91Ó°ÊÓ

About \(25 \%\) of young Americans have delayed starting a family due to the continued economic slump. Determine if the following statements are true or false, and explain your reasoning. \(^{14}\) (a) The distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump in random samples of size 12 is right skewed. (b) In order for the distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump to be approximately normal, we need random samples where the sample size is at least 40 . (c) A random sample of 50 young Americans where \(20 \%\) have delayed starting a family due to the continued economic slump would be considered unusual. (d) A random sample of 150 young Americans where \(20 \%\) have delayed starting a family due to the continued economic slump would be considered unusual. (e) Tripling the sample size will reduce the standard error of the sample proportion by one-third.

Step by step solution

01

Evaluate Skewness of Distribution for Sample Size 12

The sample proportion is based on a Binomial distribution, and a Binomial distribution is skewed when the sample size is small. Here, we use the condition \( np < 5 \) or \( n(1-p) < 5 \) to determine skewness. With \( n = 12 \) and \( p = 0.25 \), we calculate \( np = 12 \times 0.25 = 3 \) and \( n(1-p) = 12 \times 0.75 = 9 \). Since \( np < 5 \), the distribution is right skewed. Thus, (a) is **True**.

02

Determine Normality for Sample Size

The sample proportions will be approximately normally distributed if both \( np \) and \( n(1-p) \) are at least 5. We need \( np \geq 5 \) for normal approximation. With \( p = 0.25 \), we find \( n \) such that \( n \times 0.25 \geq 5 \). Solving, we find \( n \geq 20 \). Hence, (b) is **False**. A sample size of at least 20 is needed, not 40.

03

Check if 50 Sample Size is Unusual

For a sample size of 50, the standard deviation of the sampling distribution is \( \sqrt{\frac{p \times (1-p)}{n}} = \sqrt{\frac{0.25 \times 0.75}{50}} \approx 0.0612 \). The sample proportion \( \hat{p} = 0.20 \) differs from the population proportion by \( |\hat{p} - p| = |0.20 - 0.25| = 0.05 \), which is less than one standard deviation. Therefore, this is not unusual. Hence, (c) is **False**.

04

Evaluate 150 Sample Size for Unusualness

For a sample size of 150, the standard deviation of the sampling distribution is \( \sqrt{\frac{0.25 \times 0.75}{150}} \approx 0.035 \). The deviation \( |0.20 - 0.25| = 0.05 \) is more than one standard deviation \((0.035)\), making it unusual. Hence, (d) is **True**.

05

Analyze Effect of Tripling Sample Size on Standard Error

The standard error is calculated as \( SE = \sqrt{\frac{p \times (1-p)}{n}} \). Tripling \( n \) results in updated \( SE = \sqrt{\frac{p \times (1-p)}{3n}} \), which becomes \( \frac{1}{\sqrt{3}} \) times the original SE. Thus, it reduces by factor \( \frac{1}{\sqrt{3}} \), not one-third. Hence, (e) is **False**.

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Key Concepts

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

Sample Proportion

The sample proportion (\( \hat{p} \)) is a key statistic that represents the fraction or percentage of individuals in a sample that possess a particular characteristic. For instance, in a group of young Americans, it could denote those who delayed starting a family due to economic factors. The sample proportion is calculated by dividing the number of individuals with the characteristic by the total sample size.Understanding sample proportion is important because it gives us a snapshot of the population. We use this statistic to gauge population behaviors or attitudes. However, because it is based on a sample and not the entire population, it naturally contains some level of uncertainty or "sampling error." This is where other statistical concepts, like the binomial distribution and normal distribution, come into play to help us understand and manage that uncertainty.

Binomial Distribution

The binomial distribution is a fundamental probability distribution in statistics. It describes the number of successes in a fixed number of trials, where each trial has two possible outcomes: success or failure. For this exercise, considering if an individual has delayed starting a family is a binary scenario.When calculating or considering distributions for small samples, like 12 in the given scenario, the binomial distribution can often be skewed:

• If either \( np < 5 \) or \( n(1-p) < 5 \), the distribution is considered skewed.
• A right skew happens when there are fewer successes than expected success probability, resulting in a longer tail toward higher counts.

Understanding how the binomial distribution behaves for small samples helps us grasp why certain statistical approximations are needed for larger sample sizes.

Standard Error

The standard error (SE) measures the variability or dispersion of the sampling distribution of a statistic, often the mean or proportion. It provides insight into how much the sample proportion (\( \hat{p} \) ) might vary from the actual population proportion (\( p \) ).The formula for standard error of the sample proportion is:\[ SE = \sqrt{\frac{p \cdot (1-p)}{n}} \] where:

• \( p \) is the true population proportion
• \( n \) is the sample size

As seen in this exercise, increasing the sample size results in a reduced SE, indicating more precise estimates of the population parameter. Specifically, tripling the sample size doesn’t „triple" the precision; rather, it multiplies the SE by about \( \frac{1}{\sqrt{3}} \), reducing variability.

Normal Distribution

The normal distribution, often referred to as the bell curve, is critical in statistics. When certain conditions are met, sample proportions can be approximated as normal distributions. This approximation is valid under the "Central Limit Theorem," but for it to apply to sample proportions:

• Both \( np \) and \( n(1-p) \) must be at least 5.
• It ensures the sample proportion’s distribution is symmetrical, allowing the use of normal distribution properties.

In the scenario from the exercise, determining how "unusual" a sample proportion is often hinges on this normal approximation. Larger samples tend to provide better approximations, making statistical inferences and predictions feasible. Understanding this concept helps students recognize the power and limitations of using normal approximations in statistical analyses.