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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. \({ }^{7}\) (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

Determine Skewness for Small Samples

To determine if the distribution of sample proportions is right skewed for a sample size of 12, we need to consider the shape of the binomial distribution. With a proportion \( p = 0.25 \), the skewness tends to lessen as the sample size increases. For small sample sizes like 12, the distribution can indeed be skewed, especially since \( np = 3 \) and \( n(1-p) = 9 \), not meeting the rule of thumb for normal approximation (which requires both \( np \geq 10 \) and \( n(1-p) \geq 10 \)). Thus, the distribution is likely right skewed.

02

Conditions for Normal Approximation

For the distribution of sample proportions to be approximately normal, we apply the normal approximation to the binomial distribution. This requires both \( np \geq 10 \) and \( n(1-p) \geq 10 \). Solving for \( n \), we get \( n \geq 40 \) in both cases, leading to a condition that \( n \) must be at least 40 for normal approximation.

03

Evaluate Unusual Samples for n = 50

For a sample size of 50 and a sample proportion of \( \hat{p} = 0.20 \), calculate the standard error: \( \text{SE} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.25(0.75)}{50}} \approx 0.0612 \). A result is considered unusual if \( \hat{p} \) is more than 2 SE away from \( p = 0.25 \). The difference \( 0.05 = 0.25 - 0.20 \) is less than 2 SE, so it is not unusual.

04

Evaluate Unusual Samples for n = 150

Repeat the same method as in Step 3 for \( n = 150\). The standard error is \( \text{SE} = \sqrt{\frac{0.25(0.75)}{150}} \approx 0.0354 \). The difference of 0.05 is less than 2 times the SE, therefore, for \( n = 150 \), this proportion is not unusual either.

05

Impact of Tripling Sample Size on Standard Error

Doubling or tripling the sample size affects the standard error, which is given by \( \text{SE} = \sqrt{\frac{p(1-p)}{n}} \). If the sample size \( n \) is tripled, the standard error becomes \( \text{SE} = \sqrt{\frac{p(1-p)}{3n}} \). This reduces the SE to \( \frac{1}{\sqrt{3}} \) of its original value, not by one third but by approximately 0.577, though closer to a 33% reduction of its current value, it won't be exactly one-third as stated.

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Key Concepts

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

Binomial Distribution

The concept of binomial distribution revolves around the idea of conducting trials to see if an event will happen (success) or not (failure). In the context of sample proportions, we use the binomial distribution to model the probability of a certain number of successes among a set number of trials. For instance, if we consider the percentage of young Americans who delay starting a family due to economic factors, this becomes our 'success'.
The binomial distribution is denoted generally as Binomial(n, p), where:

• n is the number of trials or observations.
• p is the probability of success on a single trial.

Using the example provided, if 25% (or p = 0.25) of young Americans are delaying family, and we sample size n, each sample of their reactions follows this distribution. When the sample size is small, the distribution might be skewed, making it unsuitable for approximations to normal law.

Normal Approximation

Normal approximation is used to simplify calculations of probabilities in distributions that are binomial in nature, especially when n is large. It's because computing exact probabilities from the binomial distribution directly can be complex. However, using the normal approximation allows us to analyze these probabilities in a more straightforward manner.
For the approximation to hold effectively, some conditions have to be met, primarily that both:

• np ≥ 10
• n(1-p) ≥ 10

Once these conditions are met, the binomial distribution of the sample proportions can be approximated to a normal distribution. This is valuable as it enables using the z-score to determine the likelihood of events, greatly enhancing computation and interpretation efficiency.

Standard Error Reduction

Standard error (SE) tells us how much the sample proportion is expected to differ from the true population proportion. It's an important measure when estimating data accuracy, calculated by the formula: \( \text{SE} = \sqrt{\frac{p(1-p)}{n}} \). Here, p is the probability of success, and n is the size of the sample.
Adjusting the sample size can influence the standard error significantly. For instance, tripling the sample size results in the SE reduced by a factor related to \( \sqrt{3} \), effectively bringing it to about 0.577 of its initial value. It’s a common misconception that this would reduce SE by exactly one-third, but understanding this subtlety is crucial for accurate analysis.

Sample Size Conditions

Understanding the prerequisites for sample size is pivotal when dealing with statistical approximations and analyses. To employ the normal approximation to interpret binomial distributions reliably, both np and n(1-p) need to be greater than or equal to 10.
These conditions ensure that the sample is sufficiently large so that we don't face significant skewness in the distribution of sample proportions. Larger sample sizes stabilize the binomial distribution and make approximations more accurate. They also affect the probability bounds for observing 'unusual' outcomes, as larger samples typically provide a tighter confidence interval for estimated proportions.