$\%$ of youths ages 12 to 17 use the Internet (The Cincinnati Enquirer, February $7,2006 ) .$ Use the reported numbers as the population proportions and assume that samples of 300 adults and 300 youths will be used to learn attitudes toward Internet security. a. Show the sampling distribution of $\overline{p},$ where $\overline{p}$ is the sample proportion of adults using the Internet. b. What is the probability that the sample proportion of adults using the Internet will be within $\pm .04$ of the pooulation proportion? c. Whatis the probabilit that the sample proportion of youths using the Internet will be within $\pm .04$ of the population proportion? d. Is the probability different in parts (b) and (c)? If so, why? e. Answer part (b) for a sample of size $600 .$ Is the probability smaller? Why?

The Cincinnati Enquirer reported that, in the United States, 66$\%$ of adults and 87$\%$ of youths ages 12 to 17 use the Internet (The Cincinnati Enquirer, February $7,2006 ) .$ Use the reported numbers as the population proportions and assume that samples of 300 adults and 300 youths will be used to learn attitudes toward Internet security.
a. Show the sampling distribution of $\overline{p},$ where $\overline{p}$ is the sample proportion of adults using the Internet.
b. What is the probability that the sample proportion of adults using the Internet will be within $\pm .04$ of the pooulation proportion?
c. Whatis the probabilit that the sample proportion of youths using the Internet will be within $\pm .04$ of the population proportion?
d. Is the probability different in parts (b) and (c)? If so, why?
e. Answer part (b) for a sample of size $600 .$ Is the probability smaller? Why?

Key Concepts

Effect of Sample Size on Sampling Distribution

The sample size plays a crucial role in determining the precision of a sample proportion estimate. As the sample size increases, the standard error decreases because it is inversely related to the square root of the sample size. This reduced variability means that larger samples tend to produce estimates that are closer to the true population parameter more consistently, resulting in a higher probability that the sample proportion falls within a defined margin of error.

Standard Error of a Proportion

The standard error of a proportion quantifies the variability or dispersion of the sample proportion around the true population proportion. It is calculated using the formula sqrt[p(1-p)/n], where p is the true proportion and n is the sample size. This measurement is critical because it influences how precisely the sample proportion estimates the population proportion, with a smaller standard error indicating more precise estimates.

Margin of Error and Confidence Intervals

Margin of error represents the range within which the true population parameter is expected to fall based on the sample estimate, typically expressed in relation to the standard error. It is used when forming confidence intervals around the sample proportion. By determining the probability that a sample proportion falls within a specified margin (e.g., ±0.04) of the true value, one can assess the reliability of the estimate and the level of uncertainty in the inference.

Sampling Distribution of a Proportion

In statistics, the sampling distribution of a proportion refers to the probability distribution of sample proportions drawn from a population with a known proportion. This distribution has a mean equal to the population proportion (p) and a variance of p(1-p)/n, which describes the variability in the estimates from different samples. This concept is fundamental in inferential statistics because it allows one to make probability-based statements about where sample proportions are likely to fall relative to the true population parameter.

Central Limit Theorem (CLT) and Normal Approximation

The Central Limit Theorem states that, given a sufficiently large sample size, the distribution of the sample mean (or in this case, the sample proportion) will tend to follow a normal distribution, regardless of the shape of the underlying population distribution. This theorem enables the use of normal probability calculations to estimate the likelihood that the sample proportion deviates from the population proportion by a specified margin, making the analysis more straightforward when sample sizes are large.

Transcript

00:01 All right, so this question gives us population proportions of adult and youth populations, and it asks us to compute some sampling distribution probabilities.

00:13 So the first part wants us to draw the distribution of p -hats for the adults.

00:23 So since we're dealing with a large sample, we can guarantee that it's normal based on our large counts condition.

00:32 So we can just draw a bell curve here with a mean at the population proportion and a sigma of square root .66.

00:57 .34 over 300 based on our formula.

01:08 So now for part b, it wants us to compete the probability that a sample of 300 adults is within .0 .0 .0 .0 .0 .0 .0.

01:20 So i want probability that we are within .04, which is the same thing as asking probability that our lower bound for p hat is .62, and our upper bound is .70.

01:57 0.

02:02 And this, of course, is a normal cdf question.

02:12 So we can write that.

02:16 Normal cdf.

02:19 Our lower bound is .62.

02:22 Our upper bound is .7.

02:25 Our mean is .66.

02:28 And our standard error is the square root of our population proportion times 1 minus the population proportion.

02:43 All over the square root of our sample size.

02:52 And if you put that into your calculator, that works out to be 0 .8 ,564.

03:09 Then it wants the same question, except for the youth.

03:15 So probability that we're within 0 .04, which in this case is the same as asking probability that we're between 0 .83 and 0 .91, which is another normal cdf.

03:52 Our lower bound is 0 .83.

03:55 Our upper bound is 0 .91.

03:58 Our mean is 0 .87...

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