Entrepreneurship A Gallup survey indicated that 72% of 18- to 29-year-olds, if given a choice, would prefer to start their own business rather than work for someone else. A random sample of 600 18- to 29-year-olds is obtained today.
c. In a random sample of 600 18- to 29-year-olds, what is the probability that no more than 70% would prefer to start their own business?

Verified step by step guidance

1. Identify the population proportion \( p = 0.72 \) which represents the probability that a randomly selected 18- to 29-year-old prefers to start their own business.
2. Define the sample size \( n = 600 \) and the sample proportion of interest \( \hat{p} = 0.70 \) (no more than 70%).
3. Since the sample size is large, use the normal approximation to the binomial distribution. Calculate the mean and standard deviation of the sampling distribution of \( \hat{p} \): \[\text{mean} = p = 0.72\] \[\text{standard deviation} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.72 \times 0.28}{600}}\]
4. Convert the sample proportion \( \hat{p} = 0.70 \) to a z-score using the formula: \[z = \frac{\hat{p} - p}{\text{standard deviation}} = \frac{0.70 - 0.72}{\sqrt{\frac{0.72 \times 0.28}{600}}}\]
5. Use the standard normal distribution table or technology to find the probability corresponding to this z-score, which gives the probability that no more than 70% of the sample prefer to start their own business.

Key Concepts

Sampling Distribution of a Sample Proportion

The sampling distribution of a sample proportion describes the distribution of proportions obtained from repeated random samples of the same size from a population. It is approximately normal if the sample size is large enough, with mean equal to the population proportion and standard deviation calculated using the formula sqrt(p(1-p)/n). This concept allows us to assess probabilities related to sample proportions.

Normal Approximation to the Binomial Distribution

When the sample size is large, the binomial distribution of the number of successes can be approximated by a normal distribution. This approximation simplifies probability calculations for proportions by using the mean and standard deviation of the binomial distribution. Conditions for this approximation typically require np and n(1-p) to be at least 10.

Calculating Probabilities Using Z-Scores

Z-scores standardize a value by measuring how many standard deviations it is from the mean. By converting a sample proportion to a z-score, we can use standard normal distribution tables or software to find the probability of observing that value or less. This method is essential for answering probability questions about sample proportions.