91Ó°ÊÓ

Young Americans, Part I. 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. \(^{27}\) (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

Analyzing skewness for small sample size

For statement (a), we need to consider the sample size of 20. The distribution of sample proportions is approximately normal if both \(np\) and \(n(1-p)\) are greater than 5, where \(p\) is the true proportion (0.77 in this case). Here, \(np = 20 \times 0.77 = 15.4\) and \(n(1-p) = 20 \times 0.23 = 4.6\). Since \(n(1-p) < 5\), the distribution is not normal and more right-skewed rather than left-skewed. Therefore, the statement is **false**.

02

Assessing normality for slightly larger sample size

For statement (b), we use a sample size of 40. With \(p = 0.77\), \(np = 40 \times 0.77 = 30.8\) and \(n(1-p) = 40 \times 0.23 = 9.2\). Since both values are greater than 5, the distribution can be considered approximately normal according to the Central Limit Theorem (CLT), but the statement justifies normality due to the sample size being greater than 30 alone, which isn't solely sufficient. Hence, the statement is **false** in its justification.

03

Checking unusual event for sample size 60

For statement (c), using a sample size of 60 with 85% thinking they can achieve the American dream, we calculate the standard deviation for the sampling distribution: \( \sigma = \sqrt{ \frac{0.77 \times 0.23}{60} } \approx 0.057\). An event is typically unusual if it is beyond two standard deviations from the mean. The sample proportion is 0.85, and the mean is 0.77. The z-score is \( \frac{0.85 - 0.77}{0.057} \approx 1.40\). Since 1.40 is less than 2, the event is not unusual. Therefore, the statement is **false**.

04

Checking unusual event for larger sample size 120

For statement (d), using a sample size of 120 and 85% thinking they can achieve the American dream, the standard deviation is \( \sigma = \sqrt{ \frac{0.77 \times 0.23}{120} } \approx 0.040\). The z-score for this scenario is \( \frac{0.85 - 0.77}{0.040} \approx 2.00\). Since the z-score is exactly 2, it is unusual by typical standards, but it symbolizes the threshold of unusualness. Thus, this statement just touches upon being **true** by convention.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental concept in statistics that helps us understand the behavior of sample means. When we take samples from a population and calculate their means, the CLT states that these sample means will form a normal distribution as long as the sample size is large enough.
This applies even if the original population distribution is not normal. By "large enough," statisticians generally mean that both the product of the sample size (\(n\)) and the probability of success (\(p\)), and the product of the sample size and the probability of failure (\(1-p\)), should each be greater than 5.

Key points to remember:

• If \(np > 5\) and \(n(1-p) > 5\), the sampling distribution of the sample mean is approximately normal.
• This concept supports the answers in our exercise, ensuring the approximation of normality given certain conditions.
• Sample size is crucial in determining whether or not the sample proportions form a normal curve, as seen in statements (a) and (b).

Sampling Distribution

Sampling distribution refers to the probability distribution of a statistic, like the sample proportion, calculated from a random sample.
When we take multiple samples from a population, each sample produces a sample statistic—usually the sample mean or proportion. The distribution of these statistics is a sampling distribution.
For example, in our exercise, statements involve comparing sample proportions, where we look at the probability distribution of proportions from samples.

Understanding sampling distribution includes:

• The mean of the sampling distribution of the sample proportion, denoted as \(\hat{p}\), is equal to the true population proportion \(p\).
• The standard deviation (or standard error) of a sample proportion can be calculated as \( \sigma_{\hat{p}} = \sqrt{ \frac{p(1-p)}{n} } \).
• The use of sample distribution principles is evident in determining the z-scores for statements (c) and (d) of the exercise.

Z-score

A Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values.
When calculating a z-score, you are determining how many standard deviations away from the mean a particular value is. A higher absolute z-score indicates that the value is far from the mean. This helps identify outliers or unusual data points.
Referring to the exercise, the z-score was used to determine whether the scenarios described were unusual.

Crucial aspects of z-scores include:

• The formula for the z-score is \( Z = \frac{(x - \mu)}{\sigma} \), where \(x\) is the sample proportion, \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the sampling distribution.
• In this context, z-scores help establish whether certain proportions of young Americans believing in the American dream are unusual.
• Values beyond 2 or -2 are typically considered unusual, which is crucial in interpreting statements (c) and (d).

Sample Proportion

The sample proportion is a statistic used to estimate the proportion of a certain trait or characteristic in a population.
It is calculated as the number of favorable outcomes divided by the total number of observations in the sample. For instance, if we survey a group of people and find out how many believe in the American dream, that ratio becomes the sample proportion.
In each exercise statement, the sample proportion gives insight into what percentage of young Americans see attaining the dream as a reality.

Essentials about sample proportion include:

• Denoted as \(\hat{p}\), it is simply \( \hat{p} = \frac{x}{n} \) where \(x\) is the number of favorable outcomes, and \(n\) is the total number of trials or sample size.
• Sample proportion is used alongside the CLT to decide whether a proportion follows a normal distribution as with statements (a) and (b).
• It is vital for calculating the standard deviation of the sample's distribution, especially when determining z-scores for unusualness (as in statements (c) and (d)).