91Ó°ÊÓ

About \(77 \%\) of young adults think they can achieve the American dream. Determine if the following statements are true or false, and explain your reasoning. \({ }^{13}\) (a) The distribution of sample proportions of young Americans who think they can achieve the American dream in samples of size 20 is left skewed. (b) The distribution of sample proportions of young Americans who think they can achieve the American dream in random samples of size 40 is approximately normal since \(n \geq 30\). (c) A random sample of 60 young Americans where \(85 \%\) think they can achieve the American dream would be considered unusual. (d) A random sample of 120 young Americans where \(85 \%\) think they can achieve the American dream would be considered unusual.

Step by step solution

01

Analyze Statement (a)

To determine if the distribution is left skewed, we need to consider the sample size \(n = 20\) and the central limit theorem (CLT). According to the CLT, the sampling distribution of sample proportions can be considered approximately normal if \(np \geq 10\) and \(n(1-p) \geq 10\). Here, \(p = 0.77\). So, \(np = 20 \times 0.77 = 15.4\) and \(n(1-p) = 20 \times 0.23 = 4.6\). Since \(n(1-p) < 10\), the distribution is not normal and might be skewed. Thus, statement (a) is potentially True.

02

Analyze Statement (b)

With a sample size of \(n = 40\), we calculate \(np = 40 \times 0.77 = 30.8\) and \(n(1-p) = 40 \times 0.23 = 9.2\). Although \(np \geq 10\), \(n(1-p) < 10\), so the condition for approximate normality is not fully satisfied. Thus, the distribution may not be approximately normal, and statement (b) is False.

03

Analyze Statement (c)

To see if 85% is unusual in a sample of size \(n = 60\), calculate the standard deviation of the sampling distribution: \(\sigma = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.77 \times 0.23}{60}} \approx 0.0557\). For an 85% proportion, we compute the \(z\)-score: \(z = \frac{0.85 - 0.77}{0.0557} \approx 1.44\). Since a \(|z|\) less than about 2 is not typically considered unusual, statement (c) is False.

04

Analyze Statement (d)

With \(n = 120\), calculate \(\sigma = \sqrt{\frac{0.77 \times 0.23}{120}} \approx 0.0411\). For an 85% proportion, compute \(z\): \(z = \frac{0.85 - 0.77}{0.0411} \approx 1.95\). Since a \(z\)-score of 1.95 is still within the usual range, statement (d) is False as well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sampling distribution

When discussing the concept of a sampling distribution, it's important to understand that it involves the distribution of a statistic—usually a sample proportion—over many random samples from the same population.
This concept helps us determine what the sample results might look like if we took many samples instead of just one. Specifically, for our exercise, we're looking at the distribution of sample proportions of young adults who believe they can achieve the American dream.

• These distributions are fundamental because they allow us to understand variability between samples.
• They are particularly useful in constructing confidence intervals and performing hypothesis testing.
• The Central Limit Theorem helps us when assuming these distributions are approximately normal under certain conditions.

This brings us to the crucial aspect: the conditions that allow us to assume normality in these distributions.

Sample proportions

Sample proportions refer to the ratio of the number of successes in a sample to the total number of observations in that sample.
In our case, a success would be a young adult who thinks they can achieve the American dream, and the sample proportion reflects the percentage of people in our sample who have this belief.

• Typically denoted as \(\hat{p}\), it is calculated as \(\hat{p} = \frac{x}{n}\), where \(x\) is the number of successes, and \(n\) is the sample size.
• Sample proportions give us valuable information about the population when it's impractical to survey everyone.
• As part of hypothesis testing, the distribution of these sample proportions helps in making inferences about the population proportion \(p\).

Understanding the distribution of these sample proportions is key to determining whether a particular sample result is usual or unusual.

Normal distribution

Normal distribution is a key concept in statistics, often illustrated as a bell-shaped curve. The Central Limit Theorem is a powerful statistical theory that tells us, for sufficiently large sample sizes, the sampling distribution of the sample mean will be approximately normally distributed regardless of the original population distribution.
In our exercise, we approximated the normality of the sample proportions.

• This approximation holds best under two main conditions: the sample size must be large enough so that \(np \geq 10\) and \(n(1-p) \geq 10\), where \(p\) is the population proportion.
• If these conditions aren't met, as seen in the case with samples of size 20 and 40, the distribution might be skewed instead of normal.
• Using the properties of the normal distribution, specifically the standard deviation (or standard error), helps in understanding the spread and the expected range of our sample proportions.

Thus, understanding these conditions is critical for correctly interpreting sample results and making accurate statistical inferences.

Unusual proportion

Determining whether a sample proportion is unusual involves assessing how far it deviates from the expected sample proportion, which in this context is 77%.
To determine if a proportion is unusual, statisticians often use the \(z\)-score, representing the number of standard deviations a data point is from the mean.

• Calculate the standard deviation of the sample proportion using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size.
• The \(z\)-score is calculated as \(z = \frac{\hat{p}-p}{\sigma}\), where \(\hat{p}\) is the sample proportion.
• A \(z\)-score typically less than 2 in absolute value suggests that the proportion is not unusual.

In our specific examples with sample sizes of 60 and 120, even a seemingly high sample proportion of 85% wasn't considered statistically unusual due to their \(z\)-scores being within typical limits, showing how these statistical tools are used to make data-driven decisions.