According to a 2017 Gallup poll, 80 % of Americans report...

According to a 2017 Gallup poll, $80 \%$ of Americans report being afflicted by stress. Suppose a random sample of 1000 Americans is selected. a. What percentage of the sample would we expect to report being afflicted by stress? b. Verify that the conditions for the Central Limit Theorem are met. c. What is the standard error for this sample proportion? d. According to the Empirical Rule, there is a $95 \%$ probability that the sample proportion will fall between what two values?

According to a 2017 Gallup poll, $80 \%$ of Americans report being afflicted by stress. Suppose a random sample of 1000 Americans is selected.
a. What percentage of the sample would we expect to report being afflicted by stress?
b. Verify that the conditions for the Central Limit Theorem are met.
c. What is the standard error for this sample proportion?
d. According to the Empirical Rule, there is a $95 \%$ probability that the sample proportion will fall between what two values?

Key Concepts

Sample Proportion

The sample proportion is a statistic obtained by dividing the number of individuals in a sample with a certain attribute by the total sample size. It represents an estimate of the true population proportion and is a basic measure in statistics used to infer characteristics of a larger group from a sampled subset.

Central Limit Theorem (CLT) for Proportions

The Central Limit Theorem states that, for sufficiently large sample sizes, the sampling distribution of many estimators (such as the sample proportion) will be approximately normally distributed regardless of the population's distribution. In the context of proportions, this theorem relies on checking that both the number of successes (np) and the number of failures (n(1-p)) are at least 10 to ensure the approximation is valid.

Standard Error of the Proportion

The standard error of the proportion quantifies the variability of the sample proportion from sample to sample and is calculated using the formula sqrt[p(1-p)/n]. It is a crucial measure for understanding the precision of the sample proportion as an estimate of the population proportion.

Empirical Rule

The Empirical Rule provides approximate probabilities for observations within certain standard deviations from the mean in a normal distribution. Typically, it states that approximately 68% of data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations. This rule is used to construct confidence intervals and interpret the variability of sample estimates.

Transcript

00:01 Hello, so here in our first part, we consider that according to our 2017 gallup poll, 80 % of americans report being affected by stress, and a random sample of 1 ,000 americans is selected.

00:12 So we then use this information to find the percentage of the sample affected to report being affected by stress.

00:21 So we get that p here is 0 .80 and is equal to 1 ,000.

00:25 Therefore, we get that the percentage of the sample of 1 ,000 americans that is going to be affected by stress is going to be p is equal to 0 .80 and our sample size n is equal to 1 ,000.

00:40 So then for part b we then want to see if the central limit theorem has been met.

00:47 Well, we can go ahead here and substitute in p equals 0 .80 and equals 1 ,000 into n times p and n times 1 minus p.

01:00 So we get that n.

01:01 Times p here is going to be equal to 1 ,000 times 0 .80.

01:07 So that's going to be 800, which is certainly greater than 10.

01:12 And then if we do n times 1 minus p is going to be equal to 1 ,000 times 1 minus 0 .80.

01:22 That's going to be equal to 200, which again is greater than 10.

01:26 So the population of americans is then given by n, capital n, which is 10 times n, which is 10 times.

01:34 A thousand is 10 ,000.

01:36 So therefore, we get the conditions of the central limit theorem are met here because n p and n times 1 minus p are both greater than 10...

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